On Katugampola Fractional Multiplicative Hermite-Hadamard-Type Inequalities

Abstract

This paper presents a novel framework for Katugampola fractional multiplicative integrals, advancing recent breakthroughs in fractional calculus through a synergistic integration of multiplicative analysis. Motivated by the growing interest in fractional calculus and its applications, we address the gap in generalized inequalities for multiplicative s-convex functions by deriving a Hermite–Hadamard-type inequality tailored to Katugampola fractional multiplicative integrals. A cornerstone of our work involves the derivation of two groundbreaking identities, which serve as the foundation for midpoint- and trapezoid-type inequalities designed explicitly for mappings whose multiplicative derivatives are multiplicative s-convex. These results extend classical integral inequalities to the multiplicative fractional calculus setting, offering enhanced precision in approximating nonlinear phenomena.

1. Introduction and Foundational Concepts

1.1. Overview and Motivation

Convexity occupies a central role in mathematical analysis, particularly in the derivation of inequalities. A mapping $\mathbf{g}:I\subseteq \mathbb{R}\to \mathbb{R}$ is defined as convex on I if, for all ${\ell }_1,{\ell }_2\in I$ and $\mathfrak{f}\in [0,1]$, the inequality

$$\mathbf{g}(\mathfrak{f}{\ell }_1+(1-\mathfrak{f}){\ell }_2)\le \mathfrak{f}\mathbf{g}\left({\ell }_1\right)+(1-\mathfrak{f})\mathbf{g}\left({\ell }_2\right)$$

holds.

Convex mappings play a crucial role in the analysis of integral inequalities. A key result in this area is the Hermite–Hadamard inequality, which connects the value of a convex mapping at the midpoint of an interval with its integral mean over that interval:

$$\mathbf{g}\left(\frac{F_1+F_2}{2}\right)\le \frac{1}{F_2-F_1}\underset{F_1}{\stackrel{F_2}{\int }}\mathbf{g}(\wp )d\wp \le \frac{\mathbf{g}\left(F_1\right)+\mathbf{g}\left(F_2\right)}{2}.$$

(1)

These inequalities are indispensable in numerical analysis, approximation theory, and the formulation of error estimates for integration schemes.

In [1], Kirmaci established the following result related to the first inequality of (1):

Theorem 1

([1]). Consider a differentiable mapping $\mathbf{g}:I\subset \mathbb{R}\to \mathbb{R}$ such that $\mathbf{g}\in L_1[F_1,F_2]$, with $F_1$, $F_2\in I$ and $F_1<F_2$. If $\left|{\mathbf{g}}^{\prime }\right|$ exhibits convexity on $[F_1,F_2]$, then

$$\left|\mathbf{g}\left(\frac{F_1+F_2}{2}\right)-\frac{1}{F_2-F_1}\underset{F_1}{\stackrel{F_2}{\int }}\mathbf{g}(\wp )d\wp \right|\le \frac{F_2-F_1}{8}\left(\left|{\mathbf{g}}^{\prime }\left(F_1\right)\right|+\left|{\mathbf{g}}^{\prime }\left(F_2\right)\right|\right).$$

In [2], Dragomir and Agarwal established the following result related to the second inequality of (1):

Theorem 2

([2]). Consider a differentiable mapping $\mathbf{g}:I\subset \mathbb{R}\to \mathbb{R}$ such that $\mathbf{g}\in L_1[F_1,F_2]$, with $F_1$, $F_2\in I$ and $F_1<F_2$. If $\left|{\mathbf{g}}^{\prime }\right|$ exhibits convexity on $[F_1,F_2]$, then the following inequality holds:

$$\left|\frac{\mathbf{g}\left(F_1\right)+\mathbf{g}\left(F_2\right)}{2}-\frac{1}{F_2-F_1}\underset{F_1}{\stackrel{F_2}{\int }}\mathbf{g}(\wp )d\wp \right|\le \frac{F_2-F_1}{8}\left(\left|{\mathbf{g}}^{\prime }\left(F_1\right)\right|+\left|{\mathbf{g}}^{\prime }\left(F_2\right)\right|\right).$$

One of the most important generalizations of the concept of convexity is the class of s-convex mappings introduced in [3] as follows:

Definition 1.

A mapping $\mathbf{g}:I\subset \left[0,\infty \right)\to \mathbb{R}$ is deemed to be s-convex in the second sense for a fixed $s\in \left(0,1\right]$ if

$$\mathbf{g}(\mathfrak{f}{\ell }_1+(1-\mathfrak{f}){\ell }_2)\le {\mathfrak{f}}^s\mathbf{g}\left({\ell }_1\right)+{(1-\mathfrak{f})}^s\mathbf{g}\left({\ell }_2\right)$$

holds for all ${\ell }_1,{\ell }_2\in I$ and $\mathfrak{f}\in [0,1]$.

Dragomir and Fitzpatrick [4] generalized the outcome in (1) to the class of s-convex mappings (in the second sense) defined in $[0,\infty )$ for parameters $s\in (0,1]$, as formulated in the following inequality:

$$2^{s-1}\mathbf{g}\left(\frac{F_1+F_2}{2}\right)\le \frac{1}{F_2-F_1}\underset{F_1}{\stackrel{F_2}{\int }}\mathbf{g}(\wp )d\wp \le \frac{\mathbf{g}\left(F_1\right)+\mathbf{g}\left(F_2\right)}{s+1}.$$

1.2. Principles of Multiplicative Calculus

The concept of multiplicative calculus, originally introduced by Grosman and Katz [5] in 1972, was designed as a substitute for classical calculus to address problems related to multiplicative processes and rates of change. This specialized branch of calculus, which deals exclusively with positive-valued mappings, achieved a solid mathematical foundation through the detailed contributions of Bashirov et al. in 2008 [6]. Its value lies in its ability to model growth, decay, and proportional interactions with greater precision. Over time, it has proven particularly useful in disciplines such as finance [7], biology [8], and physics [9], offering an innovative approach to scenarios where traditional calculus might be less effective. To lay the groundwork for our analysis, we first revisit foundational concepts of multiplicative calculus that form the prerequisite framework for this study.

Proposition 1

([6]). [(The multiplicative absolute value)] Let $\mathbf{g}\in {\mathbb{R}}^{+}=(0,\infty )$. The multiplicative modulus of $\mathbf{g}$, alternatively referred to as the multiplicative absolute value, is formally defined as follows:

$${\left|\mathbf{g}\right|}^{\ast }=\left\{\begin{array}{ccc}\mathbf{g}& \mathit{if}& \mathbf{g}\ge 1\\ \frac{1}{\mathbf{g}}& \mathit{if}& \mathbf{g}<1,\end{array}\right.$$

which can also be written as ${\left|\mathbf{g}\right|}^{\ast }=exp\left\{\left|ln\left(\mathbf{g}\right)\right|\right\}$, where $|.|$ is the classical absolute value.

Remark 1

(The multiplicative triangle inequality). The multiplicative triangle inequality is formally defined as follows:

$${\left|\mathbf{g}\mathbf{h}\right|}^{\ast }\le {\left|\mathbf{g}\right|}^{\ast }{\left|\mathbf{h}\right|}^{\ast },$$

for $\mathbf{g},\mathbf{h}\in {\mathbb{R}}^{+}$.

Proposition 2

([6], Multiplicative tests for monotonicity). Given $\mathbf{g}:[F_1,F_2]\to {\mathbb{R}}^{+}$ a multiplicative differentiable mapping.

• If ${\mathbf{g}}^{\ast }(\wp )\ge 1$ for all $\wp \in [F_1,F_2]$, then $\mathbf{g}$ is multiplicative increasing on $[F_1,F_2]$.
• If ${\mathbf{g}}^{\ast }(\wp )\le 1$ for all $\wp \in [F_1,F_2]$, then $\mathbf{g}$ is multiplicative decreasing on $[F_1,F_2]$.

Definition 2

([6]). The multiplicative derivative of a positive mapping $\mathbf{g}$, represented by ${\mathbf{g}}^{\ast }$, is expressed as follows:

$$\frac{d^{\ast }\mathbf{g}}{d\wp }={\mathbf{g}}^{\ast }\left(\wp \right)=\underset{\mathfrak{k}\to 0}{lim}{\left(\frac{\mathbf{g}\left(\wp +\mathfrak{k}\right)}{\mathbf{g}\left(\wp \right)}\right)}^{{\displaystyle \frac{1}{\mathfrak{k}}}}.$$

Remark 2.

Any positive differentiable mapping $\mathbf{g}$ inherently possesses multiplicative differentiability. The interplay between ${\mathbf{g}}^{\ast }$ and the standard derivative ${\mathbf{g}}^{\prime }$ is articulated through the following identity:

$${\mathbf{g}}^{\ast }\left(\wp \right)=exp\left\{{\left(ln\mathbf{g}\left(\wp \right)\right)}^{\prime }\right\}=exp\left\{{\displaystyle \frac{{\mathbf{g}}^{\prime }\left(\wp \right)}{\mathbf{g}\left(\wp \right)}}\right\}.$$

Definition 3

([6]). Any positive mapping $\mathbf{g}$ that is Riemann integrable is necessarily multiplicative integrable, and the multiplicative integral is connected to the Riemann integral through the following relationship:

$$\underset{F_1}{\stackrel{F_2}{\int }}{\left\{\mathbf{g}\left(\wp \right)\right\}}^{d\wp }=exp\left(\underset{F_1}{\stackrel{F_2}{\int }}ln\left(\mathbf{g}\left(\wp \right)\right)d\wp \right).$$

Proposition 3

([6]). Consider two positive and multiplicative integrable mappings $\mathbf{g}$ and $\mathbf{h}$. Then, we have

• $\underset{F_1}{\stackrel{F_2}{\int }}{\left({\left[\mathbf{g}\left(\wp \right)\right]}^{\kappa }\right)}^{d\wp }={\left[\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp }\right]}^{\kappa }$, $\kappa \in \mathbb{R}$,
• $\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp }=\underset{F_1}{\stackrel{⅁}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp }\underset{⅁}{\stackrel{F_2}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp },F_1<⅁<F_2,$
• $\underset{F_1}{\stackrel{F_1}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp }=1$, and $\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp }={\left(\underset{F_2}{\stackrel{F_1}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp }\right)}^{-1},$
• $\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{g}\left(\wp \right)\mathbf{h}\left(\wp \right)\right)}^{d\wp }=\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp }\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{h}\left(\wp \right)\right)}^{d\wp },$
• $\underset{F_1}{\stackrel{F_2}{\int }}{\left({\displaystyle \frac{\mathbf{g}\left(\wp \right)}{\mathbf{h}\left(\wp \right)}}\right)}^{d\wp }={\displaystyle \frac{\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{g}\left(\wp \right)\right)}^{d\wp }}{\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{h}\left(\wp \right)\right)}^{d\wp }}}.$

Theorem 3

([6]). Consider a positive multiplicative differentiable mapping $\mathbf{g}$ on $[F_1,F_2]$, and let $\mathbf{h}:[F_1,F_2]\to \mathbb{R}$ be differentiable. Then, the mapping ${\left({\mathbf{g}}^{\ast }\right)}^{\mathbf{h}}$ is multiplicative integrable, and the result below is satisfied:

$$\underset{F_1}{\stackrel{F_2}{\int }}{\left({\mathbf{g}}^{\ast }{\left(\wp \right)}^{\mathbf{h}\left(\wp \right)}\right)}^{d\wp }=\frac{\mathbf{g}{\left(F_2\right)}^{\mathbf{h}\left(F_2\right)}}{\mathbf{g}{\left(F_1\right)}^{\mathbf{h}\left(F_1\right)}}\times \frac{1}{\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{g}{\left(\wp \right)}^{{\mathbf{h}}^{\prime }\left(\wp \right)}\right)}^{d\wp }}.$$

Lemma 1

([6]). Let $\mathbf{g}$ be a positive mapping that is differentiable in the multiplicative sense on $[F_1,F_2]$, and suppose $\mathbf{k}:J\subset \mathbb{R}\to \mathbb{R}$ and $\mathbf{h}:[F_1,F_2]\to \mathbb{R}$ are two differentiable mappings. Then, the following holds:

$$\underset{F_1}{\stackrel{F_2}{\int }}{\left({\mathbf{g}}^{\ast }{\left(\mathbf{k}\left(\wp \right)\right)}^{{\mathbf{k}}^{\prime }\left(\wp \right)\mathbf{h}\left(\wp \right)}\right)}^{d\wp }=\frac{\mathbf{g}{\left(\mathbf{k}\left(F_2\right)\right)}^{\mathbf{h}\left(F_2\right)}}{\mathbf{g}{\left(\mathbf{k}\left(F_1\right)\right)}^{\mathbf{h}\left(F_1\right)}}\times \frac{1}{\underset{F_1}{\stackrel{F_2}{\int }}{\left(\mathbf{g}{\left(\mathbf{k}\left(\wp \right)\right)}^{{\mathbf{h}}^{\prime }\left(\wp \right)}\right)}^{d\wp }}.$$

Within the framework of multiplicative calculus, the counterpart to classical convexity is referred to as multiplicative convexity or logarithmic convexity, which is defined as follows:

Definition 4

([10], Multiplicative convexity). A mapping $\mathbf{g}$ is considered multiplicative convex over the interval I if it is positive, and for all ${\ell }_1,{\ell }_2\in I$, the following inequality

$$\mathbf{g}\left(\mathfrak{f}{\ell }_1+\left(1-\mathfrak{f}\right){\ell }_2\right)\le {\left[\mathbf{g}\left({\ell }_1\right)\right]}^{\mathfrak{f}}{\left[\mathbf{g}\left({\ell }_2\right)\right]}^{1-\mathfrak{f}}$$

holds true for all $\mathfrak{f}\in [0,1]$.

The analogue of the concept of s-convexity in the context of multiplicative calculus is defined as follows:

Definition 5

([11], Multiplicative s-convexity). A mapping $\mathbf{g}:I\subset \mathbb{R}\to {\mathbb{R}}^{+}$ is considered multiplicative s-convex in the second sense on I for a fixed $s\in (0,1]$ if, for all ${\ell }_1,{\ell }_2\in I$, the inequality

$$\mathbf{g}\left(\mathfrak{f}{\ell }_1+\left(1-\mathfrak{f}\right){\ell }_2\right)\le {\left[\mathbf{g}\left({\ell }_1\right)\right]}^{{\mathfrak{f}}^s}{\left[\mathbf{g}\left({\ell }_2\right)\right]}^{{\left(1-\mathfrak{f}\right)}^s}$$

holds true for all $\mathfrak{f}\in [0,1]$.

The Hermite–Hadamard inequality for the multiplicative integral was provided by Ali et al. [12] as follows:

Theorem 4

([12]). Consider a multiplicative convex mapping $\mathbf{g}:[F_1,F_2]\to {\mathbb{R}}^{+}$. Then, the inequalities

$$\mathbf{g}\left(\frac{F_1+F_2}{2}\right)\le {\left(\underset{F_1}{\stackrel{F_2}{\int }}\mathbf{g}{\left(\wp \right)}^{d\wp }\right)}^{{\displaystyle \frac{1}{F_2-F_1}}}\le \sqrt{\mathbf{g}\left(F_1\right)\mathbf{g}\left(F_2\right)}$$

hold true.

In [13], Özcan provided a Hermite–Hadamard inequality for the multiplicative integral via multiplicative s-convexity in the following manner:

Theorem 5

([13]). Consider a multiplicative s-convex mapping $\mathbf{g}:[F_1,F_2]\to {\mathbb{R}}^{+}$. Then, the inequalities

$${\left[\mathbf{g}\left(\frac{F_1+F_2}{2}\right)\right]}^{2^{s-1}}\le {\left(\underset{F_1}{\stackrel{F_2}{\int }}\mathbf{g}{\left(\wp \right)}^{d\wp }\right)}^{{\displaystyle \frac{1}{F_2-F_1}}}\le {\left[\mathbf{g}\left(F_1\right)\mathbf{g}\left(F_2\right)\right]}^{{\displaystyle \frac{1}{s+1}}}$$

hold true.

In [14], Khan and Budak established midpoint and trapezoid inequalities pertaining to multiplicative integrals as follows:

Theorem 6

([14]). Consider the mapping $\mathbf{g}:I\subset \mathbb{R}\to {\mathbb{R}}^{+}$ that is multiplicative increasing and multiplicative differentiable on $I^0$, with $F_1$ and $F_2$ belonging to $I^0$. If ${\mathbf{g}}^{\ast }$ is multiplicative convex on $[F_1,F_2]$, then we have

$${\left|\frac{\mathbf{g}\left(\frac{F_1+F_2}{2}\right)}{{\left(\underset{F_1}{\stackrel{F_2}{\int }}\mathbf{g}(\wp )d\wp \right)}^{\frac{1}{F_2-F_1}}}\right|}^{\ast }\le {\left({\mathbf{g}}^{\ast }\left(F_1\right){\mathbf{g}}^{\ast }\left(F_2\right)\right)}^{\frac{F_2-F_1}{8}}.$$

Theorem 7

([14]). Consider the mapping $\mathbf{g}:I\subset \mathbb{R}\to {\mathbb{R}}^{+}$ that is multiplicative increasing and multiplicative differentiable on $I^0$, with $F_1$ and $F_2$ belonging to $I^0$. If ${\mathbf{g}}^{\ast }$ is multiplicative convex on $[F_1,F_2]$, then we have

$${\left|\frac{\sqrt{\mathbf{g}\left(F_1\right)\mathbf{g}\left(F_2\right)}}{{\left(\underset{F_1}{\stackrel{F_2}{\int }}\mathbf{g}(\wp )d\wp \right)}^{\frac{1}{F_2-F_1}}}\right|}^{\ast }\le {\left({\mathbf{g}}^{\ast }\left(F_1\right){\mathbf{g}}^{\ast }\left(F_2\right)\right)}^{\frac{F_2-F_1}{8}}.$$

In subsequent studies, researchers have focused on deriving error estimates for classical quadrature methods through the lens of multiplicative integrals. Notable examples include the Ostrowski and Simpson rules [15], the Maclaurin [16] and dual Simpson [17] schemes, as well as Hermite–Hadamard inequalities [18,19]. These advancements have substantially deepened the theoretical foundations of inequalities within multiplicative calculus. Comprehensive discussions on multiplicative integral inequalities can be found in [20,21].

1.3. Fundamentals of Fractional Calculus

Fractional calculus extends classical calculus by generalizing derivatives and integrals to non-integer orders, providing a powerful framework for modeling complex systems with memory and hereditary effects. Among the most celebrated fractional integral operators are the Riemann–Liouville operators, which play a foundational role in the theory and application of fractional calculus. In what follows, we provide the definition of these operators as a starting point for further exploration.

Definition 6

([22]). Let $\mathbf{g}\in L_1[F_1,F_2]$. The right-sided Riemann–Liouville fractional integral ${\mathrm{J}}_{F_2^{-}}^{\beta }\mathbf{g}$ and the left-sided one ${\mathrm{J}}_{F_1^{+}}^{\beta }\mathbf{g}$ of order $\beta >0$ with $F_1\ge 0$, are defined by

$${\mathrm{J}}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\stackrel{F_2}{\underset{\varpi }{\int }}{\left(\wp -\varpi \right)}^{\beta -1}\mathbf{g}(\wp )d\wp ,F_2>\varpi$$

(2)

and

$${\mathrm{J}}_{F_1^{+}}^{\beta }\mathbf{g}\left(\varpi \right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\stackrel{\varpi }{\underset{F_1}{\int }}{\left(\varpi -\wp \right)}^{\beta -1}\mathbf{g}(\wp )d\wp ,\varpi >F_1,$$

(3)

respectively, where $\Gamma \left(\nu \right)=\stackrel{\infty }{\underset{0}{\int }}{e}^{-u}{\wp }^{\nu -1}d\wp$ is the Gamma mapping, and ${\mathrm{J}}_{F_2^{-}}^0\mathbf{g}\left(\varpi \right)={\mathrm{J}}_{F_1^{+}}^0\mathbf{g}\left(\varpi \right)=\mathbf{g}\left(\varpi \right)$.

Definition 7

([23]). Let $\beta >0$. The right and left Hadamard fractional integrals of any order $\beta >0$ are defined by

$${\mathrm{H}}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\stackrel{F_2}{\underset{\varpi }{\int }}{\left(ln\wp -ln\varpi \right)}^{\beta -1}{\displaystyle \frac{\mathbf{g}(\wp )}{\wp }}d\wp ,F_2>\varpi$$

(4)

and

$${\mathrm{H}}_{F_1^{+}}^{\beta }\mathbf{g}\left(\varpi \right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\stackrel{\varpi }{\underset{F_1}{\int }}{\left(ln\varpi -ln\wp \right)}^{\beta -1}{\displaystyle \frac{\mathbf{g}(\wp )}{\wp }}d\wp ,\varpi >F_1,$$

(5)

respectively.

Definition 8

([24]). Let $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1,F_2\right]\right)$. The right- and left-sided Katugampola fractional integrals of order $\beta \in \mathbb{C}$ with $Re\left(\beta \right)>0$ and $\rho >0$ are expressed as

$${}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)={\displaystyle \frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}}\stackrel{F_2}{\underset{\varpi }{\int }}{\displaystyle \frac{{\wp }^{\rho -1}\mathbf{g}(\wp )}{{\left({\wp }^{\rho }-{\varpi }^{\rho }\right)}^{1-\beta }}}d\wp ,F_2>\varpi ,$$

(6)

and

$${}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(\varpi \right)={\displaystyle \frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}}\stackrel{\varpi }{\underset{F_1}{\int }}{\displaystyle \frac{{\wp }^{\rho -1}\mathbf{g}(\wp )}{{\left({\varpi }^{\rho }-{\wp }^{\rho }\right)}^{1-\beta }}}d\wp ,\varpi >F_1$$

(7)

respectively, where $X_{\ell }^{\mathfrak{m}}(F_1,F_2)$ ($\ell \in \mathbb{R}$ and the range $1\le \mathfrak{m}\le \infty$) represents the set of complex-valued mappings $\mathbf{g}$ that are Lebesgue measurable and for which the norm

$${∥\mathbf{g}∥}_{X_{\ell }^{\mathfrak{m}}}=\left\{\begin{array}{ccc}{\left(\underset{F_1}{\stackrel{F_2}{\int }}{\left|{\wp }^{\ell }\mathbf{g}(\wp )\right|}^{\mathfrak{m}}d\wp \right)}^{1/\mathfrak{m}}\hfill & \mathit{for}& 1\le \mathfrak{m}<\infty ,\\ \underset{F_1\le \wp \le F_2}{\mathit{ess}sup}\left|{\wp }^{\ell }\mathbf{g}(\wp )\right|\hfill & \mathit{for}& \mathfrak{m}=\infty \end{array}\right.$$

remains finite.

Remark 3.

Given the conditions $\beta ,\rho >0$ and for $\varpi <F_2$, it is noted that

1. 

When ρ tends to 1, the right Katugampola fractional integral, denoted as ${}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)$, converges to the Riemann–Liouville fractional integral, represented by ${\mathrm{J}}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)$.

2. 

Conversely, as ρ nears $0^{+}$, the convergence of ${}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)$ aligns with that of the Hadamard fractional integral ${\mathrm{H}}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)$.

This observation similarly applies to related results for left-sided operators.

For some results on fractional integral inequalities via different types of fractional integrals, we refer the reader to [25,26] and the references cited therein.

We now recall the beta and incomplete beta functions frequently employed in the realm of fractional calculus.

Definition 9

([22]). The incomplete beta function is defined by

$$B_F\left({⅁}_1,{⅁}_2\right)=\stackrel{F}{\underset{0}{\int }}{\wp }^{{⅁}_1-1}{\left(1-\wp \right)}^{{⅁}_2-1}d\wp ,0<F\le 1,$$

where $\Gamma \left(.\right)$ is the Gamma function, and $Re\left({⅁}_1\right)>0$ and $Re\left({⅁}_2\right)>0$.

For $F=1$, we retrieve the beta function expressed by

$$B\left({⅁}_1,{⅁}_2\right)=\stackrel{1}{\underset{0}{\int }}{\wp }^{{⅁}_1-1}{\left(1-\wp \right)}^{{⅁}_2-1}d\wp =\frac{\Gamma \left({⅁}_1\right)\Gamma \left({⅁}_2\right)}{\Gamma \left({⅁}_1+{⅁}_2\right)}.$$

1.4. Bridging the Gap: Fractional Multiplicative Calculus

The notion of fractional multiplicative integrals and derivatives was introduced in [27] in the following manner:

Definition 10

([27]). The right Riemann–Liouville fractional multiplicative integral ${}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(\varpi \right)$ of order $\beta \in \mathbb{C},Re\left(\beta \right)>0$ beginning with $F_2$ is expressed as

$${}_{\ast }\mathrm{J}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)=exp\left\{\left({\mathrm{J}}_{F_2^{-}}^{\beta }(ln\circ \mathbf{g})\right)\left(\varpi \right)\right\},\varpi <F_2,$$

and the left one is specified by

$${}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(\varpi \right)=exp\left\{\left({\mathrm{J}}_{F_1^{+}}^{\beta }(ln\circ \mathbf{g})\right)\left(\varpi \right)\right\},\varpi >F_1,$$

where ${\mathrm{J}}_{F_2^{-}}^{\beta }$ and ${\mathrm{J}}_{F_1^{+}}^{\beta }$ are the right and left Riemann–Liouville fractional integrals defined as in (2) and (3), respectively.

Following the introduction of such fractional multiplicative integrals, numerous researchers have conducted various studies aiming to establish different types of integral inequalities, including Ostrowski-type [28], Bullen-type [29], Simpson-type [30], Maclaurin-type [31,32], and Newton-type inequalities [33]. Additional relevant results can be found in [34,35,36,37] and the references cited therein.

In [38], Budak and Özçelik provided the following Hermite–Hadamard inequality for multiplicative convex mappings:

Theorem 8.

Let $\mathbf{g}:[F_1,F_2]\to {\mathbb{R}}^{+}$ such that $\mathbf{g}\in L_1[F_1,F_2]$ with $F_1<F_2$. If $\mathbf{g}$ is multiplicative convex in $[F_1,F_2]$, then the inequalities

$$\left[\mathbf{g}\left(\frac{F_1+F_2}{2}\right)\right]\le {\left[\left({}_{\ast }\mathrm{J}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1\right)\right)\left({}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2\right)\right)\right]}^{{\displaystyle \frac{\Gamma \left(\beta +1\right)}{2{\left(F_2-F_1\right)}^{\beta }}}}\le \sqrt{\mathbf{g}\left(F_1\right)\mathbf{g}\left(F_2\right)}$$

hold for $\beta >0$.

The inequalities of the midpoint and trapezoid through the Riemann–Liouville fractional multiplicative integrals for multiplicative convex mappings were provided by Budak and Ergün in [39].

Theorem 9.

Consider a multiplicative increasing and multiplicative differentiable mapping $\mathbf{g}:[F_1,F_2]\to {\mathbb{R}}^{+}$. If ${\mathbf{g}}^{\ast }$ possesses multiplicative convexity, then we have

$${\left|\frac{\mathbf{g}\left(\frac{F_1+F_2}{2}\right)}{{\left(\left({}_{\ast }\mathrm{J}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1\right)\right)\left({}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2\right)\right)\right)}^{\frac{\Gamma \left(\beta +1\right)}{2{\left(F_2-F_1\right)}^{\beta }}}}\right|}^{\ast }\le {\left[{\mathbf{g}}^{\ast }\left(F_1\right){\mathbf{g}}^{\ast }\left(F_2\right)\right]}^{\frac{F_2-F_1}{2}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\beta +1}}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)}$$

and

$${\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1\right)\right)\left(\mathbf{g}\left(F_2\right)\right)}}{{\left(\left({}_{\ast }\mathrm{J}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1\right)\right)\left({}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2\right)\right)\right)}^{\frac{\Gamma \left(\beta +1\right)}{2{\left(F_2-F_1\right)}^{\beta }}}}\right|}^{\ast }\le {\left[{\mathbf{g}}^{\ast }\left(F_1\right){\mathbf{g}}^{\ast }\left(F_2\right)\right]}^{{\displaystyle \frac{F_2-F_1}{2(\beta +1)}}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)}.$$

In [40], Zhou and Du introduced Hadamard fractional multiplicative integrals in the following manner:

Definition 11

([40]). The right-sided Hadamard fractional multiplicative integral ${}_{\ast }\mathrm{H}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)$ of order $\beta \in \mathbb{C}$ with $Re\left(\beta \right)>0$ is defined by

$${}_{\ast }\mathrm{H}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)=exp\left\{\left({\mathrm{H}}_{F_2^{-}}^{\beta }(ln\circ \mathbf{g})\right)\left(\varpi \right)\right\},\varpi <F_2,$$

and the left-sided one ${}_{\ast }\mathrm{H}_{F_1^{+}}^{\beta }\mathbf{g}\left(\varpi \right)$ is defined by

$${}_{\ast }\mathrm{H}_{F_1^{+}}^{\beta }\mathbf{g}\left(\varpi \right)=exp\left\{\left({\mathrm{H}}_{F_1^{+}}^{\beta }(ln\circ \mathbf{g})\right)\left(\varpi \right)\right\},\varpi >F_1.$$

Here, the symbols ${\mathrm{H}}_{F_2^{-}}^{\beta }$ and ${\mathrm{H}}_{F_1^{+}}^{\beta }$ denote the right- and left-sided Hadamard fractional integrals defined as in (4) and (5), respectively.

In the same study, the authors investigate a range of analytical characteristics of these operators, such as integrability, continuity, commutativity, the semigroup property, boundedness, and more. Following this, they established Hermite–Hadamard inequalities for $GG$-convex mappings. For additional research on multiplicative inequalities using different kinds of fractional multiplicative integrals, readers can consult [41,42,43,44,45,46,47,48,49,50].

The concept of Katugampola fractional multiplicative integrals was first introduced by Ai and Du in [51].

Definition 12

([51]). Let $\mathbf{g}:[F_1,F_2]\to {\mathbb{R}}^{+}$ such that $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$ with $F_1<F_2$. The right- and left-sided multiplicative Katugampola fractional integrals of order $\beta \in \mathbb{C}$ with $Re\left(\beta \right)>0$ and $\rho >0$ are defined by

$${}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(\varpi \right)=exp\left\{\left({}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }(ln\circ \mathbf{g})\right)\left(\varpi \right)\right\},\varpi <F_2,$$

and

$${}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(\varpi \right)=exp\left\{\left({}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }(ln\circ \mathbf{g})\right)\left(\varpi \right)\right\},\varpi >F_1,$$

where $(ln\circ \mathbf{g}))\left(\varpi \right)=ln\left(\mathbf{g}\left(\varpi \right)\right)$, and the symbols ${}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }$ and ${}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }$ denote the right- and left-sided Katugampola fractional integrals defined as in (6) and (7), respectively.

Remark 4

([51]). Let $\mathbf{g}:[F_1^{\rho },F_2^{\rho }]\to {\mathbb{R}}^{+}$ such that $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$ with $F_1^{\rho }<F_2^{\rho }$. The right- and left-sided multiplicative Katugampola fractional integrals of order $\beta \in \mathbb{C}$ with $Re\left(\beta \right)>0$ and $\rho >0$ can be expressed in the following alternative form:

$${}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left({\varpi }^{\rho }\right)=exp\left\{{\displaystyle \frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}}\stackrel{F_2}{\underset{\varpi }{\int }}{\displaystyle \frac{{\wp }^{\rho -1}}{{\left({\wp }^{\rho }-{\varpi }^{\rho }\right)}^{1-\beta }}}\left(ln\circ \mathbf{g}\right)\left({\wp }^{\rho }\right)d\wp \right\},F_2>\varpi ,$$

and

$${}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left({\varpi }^{\rho }\right)=exp\left\{{\displaystyle \frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}}\stackrel{\varpi }{\underset{F_1}{\int }}{\displaystyle \frac{{\wp }^{\rho -1}}{{\left({\varpi }^{\rho }-{\wp }^{\rho }\right)}^{1-\beta }}}\left(ln\circ \mathbf{g}\right)\left({\wp }^{\rho }\right)d\wp \right\},\varpi >F_1,$$

respectively.

After introducing these novel fractional multiplicative integrals, the authors derived the subsequent Newton-type inequalities for mappings with multiplicative convex second-order multiplicative derivatives:

Theorem 10

([51]). Let $\mathbf{g}:\left[F_1^{\rho },F_2^{\rho }\right]\to {\mathbb{R}}^{+}$ be a twice multiplicative differentiable mapping on $\left(F_1^{\rho },F_2^{\rho }\right)$. Under the hypothesis that ${\mathbf{g}}^{\ast \ast }$ possesses multiplicative convexity on $\left[F_1^{\rho },F_2^{\rho }\right]$ together with $\rho ,\beta >0$ and ${\mathbf{g}}^{\ast \ast }\ge 1$, then the following inequality, which relates to Katugampola fractional multiplicative integrals, holds:

$$\begin{array}{cc}\hfill & {\left|{\displaystyle \frac{{\left[\mathbf{g}\left(F_1^{\rho }\right)\right]}^{\frac{1}{8}}{\left[\mathbf{g}\left(\frac{2F_1^{\rho }+F_2^{\rho }}{3}\right)\right]}^{\frac{3}{8}}{\left[\mathbf{g}\left(\frac{F_1^{\rho }+2F_2^{\rho }}{3}\right)\right]}^{\frac{3}{8}}{\left[\mathbf{g}\left(F_2^{\rho }\right)\right]}^{\frac{1}{8}}}{{\left[{}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right).{}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right]}^{\frac{{\rho }^{\beta }\Gamma (\beta +1)}{2{(F_2^{\rho }-F_1^{\rho })}^{\beta }}}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left[{\mathbf{g}}^{\ast \ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast \ast }\left(F_2^{\rho }\right)\right]}^{{\displaystyle \frac{{\left(F_2^{\rho }-F_1^{\rho }\right)}^2}{2(\beta +1)}}\left({\Omega }_1\left(\beta \right)+{\Omega }_2\left(\beta \right)+{\Omega }_3\left(\beta \right)\right)},\hfill \end{array}$$

where

$${\Omega }_1\left(\beta \right)=\underset{0}{\stackrel{{\displaystyle \frac{1}{3}}}{\int }}\left|{\wp }^{\beta +1}+{\displaystyle \frac{3\beta -1}{4}}\wp \right|d\wp ,$$

$${\Omega }_2\left(\beta \right)=\underset{{\displaystyle \frac{1}{3}}}{\stackrel{{\displaystyle \frac{2}{3}}}{\int }}\left|{\wp }^{\beta +1}+{\displaystyle \frac{\beta -3}{4}}\wp \right|d\wp$$

and

$${\Omega }_3\left(\beta \right)={\displaystyle \frac{5}{18}}-{\displaystyle \frac{1}{\beta +2}}+{\displaystyle \frac{1}{\beta +2}}{\left({\displaystyle \frac{2}{3}}\right)}^{\beta +2}.$$

Motivated by the recently introduced operators in [51], and aiming to explore such integrals and extend several well-known results within this framework, we first derive the Hermite–Hadamard inequality for these newly proposed integrals in Section 2, specifically for multiplicative s-convex mappings. Subsequently, by introducing two novel identities based on Katugampola fractional multiplicative integrals for multiplicative differentiable mappings, we establish several Katugampola fractional multiplicative midpoint- and trapezoid-type inequalities for mappings whose multiplicative derivatives are multiplicative s-convex in Section 3 and Section 4, respectively. To validate our findings, a numerical example is presented in Section 5. In Section 6, we highlight practical applications of the obtained results. Finally, we summarize our contributions and outline potential directions for future research in Section 7.

The study conducted in this paper not only presents new results related to these novel integrals but also provides several generalized versions of well-established and celebrated findings in the existing literature. Notably, our work incorporates the multiplicative absolute value instead of the classical absolute value commonly used in prior studies. The latter appears inadequate when dealing with inherently positive quantities, whereas the multiplicative absolute value enables deriving results valid for both a quantity $\mathcal{Q}$ and its reciprocal $1/\mathcal{Q}$, thereby broadening the scope and applicability of the analysis.

2. Hermite–Hadamard Inequality

In this section, we establish the Hermite–Hadamard inequality for multiplicative s-convex mappings using the newly introduced Katugampola fractional multiplicative integrals.

Theorem 11.

Let $\mathbf{g}:[F_1^{\rho },F_2^{\rho }]\to {\mathbb{R}}^{+}$ such that $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$. If $\mathbf{g}$ possesses multiplicative s-convexity on $[F_1^{\rho },F_2^{\rho }]$ for a fixed $s\in (0,1]$, then the inequalities

$$\begin{array}{cc}\hfill {\left[\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right]}^{2^{s-1}}\le & {\left[\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right]}^{{\displaystyle \frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\hfill \\ \hfill \le & {\left[\sqrt{\mathbf{g}\left(F_1^{\rho }\right)\mathbf{g}\left(F_2^{\rho }\right)}\right]}^{\beta \rho \left({\displaystyle \frac{1}{\beta \rho +s}}+B(\beta \rho ,s+1)\right)}\hfill \end{array}$$

(8)

hold for $\beta ,\rho >0$, where $B(.,.)$ is the beta function.

Proof. 

Since $\mathbf{g}$ is multiplicative s-convex on $[F_1,F_2]$, we have

$$\begin{array}{cc}\hfill ln\left[\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right]=& ln\left[\mathbf{g}\left(\frac{\mathfrak{w}{F}_1^{\rho }+(1-\mathfrak{w})F_2^{\rho }}{2}+\frac{(1-\mathfrak{w})F_1^{\rho }+\mathfrak{w}{F}_2^{\rho }}{2}\right)\right]\hfill \\ \hfill \le & \frac{1}{2^s}\left(ln\left[\mathbf{g}\left(\mathfrak{w}{F}_1^{\rho }+(1-\mathfrak{w})F_2^{\rho }\right)\right]+ln\left[\mathbf{g}\left((1-\mathfrak{w})F_1^{\rho }+\mathfrak{w}{F}_2^{\rho }\right)\right]\right).\hfill \end{array}$$

(9)

Multiplying (9) by ${\mathfrak{w}}^{\beta \rho -1}$ then integrating the resulting inequality with respect to $\mathfrak{w}\in [0,1]$, we get

$$\begin{array}{cc}\hfill & \frac{2^s}{\beta \rho }ln\left[\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right]\hfill \\ \hfill \le & \underset{0}{\stackrel{1}{\int }}{\mathfrak{w}}^{\beta \rho -1}ln\left[\mathbf{g}\left(\mathfrak{w}{F}_1^{\rho }+(1-\mathfrak{w})F_2^{\rho }\right)\right]d\mathfrak{w}+\underset{0}{\stackrel{1}{\int }}{\mathfrak{w}}^{\beta \rho -1}ln\left[\mathbf{g}\left((1-\mathfrak{w})F_1^{\rho }+\mathfrak{w}{F}_2^{\rho }\right)\right]d\mathfrak{w}\hfill \\ \hfill =& \underset{F_1}{\stackrel{F_2}{\int }}{\left(\frac{F_2^{\rho }-{\wp }^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}\frac{{\wp }^{\rho -1}}{F_2^{\rho }-F_1^{\rho }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp +\underset{F_1}{\stackrel{F_2}{\int }}{\left(\frac{{\wp }^{\rho }-F_1^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}\frac{{\wp }^{\rho -1}}{F_2^{\rho }-F_1^{\rho }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp \hfill \\ \hfill =& \frac{{\rho }^{\beta -1}\Gamma \left(\beta \right)}{{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}\left[\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left(F_2^{\rho }-{\wp }^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp +\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left({\wp }^{\rho }-F_1^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp \right],\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill & 2^{s-1}ln\left[\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right]\hfill \\ \hfill \le & \frac{{\rho }^{\beta }\Gamma (\beta +1)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}\left[\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left(F_2^{\rho }-{\wp }^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp +\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left({\wp }^{\rho }-F_1^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp \right].\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill & {\left[\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right]}^{2^{s-1}}\hfill \\ \hfill \le & exp\left\{\frac{{\rho }^{\beta }\Gamma (\beta +1)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}\left[\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left(F_2^{\rho }-{\wp }^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp +\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left({\wp }^{\rho }-F_1^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp \right]\right\}\hfill \\ \hfill =& {\left[exp\left\{\left[\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left(F_2^{\rho }-{\wp }^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp +\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left({\wp }^{\rho }-F_1^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp \right]\right\}\right]}^{{\displaystyle \frac{{\rho }^{\beta }\Gamma (\beta +1)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\hfill \\ \hfill =& {\left[\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right]}^{{\displaystyle \frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}.\hfill \end{array}$$

(10)

This establishes the left inequality of (8). Now, since $\mathbf{g}$ possesses multiplicative s-convexity, we have

$$ln\left[\mathbf{g}\left(\mathfrak{w}{F}_1^{\rho }+(1-\mathfrak{w})F_2^{\rho }\right)\right]\le {\mathfrak{w}}^{s}ln\left[\mathbf{g}\left(F_1^{\rho }\right)\right]+{(1-\mathfrak{w})}^{s}ln\left[\mathbf{g}\left(F_2^{\rho }\right)\right]$$

(11)

and

$$ln\left[\mathbf{g}\left((1-\mathfrak{w})F_1^{\rho }+\mathfrak{w}{F}_2^{\rho }\right)\right]\le {(1-\mathfrak{w})}^{s}ln\left[\mathbf{g}\left(F_1^{\rho }\right)\right]+{\mathfrak{w}}^{s}ln\left[\mathbf{g}\left(F_2^{\rho }\right)\right].$$

(12)

By adding (11) and (12), we obtain

$$\begin{array}{cc}\hfill & ln\left[\mathbf{g}\left(\mathfrak{w}{F}_1^{\rho }+(1-\mathfrak{w})F_2^{\rho }\right)\right]+ln\left[\mathbf{g}\left((1-\mathfrak{w})F_1^{\rho }+\mathfrak{w}{F}_2^{\rho }\right)\right]\hfill \\ \hfill \le & \left[{\mathfrak{w}}^s+{(1-\mathfrak{w})}^s\right]\left(ln\left[\mathbf{g}\left(F_1^{\rho }\right)\right]+ln\left[\mathbf{g}\left(F_2^{\rho }\right)\right]\right).\hfill \end{array}$$

(13)

By applying the factor ${\mathfrak{w}}^{\beta \rho -1}$ to both sides of inequality (13) and performing integration over $\mathfrak{w}\in [0,1]$, the following result is derived:

$$\begin{array}{cc}\hfill & \underset{F_1}{\stackrel{F_2}{\int }}{\left(\frac{F_2^{\rho }-{\wp }^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}\frac{{\wp }^{\rho -1}}{F_2^{\rho }-F_1^{\rho }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp +\underset{F_1}{\stackrel{F_2}{\int }}{\left(\frac{{\wp }^{\rho }-F_1^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}\frac{{\wp }^{\rho -1}}{F_2^{\rho }-F_1^{\rho }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp \hfill \\ \hfill \le & \left(ln\left[\mathbf{g}\left(F_1^{\rho }\right)\right]+ln\left[\mathbf{g}\left(F_2^{\rho }\right)\right]\right)\underset{0}{\stackrel{1}{\int }}{\mathfrak{w}}^{\beta \rho -1}\left[{\mathfrak{w}}^s+{(1-\mathfrak{w})}^s\right]d\mathfrak{w}\hfill \\ \hfill =& ln\left[\mathbf{g}\left(F_1\right)\mathbf{g}\left(F_2\right)\right]\left({\displaystyle \frac{1}{\rho \beta +s}}+B(\rho \beta ,s+1)\right),\hfill \end{array}$$

which implies

$$\begin{array}{cc}\hfill & \frac{{\rho }^{\beta }\Gamma (\beta +1)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}\left[\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left(F_2^{\rho }-{\wp }^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp +\frac{{\rho }^{1-\beta }}{\Gamma \left(\beta \right)}\underset{F_1}{\stackrel{F_2}{\int }}\frac{{\wp }^{\rho -1}}{{\left({\wp }^{\rho }-F_1^{\rho }\right)}^{1-\beta }}ln\left[\mathbf{g}\left({\wp }^{\rho }\right)\right]d\wp \right]\hfill \\ \hfill \le & {\displaystyle \frac{\beta \rho }{2}}ln\left[\mathbf{g}\left(F_1\right)\mathbf{g}\left(F_2\right)\right]\left({\displaystyle \frac{1}{\beta \rho +s}}+B(\beta \rho ,s+1)\right).\hfill \end{array}$$

(14)

Thus, we obtain

$${\left[\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right]}^{{\displaystyle \frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\le {\left[\sqrt{\mathbf{g}\left(F_1^{\rho }\right)\mathbf{g}\left(F_2^{\rho }\right)}\right]}^{\beta \rho \left({\displaystyle \frac{1}{\beta \rho +s}}+B(\beta \rho ,s+1)\right)}.$$

(15)

This establishes the right inequality of (8). Thus, we complete the proof.   □

Corollary 1.

By setting $s=1$, Theorem 11 yields

$$\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\le {\left[\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right]}^{{\displaystyle \frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\le \sqrt{\mathbf{g}\left(F_1\right)\mathbf{g}\left(F_2\right)}.$$

Moreover, if we take $\rho =1$, we retrieve Theorem 8 from [38].

Remark 5.

By setting $\beta =\rho =1$, Theorem 11 coincides with Theorem 5 obtained by Özcan in [13].

3. Midpoint-Type Inequalities

We begin by introducing a novel identity involving Katugampola fractional multiplicative integrals, which we then utilize to derive midpoint-type inequalities for differentiable multiplicative s-convex mappings.

Lemma 2.

Consider a multiplicative differentiable mapping $\mathbf{g}:\left[F_1^{\rho },F_2^{\rho }\right]\to {\mathbb{R}}^{+}$, such that $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$ and ${\mathbf{g}}^{\ast }\in L^1\left[F_1^{\rho },F_2^{\rho }\right]$. Then, the following equality holds:

$$\begin{array}{cc}\hfill & \frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\hfill \\ \hfill =& {\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}}\right)}^{d\mathfrak{f}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}\hfill \\ \hfill & \times {\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}}\right)}^{d\mathfrak{f}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}.\hfill \end{array}$$

Proof. 

Let

$${\mathcal{I}}_1={\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}}\right)}^{d\mathfrak{f}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}$$

and

$${\mathcal{I}}_2={\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left(\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right)\right){\mathfrak{f}}^{\rho -1}}\right)}^{d\mathfrak{f}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}.$$

Integrating ${\mathcal{I}}_1$ by parts, we get

$$\begin{array}{cc}\hfill {\mathcal{I}}_1=& {\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}}\right)}^{d\mathfrak{f}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}\hfill \\ \hfill =& exp\left\{\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}ln\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}}\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}{\left[ln\left(\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\prime }d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{{\left.\frac{1}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right)\left(ln\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right|}_0^{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}\right.\hfill \\ \hfill & -\left.\frac{\beta \rho }{2}\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\mathfrak{f}}^{\rho \left(\beta -1\right)}+{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta -1}\right){\mathfrak{f}}^{\rho -1}\left(ln\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{1}{2}\left(ln\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right)-\frac{\beta \rho }{2}\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\mathfrak{f}}^{\rho \left(\beta -1\right)}+{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta -1}\right){\mathfrak{f}}^{\rho -1}\left(ln\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{1}{2}\left(ln\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right)-\frac{\beta \rho }{2\left(F_2^{\rho }-F_1^{\rho }\right)}\underset{F_1}{\stackrel{{\left({\displaystyle \frac{F_1^{\rho }+F_2^{\rho }}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\left(\frac{{\wp }^{\rho }-F_1^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}+{\left(\frac{F_2^{\rho }-{\wp }^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}\right){\wp }^{\rho -1}\left(ln\mathbf{g}\left({\wp }^{\rho }\right)\right)d\wp \right\}.\hfill \end{array}$$

(16)

Similarly, we have

$$\begin{array}{cc}\hfill {\mathcal{I}}_2=& {\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}}\right)}^{d\mathfrak{f}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}\hfill \\ \hfill =& exp\left\{\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}ln\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}}\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}{\left[ln\left(\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\prime }d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{{\left.\frac{1}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right)\left(ln\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right|}_{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}^1\right.\hfill \\ \hfill & -\left.\frac{\beta \rho }{2}\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\rho \left(\beta -1\right)}+{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta -1}\right){\mathfrak{f}}^{\rho -1}\left(ln\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{1}{2}\left(ln\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right)-\frac{\beta \rho }{2}\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\rho \left(\beta -1\right)}+{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta -1}\right){\mathfrak{f}}^{\rho -1}\left(ln\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{1}{2}\left(ln\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)\right)-\frac{\beta \rho }{2\left(F_2^{\rho }-F_1^{\rho }\right)}\underset{{\left({\displaystyle \frac{F_1^{\rho }+F_2^{\rho }}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{F_2}{\int }}\left({\left(\frac{{\wp }^{\rho }-F_1^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}+{\left(\frac{F_2^{\rho }-{\wp }^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}\right){\wp }^{\rho -1}\left(ln\mathbf{g}\left({\wp }^{\rho }\right)\right)d\wp \right\}.\hfill \end{array}$$

(17)

Multiplying (16) by (17) and using Remark 4, we get the desired result. This completes the proof.   □

Theorem 12.

Consider a multiplicative increasing mapping $\mathbf{g}:[F_1^{\rho },F_2^{\rho }]\to {\mathbb{R}}^{+}$ and assume that it is multiplicative differentiable on $[F_1^{\rho },F_2^{\rho }]$ with $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$. If ${\mathbf{g}}^{\ast }$ is multiplicative s-convex on $\left[F_1^{\rho },F_2^{\rho }\right]$ where $s\in \left(0,1\right]$, then the inequality

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{\left({\displaystyle \frac{1}{s+1}}+{\displaystyle \frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)\frac{F_2^{\rho }-F_1^{\rho }}{2}}\hfill \end{array}$$

holds, where $B_{⅁}$ is the incomplete beta function.

Proof. 

By applying the multiplicative absolute value for the equality given in Lemma 2, then using the multiplicative triangle inequality, we get

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & \left|exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}ln{\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}}d\mathfrak{f}\right\}\right.\hfill \\ \hfill & \times {\left.exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}ln{\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}}d\mathfrak{f}\right\}\right|}^{\ast }\hfill \\ \hfill \le & {\left|exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}ln{\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}}d\mathfrak{f}\right\}\right|}^{\ast }\hfill \\ \hfill & \times {\left|exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}ln{\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}}d\mathfrak{f}\right\}\right|}^{\ast }.\hfill \end{array}$$

(18)

By leveraging Propositions 1 and 3 and the property that $\mathbf{g}$ is multiplicative increasing (i.e., ${\mathbf{g}}^{\ast }\ge 1$), (18) implies

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left|\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}ln{\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}}d\mathfrak{f}\right|\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left|\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}ln{\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}}d\mathfrak{f}\right|\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left|\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left(\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right)ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right|\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left|\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left(\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right)ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right|\right\}\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}.\hfill \end{array}$$

(19)

Now, since ${\mathbf{g}}^{\ast }$ is multiplicative s-convex, inequality (19) yields

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left|{\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right|{\mathfrak{f}}^{\rho -1}ln{\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left(1-{\mathfrak{f}}^{\rho }\right)}^s}{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\mathfrak{f}}^{\rho s}}d\mathfrak{f}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left|{\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right|{\mathfrak{f}}^{\rho -1}\left(ln{\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left(1-{\mathfrak{f}}^{\rho }\right)}^s}{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\mathfrak{f}}^{\rho s}}\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|\left({\left(1-\wp \right)}^{s}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\wp }^{s}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)d\wp \right\}\hfill \\ \hfill & \times exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\underset{{\displaystyle \frac{1}{2}}}{\stackrel{1}{\int }}\left|{\wp }^{\beta }-{\left(1-\wp \right)}^{\beta }-1\right|\left({\left(1-\wp \right)}^{s}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\wp }^{s}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)d\wp \right\}.\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\left(\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}{\left(1-\wp \right)}^s\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|d\wp \right)ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right.\hfill \\ \hfill & +\left.\frac{F_2^{\rho }-F_1^{\rho }}{2}\left(\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}{\wp }^s\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|d\wp \right)ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right\}\hfill \\ \hfill & \times exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\left(\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}{\wp }^s\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|d\wp \right)ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right.\hfill \\ \hfill & +\left.\frac{F_2^{\rho }-F_1^{\rho }}{2}\left(\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}{\left(1-\wp \right)}^s\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|d\wp \right)ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right\}\hfill \\ \hfill =& {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{\left(\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left[{\wp }^s+{\left(1-\wp \right)}^s\right]\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|d\wp \right)\frac{F_2^{\rho }-F_1^{\rho }}{2}}\hfill \\ \hfill =& {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{\left({\displaystyle \frac{1}{s+1}}+{\displaystyle \frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)\frac{F_2^{\rho }-F_1^{\rho }}{2}}.\hfill \end{array}$$

We have thus proved the theorem.   □

Corollary 2.

By setting $s=1$ in Theorem 12, we obtain the following Katugampola fractional multiplicative midpoint-type inequality via multiplicative convexity:

$${\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\le {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{\frac{F_2^{\rho }-F_1^{\rho }}{2}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\beta +1}}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)}.$$

Corollary 3.

By setting $\rho =1$ in Theorem 12, we obtain the following midpoint-type inequality via Riemann–Liouville fractional multiplicative integrals:

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1+F_2}{2}\right)}{{\left(\left({}_{\ast }\mathrm{J}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1\right)\right)\left({}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2\right)\right)\right)}^{\frac{\Gamma \left(\beta +1\right)}{2{\left(F_2-F_1\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left({\mathbf{g}}^{\ast }\left(F_1\right){\mathbf{g}}^{\ast }\left(F_2\right)\right)}^{\left({\displaystyle \frac{1}{s+1}}+{\displaystyle \frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)\frac{F_2-F_1}{2}}.\hfill \end{array}$$

Moreover, if we take $s=1$, we get

$${\left|\frac{\mathbf{g}\left(\frac{F_1+F_2}{2}\right)}{{\left(\left({}_{\ast }\mathrm{J}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1\right)\right)\left({}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2\right)\right)\right)}^{\frac{\Gamma \left(\beta +1\right)}{2{\left(F_2-F_1\right)}^{\beta }}}}\right|}^{\ast }\le {\left[{\mathbf{g}}^{\ast }\left(F_1\right){\mathbf{g}}^{\ast }\left(F_2\right)\right)}^{\frac{F_2-F_1}{2}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\beta +1}}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)},$$

which coincides with the first inequality presented in Theorem 9, obtained by Budak and Ergün in [39].

Remark 6.

Setting $\beta =\rho =s=1$, Theorem 12 simplifies to Theorem 6 established in [14].

Theorem 13.

Consider a multiplicative increasing mapping $\mathbf{g}:[F_1^{\rho },F_2^{\rho }]\to {\mathbb{R}}^{+}$, and assume that it is multiplicative differentiable on $[F_1^{\rho },F_2^{\rho }]$ with $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$. If ${\left(ln{\mathbf{g}}^{\ast }\right)}^{\mathfrak{p}}$ is s-convex on $[F_1^{\rho },F_2^{\rho }]$, then we have

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta -1}}}}\right|}^{\ast }\hfill \\ \hfill \le & \left({\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\frac{1}{\rho }{B}_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}+{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\frac{1}{\rho }\left(B\left(\frac{1}{\rho },s+1\right)-B_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right.\hfill \\ \hfill & \times {\left.{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}+{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{2^{s+\frac{1}{\rho }}-1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}},\hfill \end{array}$$

where $s\in \left(0,1\right]$ and $\mathfrak{q},\mathfrak{p}>1$ with ${\displaystyle \frac{1}{\mathfrak{q}}}+{\displaystyle \frac{1}{\mathfrak{p}}}=1$ and

$${\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)=\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|}^{\mathfrak{q}}d\mathfrak{f}$$

(20)

and

$${\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)=\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|}^{\mathfrak{q}}d\mathfrak{f}.$$

(21)

Proof. 

By using Hölder’s inequality along with the s-convexity of ${\left(ln{\mathbf{g}}^{\ast }\right)}^{\mathfrak{p}}$, inequalty (19) yields

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta -1}}}}\right|}^{\ast }\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left[ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left[ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\right.\hfill \\ \hfill & \times \left.{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left[{\left(1-{\mathfrak{f}}^{\rho }\right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\mathfrak{f}}^{\rho s}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\right.\hfill \\ \hfill & \times \left.{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left[{\left(1-{\mathfrak{f}}^{\rho }\right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\mathfrak{f}}^{\rho s}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\right.\hfill \\ \hfill & \times \left.{\left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left[{\left(1-\wp \right)}^s{\wp }^{{\displaystyle \frac{1}{\rho }}-1}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\wp }^{s+{\displaystyle \frac{1}{\rho }}-1}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\wp \right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\right.\hfill \\ \hfill & \times \left.{\left(\frac{1}{\rho }\underset{{\displaystyle \frac{1}{2}}}{\stackrel{1}{\int }}\left[{\left(1-\wp \right)}^s{\wp }^{{\displaystyle \frac{1}{\rho }}-1}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\wp }^{s+{\displaystyle \frac{1}{\rho }}-1}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\wp \right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\left[{\left(\frac{1}{\rho }{B}_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right]}^{\mathfrak{p}}+{\left[{\left(\frac{1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\left({\left[{\left(\frac{1}{\rho }\left(B\left(\frac{1}{\rho },s+1\right)-B_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right]}^{\mathfrak{p}}\right.\right.\hfill \\ \hfill & {\left.+{\left[{\left(\frac{2^{s+\frac{1}{\rho }}-1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\left({\left(\frac{1}{\rho }{B}_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\left(\frac{1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\left({\left(\frac{1}{\rho }\left(B\left(\frac{1}{\rho },s+1\right)-B_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left({\rho }^{\rho }\right)\right)+{\left(\frac{2^{s+\frac{1}{\rho }}-1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\left(ln{\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left(\frac{1}{\rho }{B}_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\left({\displaystyle \frac{1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right)\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\left(ln{\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left(\frac{1}{\rho }\left(B\left(\frac{1}{\rho },s+1\right)-B_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\left({\displaystyle \frac{2^{s+\frac{1}{\rho }}-1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right)\right\}\hfill \\ \hfill =& {\left({\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\frac{1}{\rho }{B}_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right)}^{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}\hfill \\ \hfill & \times {\left({\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\frac{1}{\rho }\left(B\left(\frac{1}{\rho },s+1\right)-B_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{2^{s+\frac{1}{\rho }}-1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}\hfill \\ \hfill =& \left({\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\frac{1}{\rho }{B}_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}+{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\frac{1}{\rho }\left(B\left(\frac{1}{\rho },s+1\right)-B_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right.\hfill \\ \hfill & \times {\left.{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\left({\mathcal{C}}_1\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}+{\left({\mathcal{C}}_2\left(\beta ,\rho ,\mathfrak{q}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{2^{s+\frac{1}{\rho }}-1}{2^{s+\frac{1}{\rho }}\left(\rho s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}},\hfill \end{array}$$

where we have used the fact that ${{\rm Y}}_1^{\kappa }+{{\rm Y}}_2^{\kappa }\le {({{\rm Y}}_1+{{\rm Y}}_2)}^{\kappa }$ for ${{\rm Y}}_1,{{\rm Y}}_2\ge 0$ with $\kappa \ge 1$. The proof is finished.   □

Theorem 14.

Consider a multiplicative increasing mapping $\mathbf{g}:[F_1^{\rho },F_2^{\rho }]\to {\mathbb{R}}^{+}$, and assume that it is multiplicative differentiable on $[F_1^{\rho },F_2^{\rho }]$ with $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$. If ${\left(ln{\mathbf{g}}^{\ast }\right)}^{\mathfrak{p}}$ is s-convex on $[F_1^{\rho },F_2^{\rho }]$, where $\mathfrak{p}>1$, then we have

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta -1}}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left(\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\frac{F_2^{\rho }-F_1^{\rho }}{2}{\left(\frac{1}{2}-\frac{1}{\beta +1}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}{\left(\frac{1}{s+1}+\frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}.\hfill \end{array}$$

Proof. 

By utilizing the power mean inequality along with the s-convexity of ${\left(ln{\mathbf{g}}^{\ast }\right)}^{\mathfrak{p}}$, inequality (19) yields

$$\begin{array}{cc}\hfill & {\left|\frac{\mathbf{g}\left(\frac{F_1^{\rho }+F_2^{\rho }}{2}\right)}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta -1}}}}\right|}^{\ast }\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|d\mathfrak{f}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left.{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|{\left[ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|d\mathfrak{f}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left.{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|{\left[ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|d\mathfrak{f}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }+1\right){\mathfrak{f}}^{\rho -1}\right|{\left.\left[{\left(1-{\mathfrak{f}}^{\rho }\right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\mathfrak{f}}^{\rho s}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|d\mathfrak{f}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-1\right){\mathfrak{f}}^{\rho -1}\right|{\left.\left[{\left(1-{\mathfrak{f}}^{\rho }\right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\mathfrak{f}}^{\rho s}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|d\wp \right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|{\left.\left[{\left(1-\wp \right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\wp }^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\wp \right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|d\wp \right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left|{\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }-1\right|{\left.\left[{\left(1-\wp \right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\wp }^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\wp \right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}{\left(\frac{1}{2}-\frac{1}{\beta +1}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left({\left.\left[{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left|{\left(1-2\wp \right)}^{\beta }-1\right|\left({\left(1-\wp \right)}^s+{\wp }^s\right)d\wp \right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}{\left(\frac{1}{2}-\frac{1}{\beta +1}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\left(\left(\frac{1}{s+1}+\frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)\right.\hfill \\ \hfill & \left.{\left.\times \left[{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}{\left(\frac{1}{2}-\frac{1}{\beta +1}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}{\left(\frac{1}{s+1}+\frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \left.\times {\left[{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill \le & exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}{\left(\frac{1}{2}-\frac{1}{\beta +1}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}{\left(\frac{1}{s+1}+\frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \left.\times \left[ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right]\right\}\hfill \\ \hfill =& {\left(\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\frac{F_2^{\rho }-F_1^{\rho }}{2}{\left(\frac{1}{2}-\frac{1}{\beta +1}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}{\left(\frac{1}{s+1}+\frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}},\hfill \end{array}$$

where we have used the fact that ${{\rm Y}}_1^{\kappa }+{{\rm Y}}_2^{\kappa }\le {({{\rm Y}}_1+{{\rm Y}}_2)}^{\kappa }$ for ${{\rm Y}}_1,{{\rm Y}}_2\ge 0$ with $\kappa \ge 1$. The proof is finished.   □

4. Trapezoid-Type Inequalities

Here, we present another new identity based on Katugampola fractional multiplicative integrals and subsequently establish trapezoid-type inequalities via multiplicative s-convexity.

Lemma 3.

Consider a multiplicative differentiable mapping $\mathbf{g}:\left[F_1^{\rho },F_2^{\rho }\right]\to {\mathbb{R}}^{+}$ such that $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$ and ${\mathbf{g}}^{\ast }\in L^1\left[F_1^{\rho },F_2^{\rho }\right]$. Then, the following equality holds:

$$\begin{array}{cc}\hfill & \frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\hfill \\ \hfill =& {\left(\underset{0}{\stackrel{1}{\int }}{\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}}\right)}^{d\mathfrak{f}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}.\hfill \end{array}$$

(22)

Proof. 

By integrating the right side of (22), we get

$$\begin{array}{cc}\hfill & {\left(\underset{0}{\stackrel{1}{\int }}{\left({\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}}\right)}^{d\mathfrak{f}}\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}}\hfill \\ \hfill =& exp\left\{\underset{0}{\stackrel{1}{\int }}ln\left({\left({\mathbf{N}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{{\displaystyle \frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}}\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\underset{0}{\stackrel{1}{\int }}\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\underset{0}{\stackrel{1}{\int }}\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}{\left[ln\left(\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\prime }d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{{\left.\frac{1}{2}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right)ln\left(\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right|}_0^1\right.\hfill \\ \hfill & -\left.\frac{\beta \rho }{2}\underset{0}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\rho \left(\beta -1\right)}+{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta -1}\right){\mathfrak{f}}^{\rho -1}ln\left(\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{1}{2}\left(ln\mathbf{g}\left(F_2^{\rho }\right)\right)+\frac{1}{2}\left(ln\mathbf{g}\left(F_1^{\rho }\right)\right)\right.\hfill \\ \hfill & -\left.\frac{\beta \rho }{2}\underset{0}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\rho \left(\beta -1\right)}+{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta -1}\right){\mathfrak{f}}^{\rho -1}ln\left(\mathbf{g}\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{1}{2}\left(ln\mathbf{g}\left(F_2^{\rho }\right)\right)+\frac{1}{2}\left(ln\mathbf{g}\left(F_1^{\rho }\right)\right)\right.\hfill \\ \hfill & -\left.\frac{\beta \rho }{2\left(F_2^{\rho }-F_1^{\rho }\right)}\underset{F_1}{\stackrel{F_2}{\int }}\left({\left(\frac{{\wp }^{\rho }-F_1^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}+{\left(\frac{F_2^{\rho }-{\wp }^{\rho }}{F_2^{\rho }-F_1^{\rho }}\right)}^{\beta -1}\right){\wp }^{\rho -1}\left(ln\mathbf{g}\left({\wp }^{\rho }\right)\right)d\wp \right\}\hfill \\ \hfill =& exp\left\{\frac{1}{2}\left(ln\mathbf{g}\left(F_2^{\rho }\right)\right)+\frac{1}{2}\left(ln\mathbf{g}\left(F_1^{\rho }\right)\right)\right.\hfill \\ \hfill & -\left.\frac{\beta \rho }{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}\left(\underset{F_1}{\stackrel{F_2}{\int }}{\left({\wp }^{\rho }-F_1^{\rho }\right)}^{\beta -1}{\wp }^{\rho -1}\left(ln\mathbf{g}\left({\wp }^{\rho }\right)\right)d\wp +\underset{F_1}{\stackrel{F_2}{\int }}{\left(F_2^{\rho }-{\wp }^{\rho }\right)}^{\beta -1}{\wp }^{\rho -1}\left(ln\mathbf{N}\left({\wp }^{\rho }\right)\right)d\wp \right)\right\}\hfill \\ \hfill =& \frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}.\hfill \end{array}$$

The proof is completed.   □

Theorem 15.

Consider a multiplicative increasing mapping $\mathbf{g}:[F_1^{\rho },F_2^{\rho }]\to {\mathbb{R}}^{+}$, and assume that it is multiplicative differentiable on $[F_1^{\rho },F_2^{\rho }]$ with $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$. If ${\mathbf{g}}^{\ast }$ is multiplicative s-convex on $\left[F_1^{\rho },F_2^{\rho }\right]$, then the inequality

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\displaystyle \frac{F_2^{\rho }-F_1^{\rho }}{2}}\left(B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+{\displaystyle \frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}}\right)}\hfill \end{array}$$

holds, where $B_{⅁}$ is the incomplete beta function, respectively.

Proof. 

By applying the multiplicative absolute value for both sides of the equality given in Lemma 3, we get

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left|exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{1}{\int }}ln{\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)}^{\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}}d\mathfrak{f}\right\}\right|}^{\ast }\hfill \\ \hfill =& exp\left\{\left|\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right|\right\}\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{1}{\int }}\left|\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}\right|ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)d\mathfrak{f}\right\}.\hfill \end{array}$$

(23)

Now, using the fact that ${\mathbf{g}}^{\ast }$ is multiplicative s-convex, inequality (23) gives

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}\right.\left.ln{\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left(1-{\mathfrak{f}}^{\rho }\right)}^s}{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\mathfrak{f}}^{\rho s}}d\mathfrak{f}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}\left(ln{\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{{\left(1-{\mathfrak{f}}^{\rho }\right)}^s}{\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\mathfrak{f}}^{\rho s}}\right)d\mathfrak{f}\right\}\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right)\left({\left(1-\wp \right)}^{s}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\wp }^{s}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)d\wp \right\}\hfill \\ \hfill & \times exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\underset{{\displaystyle \frac{1}{2}}}{\stackrel{1}{\int }}\left({\wp }^{\beta }-{\left(1-\wp \right)}^{\beta }\right)\left({\left(1-\wp \right)}^{s}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\wp }^{s}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)d\wp \right\}\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right)\left({\left(1-\wp \right)}^{s}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\wp }^{s}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)d\wp \right\}\hfill \\ \hfill & \times exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right)\left({\wp }^{s}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\left(1-\wp \right)}^{s}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)d\wp \right\}\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\left[ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right]\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right)\left({\wp }^s+{\left(1-\wp \right)}^s\right)d\wp \right\}\hfill \\ \hfill =& exp\left\{\frac{F_2^{\rho }-F_1^{\rho }}{2}\left(B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+\frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}\right)\left[ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right]\right\}\hfill \\ \hfill =& {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\displaystyle \frac{F_2^{\rho }-F_1^{\rho }}{2}}\left(B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+{\displaystyle \frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}}\right)},\hfill \end{array}$$

where we have used

$$\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right)\left({\wp }^s+{\left(1-\wp \right)}^s\right)d\wp =B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+\frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}.$$

(24)

The proof is completed.   □

Corollary 4.

By setting $s=1$ in Theorem 15, we obtain the following Katugampola fractional multiplicative trapezoid-type inequalities via multiplicative convexity:

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{{\displaystyle \frac{F_2^{\rho }-F_1^{\rho }}{2(\beta +1)}}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)}.\hfill \end{array}$$

Corollary 5.

By setting $\rho =1$ in Theorem 15, we obtain the following trapezoid-type inequality via Riemann–Liouville fractional multiplicative integrals:

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1\right)\right)\left(\mathbf{g}\left(F_2\right)\right)}}{{\left(\left({}_{\ast }\mathrm{J}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1\right)\right)\left({}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2\right)\right)\right)}^{\frac{\Gamma \left(\beta +1\right)}{2{\left(F_2-F_1\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left({\mathbf{g}}^{\ast }\left(F_1\right){\mathbf{g}}^{\ast }\left(F_2\right)\right)}^{{\displaystyle \frac{F_2-F_1}{2}}\left(B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+{\displaystyle \frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}}\right)}.\hfill \end{array}$$

Moreover, if we take $s=1$, we get

$${\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1\right)\right)\left(\mathbf{g}\left(F_2\right)\right)}}{{\left(\left({}_{\ast }\mathrm{J}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1\right)\right)\left({}_{\ast }\mathrm{J}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2\right)\right)\right)}^{\frac{\Gamma \left(\beta +1\right)}{2{\left(F_2-F_1\right)}^{\beta }}}}\right|}^{\ast }\le {\left({\mathbf{g}}^{\ast }\left(F_1\right){\mathbf{g}}^{\ast }\left(F_2\right)\right)}^{{\displaystyle \frac{F_2-F_1}{2(\beta +1)}}\left(1-{\displaystyle \frac{1}{2^{\beta }}}\right)},$$

which coincides with the second inequality presented in Theorem 9 provided by Budak and Ergün in [39].

Remark 7.

Setting $\beta =\rho =s=1$, Theorem 15 simplifies to Theorem 7 from [14].

Theorem 16.

Consider a multiplicative increasing mapping $\mathbf{g}:[F_1^{\rho },F_2^{\rho }]\to {\mathbb{R}}^{+}$ and assume that it is multiplicative differentiable on $[F_1^{\rho },F_2^{\rho }]$ with $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$. If ${\left(ln{\mathbf{g}}^{\ast }\right)}^{\mathfrak{p}}$ is s-convex on $[F_1^{\rho },F_2^{\rho }]$, where $\mathfrak{q},\mathfrak{p}>1$ with ${\displaystyle \frac{1}{\mathfrak{q}}}+{\displaystyle \frac{1}{\mathfrak{p}}}=1$, then we have

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{\left({\left({\displaystyle \frac{1}{2^{\frac{1}{\rho }}}}B\left({\displaystyle \frac{1}{\rho }},\beta \mathfrak{q}+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{2^{s+1}-1}{2^{s+1}\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}+{\left({\displaystyle \frac{\left({}_{2}F_1\left(1-\frac{1}{\rho },1,\beta \mathfrak{q}+2;\frac{1}{2}\right)\right)}{2\left(\beta \mathfrak{q}+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{1}{2^{s+1}\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right){\displaystyle \frac{F_2^{\rho }-F_1^{\rho }}{2}}}\hfill \\ \hfill & \times {\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{\left({\left({\displaystyle \frac{1}{2^{\frac{1}{\rho }}}}B\left({\displaystyle \frac{1}{\rho }},\beta \mathfrak{q}+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{1}{2^{s+1}\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}+{\left({\displaystyle \frac{\left({}_{2}F_1\left(1-\frac{1}{\rho },1,\beta \mathfrak{q}+2;\frac{1}{2}\right)\right)}{2\left(\beta \mathfrak{q}+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{2^{s+1}-1}{2^{s+1}\left(s+1\right)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right){\displaystyle \frac{F_2^{\rho }-F_1^{\rho }}{2}}},\hfill \end{array}$$

where ${\mathcal{C}}_3(\beta ,\rho ,\mathfrak{q})$ and ${\mathcal{C}}_4(\beta ,\rho ,\mathfrak{q})$ are defined as in (25) and (26), respectively, and $B_{⅁}$ is the incomplete beta function.

Proof. 

By applying Hölder’s inequality along with the s-convexity of ${\left(ln{\mathbf{g}}^{\ast }\right)}^{\mathfrak{p}}$, inequality (23) gives

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left[\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left[ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left[\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left[ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left[\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left[{\left(1-{\mathfrak{f}}^{\rho }\right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\mathfrak{f}}^{\rho s}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left[\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left[{\left(1-{\mathfrak{f}}^{\rho }\right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\mathfrak{f}}^{\rho s}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left[\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\right.\hfill \\ \hfill & \times \left.{\left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left[{\left(1-\wp \right)}^s{\wp }^{{\displaystyle \frac{1}{\rho }}-1}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\wp }^{s+{\displaystyle \frac{1}{\rho }}-1}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left[\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\right.\hfill \\ \hfill & \times \left.{\left({\displaystyle \frac{1}{\rho }}\underset{{\displaystyle \frac{1}{2}}}{\stackrel{1}{\int }}\left[{\left(1-\wp \right)}^s{\wp }^{{\displaystyle \frac{1}{\rho }}-1}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\wp }^{s+{\displaystyle \frac{1}{\rho }}-1}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left[\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\right.\hfill \\ \hfill & \times \left.\frac{1}{{\rho }^{\frac{1}{\mathfrak{p}}}}{\left[B_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right){\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+\frac{{\left(\frac{1}{2}\right)}^{s+\frac{1}{\rho }}}{s+\frac{1}{\rho }}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left[\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\right.\hfill \\ \hfill & \times \left.\frac{1}{{\rho }^{\frac{1}{\mathfrak{p}}}}{\left[B_{{\displaystyle \frac{1}{2}}}\left(s+1,\frac{1}{\rho }\right){\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+\frac{1-{\left(\frac{1}{2}\right)}^{s+\frac{1}{\rho }}}{s+\frac{1}{\rho }}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill \le & exp\left\{\frac{{\rho }^{1-\frac{1}{\mathfrak{p}}}\left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left({\mathcal{C}}_3(\beta ,\rho ,\mathfrak{q})\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\left[{\left(B_{{\displaystyle \frac{1}{2}}}\left(\frac{1}{\rho },s+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\left(\frac{{\left(\frac{1}{2}\right)}^{s+\frac{1}{\rho }}}{s+\frac{1}{\rho }}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right]\right\}\hfill \\ \hfill & \times exp\left\{\frac{{\rho }^{1-\frac{1}{\mathfrak{p}}}\left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left({\mathcal{C}}_4(\beta ,\rho ,\mathfrak{q})\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}\left[{\left(B_{{\displaystyle \frac{1}{2}}}\left(s+1,\frac{1}{\rho }\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+{\left(\frac{1-{\left(\frac{1}{2}\right)}^{s+\frac{1}{\rho }}}{s+\frac{1}{\rho }}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right]\right\}\hfill \\ \hfill \le & {\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)}^{\left({\left({\mathcal{C}}_3(\beta ,\rho ,\mathfrak{q})\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(B_{{\displaystyle \frac{1}{2}}}\left({\displaystyle \frac{1}{\rho }},s+1\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}+{\left({\mathcal{C}}_4(\beta ,\rho ,\mathfrak{q})\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left(B_{{\displaystyle \frac{1}{2}}}\left(s+1,{\displaystyle \frac{1}{\rho }}\right)\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right){\displaystyle \frac{{\rho }^{1-\frac{1}{\mathfrak{p}}}{F}_2^{\rho }-F_1^{\rho }}{2}}}\hfill \\ \hfill & \times {\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)}^{\left({\left({\mathcal{C}}_3(\beta ,\rho ,\mathfrak{q})\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{{\left(\frac{1}{2}\right)}^{s+\frac{1}{\rho }}}{s+\frac{1}{\rho }}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}+{\left({\mathcal{C}}_4(\beta ,\rho ,\mathfrak{q})\right)}^{{\displaystyle \frac{1}{\mathfrak{q}}}}{\left({\displaystyle \frac{1-{\left(\frac{1}{2}\right)}^{s+\frac{1}{\rho }}}{s+\frac{1}{\rho }}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right){\displaystyle \frac{{\rho }^{1-\frac{1}{\mathfrak{p}}}{F}_2^{\rho }-F_1^{\rho }}{2}}},\hfill \end{array}$$

where we have used

$${\mathcal{C}}_3(\beta ,\rho ,\mathfrak{q})=\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}{\left[\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f}$$

(25)

and

$${\mathcal{C}}_4(\beta ,\rho ,\mathfrak{q})=\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}{\left[\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}\right]}^{\mathfrak{q}}d\mathfrak{f},$$

(26)

and the fact that ${{\rm Y}}_1^{\kappa }+{{\rm Y}}_2^{\kappa }\le {({{\rm Y}}_1+{{\rm Y}}_2)}^{\kappa }$ for ${{\rm Y}}_1,{{\rm Y}}_2\ge 0$ with $\kappa \ge 1$. The proof is finished.   □

Theorem 17.

Consider a multiplicative increasing mapping $\mathbf{g}:[F_1^{\rho },F_2^{\rho }]\to {\mathbb{R}}^{+}$ and assume that it is multiplicative differentiable on $[F_1^{\rho },F_2^{\rho }]$ with $\mathbf{g}\in X_{\ell }^{\mathfrak{m}}\left(\left[F_1^{\rho },F_2^{\rho }\right]\right)$. If ${\left(ln{\mathbf{g}}^{\ast }\right)}^{\mathfrak{p}}$ is s-convex on $[F_1^{\rho },F_2^{\rho }]$, where $\mathfrak{p}>1$, then we have

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & \left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }{\left(F_2^{\rho }\right)}^{\frac{{\rho }^{1-\frac{1}{\mathfrak{p}}}\left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\frac{1-{\left(\frac{1}{2}\right)}^{\beta }}{\rho (\beta +1)}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}{\left(B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+\frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right),\hfill \end{array}$$

where $B_z\left(.,.\right)$ is the incomplete beta function.

Proof. 

By using the power mean inequality and the s-convexity of ${\left(ln{\mathbf{g}}^{\ast }\right)}^{\mathfrak{p}}$, inequality (23) yields

$$\begin{array}{cc}\hfill & {\left|\frac{\sqrt{\left(\mathbf{g}\left(F_1^{\rho }\right)\right)\left(\mathbf{g}\left(F_2^{\rho }\right)\right)}}{{\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^{\beta }\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^{\beta }\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{\frac{{\rho }^{\beta }\Gamma \left(\beta +1\right)}{2{\left(F_2^{\rho }-F_1^{\rho }\right)}^{\beta }}}}\right|}^{\ast }\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}d\mathfrak{f}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left.{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}{\left[ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}d\mathfrak{f}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left.{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}{\left[ln\left({\mathbf{g}}^{\ast }\left(\left(1-{\mathfrak{f}}^{\rho }\right)F_1^{\rho }+{\mathfrak{f}}^{\rho }{F}_2^{\rho }\right)\right)\right]}^{\mathfrak{p}}d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill \le & exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}d\mathfrak{f}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left(\underset{0}{\stackrel{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\int }}\left({\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }-{\mathfrak{f}}^{\beta \rho }\right){\mathfrak{f}}^{\rho -1}{\left.\left[{\left(1-{\mathfrak{f}}^{\rho }\right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\mathfrak{f}}^{\rho s}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}d\mathfrak{f}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left(\underset{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{1}{\rho }}}}{\stackrel{1}{\int }}\left({\mathfrak{f}}^{\beta \rho }-{\left(1-{\mathfrak{f}}^{\rho }\right)}^{\beta }\right){\mathfrak{f}}^{\rho -1}{\left.\left[{\left(1-{\mathfrak{f}}^{\rho }\right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\mathfrak{f}}^{\rho s}{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\mathfrak{f}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right)d\wp \right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right){\left.\left[{\left(1-\wp \right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\wp }^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\wp \right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill & \times exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\frac{1}{\rho }\underset{{\displaystyle \frac{1}{2}}}{\stackrel{1}{\int }}\left({\wp }^{\beta }-{\left(1-\wp \right)}^{\beta }\right)d\wp \right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \times \left(\frac{1}{\rho }\underset{{\displaystyle \frac{1}{2}}}{\stackrel{1}{\int }}\left({\wp }^{\beta }-{\left(1-\wp \right)}^{\beta }\right){\left.\left[{\left(1-\wp \right)}^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\wp }^s{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right]d\wp \right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill =& exp\left\{\frac{\rho \left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\frac{1}{\rho }\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right)d\wp \right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \left.\times {\left(\frac{1}{\rho }\left({\left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)\right)}^{\mathfrak{p}}+{\left(ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)}^{\mathfrak{p}}\right)\underset{0}{\stackrel{{\displaystyle \frac{1}{2}}}{\int }}\left({\left(1-\wp \right)}^{\beta }-{\wp }^{\beta }\right)\left[{\left(1-\wp \right)}^s+{\wp }^s\right]d\wp \right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right\}\hfill \\ \hfill \le & exp\left\{\frac{{\rho }^{1-\frac{1}{\mathfrak{p}}}\left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\frac{1-{\left(\frac{1}{2}\right)}^{\beta }}{\rho (\beta +1)}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}{\left(B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+\frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}\right.\hfill \\ \hfill & \left.\times \left(ln\left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right)\right)+ln\left({\mathbf{g}}^{\ast }\left(F_2^{\rho }\right)\right)\right)\right\}\hfill \\ \hfill =& \left({\mathbf{g}}^{\ast }\left(F_1^{\rho }\right){\mathbf{g}}^{\ast }{\left(F_2^{\rho }\right)}^{\frac{{\rho }^{1-\frac{1}{\mathfrak{p}}}\left(F_2^{\rho }-F_1^{\rho }\right)}{2}{\left(\frac{1-{\left(\frac{1}{2}\right)}^{\beta }}{\rho (\beta +1)}\right)}^{1-{\displaystyle \frac{1}{\mathfrak{p}}}}{\left(B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+\frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}}}\right),\hfill \end{array}$$

where we have used (24) and the fact that ${{\rm Y}}_1^{\kappa }+{{\rm Y}}_2^{\kappa }\le {({{\rm Y}}_1+{{\rm Y}}_2)}^{\kappa }$ for ${{\rm Y}}_1,{{\rm Y}}_2\ge 0$ with $\kappa \ge 1$. The proof is finished.   □

5. Illustrative Example

To verify the theoretical results obtained thus far, we present a numerical example that demonstrates their accuracy and applicability.

Example 1.

Let us consider the mapping $\mathbf{g}:[F_1,F_2]=[0,1]\to {\mathbb{R}}^{+}$ defined by $\mathbf{g}(\wp )=exp\left\{{\wp }^{s+1}\right\}$, where $s\in (0,1]$ is a fixed parameter. The multiplicative derivative of $\mathbf{g}$ is characterized by ${\mathbf{g}}^{\ast }(\wp )=exp\left\{(s+1){\wp }^s\right\}$, which exhibits multiplicative s-convexity over $[0,1]$.

Case 1.

By invoking Theorem 12, we derive the following inequality:

$$\begin{array}{cc}\hfill & exp\left\{\beta \frac{B(s+2,\beta )(s+\beta +1)+1}{2(s+\beta +1)}-\frac{1}{2^{s+1}}\right\}\hfill \\ \hfill \le & exp\left\{\frac{s+1}{2}\left(\frac{1}{s+1}+\frac{{\left(\frac{1}{2}\right)}^{s+\beta }-1}{s+\beta +1}+B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)-B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)\right)\right\},\hfill \end{array}$$

(27)

provided that

$${\left|exp\left\{\frac{1}{2^{s+1}}-\beta \frac{B(s+2,\beta )(s+\beta +1)+1}{2(s+\beta +1)}\right\}\right|}^{\ast }=exp\left\{\beta \frac{B(s+2,\beta )(s+\beta +1)+1}{2(s+\beta +1)}-\frac{1}{2^{s+1}}\right\}.$$

Figure 1 provides a visual comparison of both sides of inequality (1). As illustrated, the left-hand side consistently remains beneath the right-hand side, thereby validating the accuracy of the conclusion established in Theorem 12.

Case 2.

Through the application of Theorem 15, we establish the inequality

$$\begin{array}{cc}\hfill & exp\left\{\frac{1}{2}-\beta \frac{B(s+2,\beta )(s+\beta +1)+1}{2(s+\beta +1)}\right\}\hfill \\ \hfill \le & exp\left\{\frac{s+1}{2}\left(B_{{\displaystyle \frac{1}{2}}}(s+1,\beta +1)-B_{{\displaystyle \frac{1}{2}}}(\beta +1,s+1)+\frac{1-{\left(\frac{1}{2}\right)}^{\beta +s}}{\beta +s+1}\right)\right\},\hfill \end{array}$$

(28)

given that

$${\left|exp\left\{\frac{1}{2}-\beta \frac{B(s+2,\beta )(s+\beta +1)+1}{2(s+\beta +1)}\right\}\right|}^{\ast }=exp\left\{\frac{1}{2}-\beta \frac{B(s+2,\beta )(s+\beta +1)+1}{2(s+\beta +1)}\right\}.$$

Figure 2 illustrates a graphical comparison of both sides of (2). As depicted, the left-hand side remains uniformly bounded above by the right-hand side, thereby confirming the validity of the theoretical result derived in Theorem 15.

6. Applications of Derived Inequalities to Special Means

In this section, we investigate the practical implications of the derived inequalities and emphasize their significance in the context of special means. To begin, we revisit several well-known means for arbitrary real numbers $F_1$ and $F_2$, defined as follows:

-The arithmetic mean is expressed as

$$\mathcal{A}(F_1,F_2)={\displaystyle \frac{F_1+F_2}{2}}.$$

-The geometric mean is given by

$$\mathcal{G}(F_1,F_2)=\sqrt{F_1{F}_2}.$$

-The logarithmic mean is defined as

$$\mathcal{L}(F_1,F_2)={\displaystyle \frac{F_2-F_1}{ln{F}_2-ln{F}_1}},$$

where $F_1,F_2>0$ and $F_1\ne F_2$.

-The $\mathfrak{p}$-logarithmic mean is formulated as

$${\mathcal{L}}_{\mathfrak{p}}(F_1,F_2)={\left({\displaystyle \frac{F_2^{\mathfrak{p}+1}-F_1^{\mathfrak{p}+1}}{(\mathfrak{p}+1)(F_2-F_1)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{p}}}},$$

valid for $F_1,F_2>0$, $F_1\ne F_2$, and $\mathfrak{p}\in \mathbb{R}\setminus \{-1,0\}$.

Proposition 4.

For real numbers $F_1^{\rho },F_2^{\rho }$ such that $0<F_1^{\rho }<F_2^{\rho }$ with $\rho >0$, we have

$${\left|exp\left\{\mathcal{A}\left(F_1^{\rho },F_2^{\rho }\right)-{\mathcal{L}}_2^2\left(F_1^{\rho },F_2^{\rho }\right)\right\}\right|}^{\ast }\le exp\left\{\frac{\left(F_1^{\rho }+F_2^{\rho }\right)\left(F_2^{\rho }-F_1^{\rho }\right)}{4}\right\}.$$

Proof. 

By applying Theorem 12 with $\beta =s=1$ to the mapping $\mathbf{g}(\wp )=exp\left\{{\wp }^2\right\}$, whose multiplicative derivative is given by ${\mathbf{g}}^{\ast }(\wp )=exp\left\{2\wp \right\}$, we validate the previously stated outcome. Specifically, the expression

$${\left(\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_2^{-}}^1\mathbf{g}\left(F_1^{\rho }\right)\right)\left({}_{\ast }{}^{\rho }\mathrm{K}_{F_1^{+}}^1\mathbf{g}\left(F_2^{\rho }\right)\right)\right)}^{{\displaystyle \frac{\rho }{2\left(F_2^{\rho }-F_1^{\rho }\right)}}}={\left(\underset{F_1^{\rho }}{\stackrel{F_2^{\rho }}{\int }}\mathbf{g}\left(\wp \right)d\wp \right)}^{{\displaystyle \frac{1}{F_2^{\rho }-F_1^{\rho }}}}=exp\left\{{\mathcal{L}}_2^2\left(F_1^{\rho },F_2^{\rho }\right)\right\}$$

leads directly to the desired conclusion.   □

Proposition 5.

For real numbers $F_1^{\rho },F_2^{\rho }$ such that $0<F_1^{\rho }<F_2^{\rho }$ with $\rho >0$, we have

$${\left|exp\left\{\mathcal{A}\left(F_1^{\rho },F_2^{\rho }\right)-{\mathcal{G}}^2\left(F_1^{\rho },F_2^{\rho }\right){\mathcal{L}}^{-1}\left(F_1^{\rho },F_2^{\rho }\right)\right\}\right|}^{\ast }\le exp\left\{-\frac{{\left(F_1^{\rho }+F_2^{\rho }\right)}^2}{8F_1^{\rho }{F}_2^{\rho }}\right\}.$$

Proof. 

By applying Theorem 15 with $\beta =s=1$ to the mapping $\mathbf{g}(\wp )=exp\left\{{\displaystyle \frac{1}{\wp }}\right\}$ defined over the interval $\left[{\displaystyle \frac{1}{F_2^{\rho }}},{\displaystyle \frac{1}{F_1^{\rho }}}\right]$, whose multiplicative derivative is given by ${\mathbf{g}}^{\ast }(\wp )=exp\left\{-{\displaystyle \frac{1}{{\wp }^2}}\right\}$, we establish the following. Specifically, the expression

$$\begin{array}{cc}\hfill & {\left(\left({{}_{\ast }{}^{\rho }\mathrm{K}^1}_{{\left({\displaystyle \frac{1}{F_1}}\right)}^{-}}\mathbf{g}\left({\displaystyle \frac{1}{F_2^{\rho }}}\right)\right)\left({{}_{\ast }{}^{\rho }\mathrm{K}^1}_{{\left({\displaystyle \frac{1}{F_2}}\right)}^{+}}\mathbf{g}\left({\displaystyle \frac{1}{F_1^{\rho }}}\right)\right)\right)}^{{\displaystyle \frac{\rho }{2\left(\frac{1}{F_1^{\rho }}-\frac{1}{F_2^{\rho }}\right)}}}\hfill \\ \hfill =& {\left({\int }_{{\displaystyle \frac{1}{F_2^{\rho }}}}^{{\displaystyle \frac{1}{F_1^{\rho }}}}\mathbf{g}(\wp )d\wp \right)}^{{\displaystyle \frac{F_1^{\rho }{F}_2^{\rho }}{F_2^{\rho }-F_1^{\rho }}}}\hfill \\ \hfill =& exp\left\{{\mathcal{G}}^2\left(F_1^{\rho },F_2^{\rho }\right){\mathcal{L}}^{-1}\left(F_1^{\rho },F_2^{\rho }\right)\right\}\hfill \end{array}$$

validates the previously stated relationship.   □

7. Conclusions

In this study, we have introduced and explored new inequalities within the framework of multiplicative s-convex mappings and Katugampola fractional multiplicative integrals. Starting with the derivation of the Hermite–Hadamard inequality for multiplicative s-convex mappings, we extended our analysis to establish novel midpoint and trapezoid inequalities using Katugampola fractional multiplicative integrals. The theoretical findings were validated through a numerical example, demonstrating their accuracy and applicability. Furthermore, we discussed practical applications of these results, emphasizing their potential impact in various fields.

This work not only enriches the theory of fractional calculus and convex analysis but also opens up new avenues for further investigation. Future research could focus on generalizing these results to broader classes of mappings, exploring additional applications, or extending the framework to other types of fractional operators. By building on the foundation laid here, researchers can continue to advance the understanding of inequalities and their role in mathematical analysis and beyond.