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Article

Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures

1
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
2
Department of Mathematics, Universiti of Balochistan, Quetta 87300, Pakistan
3
Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(9), 617; https://doi.org/10.3390/fractalfract9090617
Submission received: 20 August 2025 / Revised: 11 September 2025 / Accepted: 19 September 2025 / Published: 22 September 2025

Abstract

The aim of this manuscript is to introduce the fractional integral inequalities of H-H types via multiplicative (Antagana-Baleanu) A-B fractional operators. We also provide the fractional version of the H-H type of the product and quotient of multiplicative superquadratic and multiplicative subquadratic functions via the same operators. Superquadratic functions, have stronger convexity-like behavior. They provide sharper bounds and more refined inequalities, which are valuable in optimization, information theory, and related fields. The use of multiplicative fractional operators establishes a nonlinear fractional structure, enhancing the analytical tools available for studying dynamic and nonlinear systems. The authenticity of the obtained results are verified by graphical and numerical illustrations by taking into account some examples. Additionally, the study explores applications involving special means, special functions and moments of random variables resulting in new fractional recurrence relations within the multiplicative calculus framework. These contributions not only generalize existing inequalities but also pave the way for future research in both theoretical mathematics and real-world modeling scenarios.

1. Introduction

Convexity is a intrinsic mathematical notion used in mathematics and engineering to describe and study a wide range of phenomena. Convex functions and convex sets, which are the primary instruments for reducing tough mathematical structures to simpler forms and enabling optimisation of intricate models, form the base of convexity theory. The existence of one global minimum for every problem at all times is the very essence of convex problems, and it makes for enormously simpler tractability and solvability of optimisation problems. It is only because convexity holds this property that convexity is at the heart of optimisation theory, which marries computational complexity with theoretical acuteness. The applications of convexity are not limited to optimisation and spread across various everyday applications like systems engineering, economics, and control theory. Convexity leads to stability and robustness in control theory to allow systems to be built that will behave in a predictable manner under a certain set of uncertain conditions. Convex functions are also widely used in economics to describe utility functions, production sets, and consumer preferences. This helps market equilibrium analysis and policy-making for optimal resource allocation. Convexity is hence a practical need for the building of stable and effective systems rather than a theoretical idea. Moreover, the convexity theory is closely related to the study of integral inequalities, rendering one of the most vibrant and best-studied areas in mathematical analysis. More specifically, convex functions appear most frequently in the context and definition of integral inequalities, which are utilized to find limits for integral expressions and derive basic results in a large number of areas of analysis. One prominent example is the H-H inequality, which has been widely used in many analytical contexts and gives an estimate of the mean value of a convex function over an interval [1,2,3].
In order to describe systems with memory and hereditary properties, fractional calculus, an extension of classical calculus to non-integer order integrals and derivatives—has become more popular during the past ten years. As a result, it is crucial for computer science as well as physics and engineering, which may surprise some people [4]. As inequality theory has grown in significance outside of the practical sciences, fractional calculus has played a significant role in its development. Specifically, fractional integral inequalities have been a highly active area of research in the last decade. One such significant contribution was made by Sarikaya et al. (2013) [5], who extended the standard H-H inequality using Riemann–Liouville fractional integrals. This extension provided new insights and more accurate bounds for convex functions, which were useful for the theory of inequalities as well as the analytical toolkit for dynamic system analysis.
Since the introduction of the H-H inequality in the framework of R-L fractional integrals, researchers have extensively explored H-H type inequalities using a wide variety of other fractional integral operators. Notable among these are Sarikaya fractional integrals [6], generalized proportional fractional integrals [7], ( k p ) fractional integrals [8], k-fractional integrals [9], Katugampola fractional integrals [10], generalized R-L integrals [11], ψ -R-L fractional integrals [12], and conformable fractional integrals [13]. In addition to these, several other classes of fractional inequalities have been developed, including fractal Hadamard–Mercer [14], Simpson [15], Euler–Maclaurin [16], Bullen [17], and Ostrowski type inequalities [18]. For those interested in the latest developments and comprehensive surveys in this evolving field, recent studies such as [19,20,21] and the references therein offer valuable insights into the current state-of-the-art research in fractional integral inequalities.
Superquadraticity is a refinement of the accuracy of integral inequalities and is better than the typical convexity. The refinement in this sense is especially useful for optimisation, where precise boundaries result in optimum solutions directly, and applied mathematics, where exact estimates allow for improved modelling. For theoretical as well as practical reasons, superquadraticity is a solid theoretical basis that makes the derivation and implementation of integral inequalities more efficient.
The concept of superquadratic functions, which generalizes and extends the classical class of convex functions, was initially introduced by Abramovich et al. in [22]. A more rigorous formalization and foundational theoretical framework were later provided by the same authors in [23], laying the groundwork for subsequent developments. Building upon this foundation, Li and Chen [24] explored a fractional version of H-H type inequalities using R-L fractional integrals, offering new insights into superquadratic behavior in the context of fractional calculus. Alomari and Chesneau [25] added to the theory by introducing h-superquadratic functions, their structural features and analytic significance. Logarithmically or multiplicatively superquadratic functions, introduced by Krnić, Moradi, and Sababheh, are shown to inherit key properties of superquadratic functions, including convexity, logarithmic superadditivity, and refined Jensen-type inequalities in both scalar and operator forms [26]. Butt and Khan [27] advanced this line of work by proving Fejér and H-H type inequalities for h-superquadratic functions via R-L fractional operators, contributing to the volume of fractional inequality studies. In another landmark work, Khan et al. [28] proposed the ( P , m ) -superquadratic function as a generalization of many subclasses, its graphical examples, analytical properties, and some integral inequalities to signify prospects for applications. As a complement to this, Butt et al. [29] derived Fejér and H-H type inequalities for superquadratic functions by employing Atangana–Baleanu fractional integrals, thereby incorporating non-singular kernels in the study. A novel extension was introduced in [30], where the authors studied fuzzy interval-valued superquadratic functions, which brought new possibilities for uncertainty modeling in functional analysis. Banić et al. [31] gave a foundational contribution by constructing classical inequalities from the perspective of superquadraticity, establishing a rigorous theoretical framework that has inspired many recent studies. Based on this, Khan et al. [32] considered multiplicatively ( P , m ) -superquadratic functions to get fractional integral inequalities that merge multiplicative calculus with generalized convexity. There was another extension by Khan et al. [33], who considered P-superquadratic functions for multiplicative R-L integrals, obtaining more precise bounds and establishing broader applicability within inequality theory. Most recently, Butt et al. [34] introduced a fractal generalization to superquadratic functions, such as generalized probability distributions and opening up new doors to stochastic modeling and analysis. Collectively, these articles underscore the increasing applicability and importance of superquadratic functions to a broad spectrum of disciplines—ranging from inequality theory, fractional calculus, fuzzy analysis, multiplicative structures, and stochastic processes. This growing body of literature continues to lay the groundwork for ongoing theoretical development and practical applications in current mathematical analysis.
The theory of multiplicative calculus has also garnered a lot of attention lately, particularly in light of the work of Ali et al. [35]. Their work sparked a flurry of interest in the use of multiplicative calculus, particularly in the context of integral inequalities. Because multiplicative calculus is based on a different kind of foundation structure than traditional calculus, it is very good at solving problems involving growth procedures and ratio-based systems. Examining inequalities of other function classes has been made easier with the help of this structure. Several multiplicative inequalities of integer order for other classes of functions have been discovered by researchers for this purpose. Notably, considerable progress has been made in determining inequalities with multiplicative harmonically convex functions [36] and multiplicative preinvex P-convex functions [37], both of which apply classical convexity principles to the multiplicative setting. These findings enhance the theoretical foundation of multiplicative calculus and offer useful resources for mathematical analysis and optimisation applications. In addition, there has been much research on additional kinds of integer order multiplicative inequalities. For instance, in the context of multiplicative integrals, Khan and Budak [38] developed inequalities of H-H’s type for *differentiable function. In a corresponding development, Xie et al. [39] addressed parallel inequalities for **differentiable functions and thus extended the area of multiplicative inequalities into wider classes of differentiable functions. To an interested reader seeking to gain a wider picture of multiplicative integer-order inequalities, particularly Ostrowski, Simpson, and Maclaurin type, additional information and progress are available from the references cited in [40,41] and references therein. These papers continue to widen the horizons of this emerging discipline and bear testimony to the flexibility of multiplicative calculus in solving problems of traditional mathematics in a novel perspective.
Growing academic interest has emerged in recent years regarding integer order inequalities in the setting of multiplicative calculus, producing an array of results for different types of functions. However, there has not been much research done on fractional variants, particularly those that use multiplicative fractional operators. Despite the promising potential of fractional calculus within multiplicative frameworks, only a limited number of studies have delved into this area. In 2020, Budak and Özçelik [42], achieved a significant breakthrough by presenting an innovative method for establishing new H-H type inequalities using multiplicative R-L fractional integrals. This pioneering work was a milestone in the field, stimulating a great deal of interest in the mathematical research community due to the new methodology and the potential for broader application. Fu et al. [43] expanded on this basis by considering a family of operators known as multiplicative tempered fractional integrals. They extended the scope of H-H-type inequalities to multiplicatively convex functions, enriching the theoretical foundation of multiplicative fractional inequalities. Long and Du [44] conducted a comprehensive study on multiplicative k-Atangana–Baleanu fractional integrals, introducing a novel class of non-local operators defined within the multiplicative calculus framework. Their work not only established foundational properties of these integrals but also applied them to derive several Mercer-type integral inequalities. By bridging multiplicative calculus with generalized fractional models, this study provides valuable insights into the behavior of complex dynamical systems and extends the analytical toolkit available for dealing with nonlinear, non-local problems. These findings were expanded upon by Peng and Du [45], who introduced the concepts of differentiable multiplicative m-preinvex and (s,m)-preinvex. In the framework of multiplicative tempered fractional integrals, they established new H-H-type inequalities that expand the application scope of this theory to more generalized convexity frameworks. These are important contributions to the systematic development of inequality theory in the context of fractional multiplicative calculus.
A major advance in multiplicative fractional calculus was made in 2022 when Peng et al. [46] established a new family of operators known as multiplicative fractional operators with exponential kernels. Through these newly proposed operators, they were able to derive various integral identities and establish upper bound bounds for certain functions. These results favored the ongoing research efforts that were focused on extending classical integral inequalities to the multiplicative fractional scenario. Building from this foundation, Kashuri et al. [47] investigated this particular family of operators within the framework of H-H type inequalities. Their works focused on multiplicative Sarikaya fractional integral inequalities, hence expanding the working fields of such operators to more classes of functions and providing a deeper insight into multiplicative convexity structure. In addition to these specific advancements, several other recent publications have explored various aspects of multiplicative fractional integrals. For a broader view of the current state of research in this rapidly evolving area, readers are encouraged to consult references [48,49], which provide further developments, extensions, and applications of multiplicative–fractional calculus in the context of integral inequalities and beyond.
Building upon the foundational research in convexity, fractional calculus, superquadraticity, and multiplicative calculus, we identified a notable gap in the literature: the absence of fractional versions of inequalities associated with superquadratic functions from the perspective of multiplicative calculus. This observation served as the primary motivation for our investigation. In response, we directed our efforts toward this unexplored area and successfully developed novel integral inequalities by employing multiplicative A-B fractional integral operators. It is well established that superquadratic functions and their corresponding inequalities represent a refinement of the classical theory of convex functions. In alignment with this idea, the concept we introduce may be regarded as a refinement of multiplicative convexity and its associated results. To the best of our knowledge, this is the first work to provide explicit examples of functions that satisfy the conditions of multiplicative superquadraticity. Since this new class is a strict extension of multiplicatively convex functions, the constructed examples inherently qualify as examples of multiplicatively convex functions as well. Moreover, we explored potential applications of this new theory in the context of special means and special functions. Notably, these applications have not yet been investigated even within the broader scope of multiplicative convexity, highlighting the novelty and potential impact of our contributions.
In recent years, the applications of fractional calculus in the field of integral inequalities have witnessed a significant surge, underscoring its growing importance in modern mathematical analysis. Continuing along this promising trajectory, it becomes essential to integrate the tools of fractional calculus with emerging concepts such as multiplicative superquadratic functions in order to advance the theory of inequalities. As previously discussed, the fractional formulation of inequalities related to superquadraticity within the framework of multiplicative calculus remains an unexplored area in the existing literature. Recognizing this gap, our objective is to develop and extend H-H-type inequalities by leveraging the combined structure of fractional and multiplicative calculus. This approach not only fills a crucial void in the current research landscape but also opens up new avenues for further analytical exploration and applications.
The organization of the paper is outlined as follows:
Section 1 presents the essential background and foundational concepts related to convex and superquadratic functions, as well as fractional and multiplicative calculus along with their associated inequalities. In Section 2, we summarize key formulas and fundamental results pertaining to convexity, superquadraticity, and both types of calculus. Section 3 presents novel fractional order inequalities via multiplicative calculus, employing multiplicatively superquadratic functions. To assess the utility of these findings, graphical illustrations and relevant examples are provided. Section 4 explains the applications and implications. Section 5 wraps up the discussion with a succinct summary and points to possible lines of future inquiry inspired by the results.

2. Preliminaries

2.1. Convexity and Related Integral Inequalities

Definition 1
([50]). The function ψ : I is said to be convex, if it holds
ψ ς μ o + ( 1 ς ) ν o ς ψ ( μ o ) + ( 1 ς ) ψ ( ν o ) ,
ς [ 0 , 1 ] and μ o , ν o I .
One important foundational result in the study of convex functions is the H-H inequality, which is credited to Jacques Hadamard and Charles Hermite [1]. The integral average of a convex function defined on a closed and bounded interval has lower and upper bounds given by this well-known inequality. It is important because it provides a useful way to estimate the integral of convex functions using only the values at the interval’s midpoint and endpoints. The foundation for convex analysis was established by Hermite and Hadamard, who independently arrived at this inequality in the past. Because of its sophisticated formulation and wide range of applications, the H-H inequality has since evolved into a traditional tool in mathematical analysis. In addition to its theoretical significance, the inequality is crucial for numerical integration, approximation theory, and optimisation problems where it is necessary to comprehend how convex functions behave. The H-H inequality is a potent analytical tool in both pure and applied mathematics because it captures important convexity properties, which not only make accurate estimations easier but also help us better understand the structural features of convex functions across intervals.
H-H inequalities for a convex function ψ : I on I , where μ o , ν o I with μ o < ν o are given by
ψ μ o + ν o 2 1 ν o μ o μ o ν o ψ ( z ) d z ψ ( μ o ) + ψ ( ν o ) 2 .
By evaluating the function at the midpoint, the first term in the H-H inequality, known as the inequality of midpoint-type, provides a lower estimate of the integral. Using the average of the function’s values at the two endpoints, the second term known as the inequality of trapezoid-type provides an upper estimate.
Definition 2
([50]). A function ψ : I is said to be a multiplicatively convex, if it holds
ψ ( ς μ o + ( 1 ς ) ν o ) [ ψ ( μ o ) ] ς [ ψ ( ν o ) ] ( 1 ς ) ,
ς [ 0 , 1 ] and μ o , ν o I .
H-H-type inequalities for such a function was first proved by Ali et al. [35].
Theorem 1.
Let the function ψ be a multiplicatively convex on [ μ o , ν o ] , then it holds
ψ μ o + ν o 2 μ o ν o ( ψ ( z ) ) d z 1 ν o μ o G ( ψ ( μ o ) , ψ ( ν o ) ) ,
where G ( . , . ) is geometric mean.

2.2. Fractional Integral Operators

Let us now introduce the concept of R-L and A-B fractional integrals, which will serve as a foundational tool in the subsequent analysis.
Definition 3.
The R-L fractional integrals (right and left-sided) of order ς 0 with μ o 0 are given by I ν o ς ψ ( z ) and I μ o + ς ψ ( z ) correspondingly and defined by
I μ o + ς ψ ( z ) = 1 Γ ( ς ) μ o z ( z ς ) ς 1 ψ ( ς ) d ς , z > μ o ,
and
I ν o ς ψ ( z ) = 1 Γ ( ς ) z ν o ( ς z ) ς 1 ψ ( ς ) d ς , z < ν o .
where Γ ( ς ) is a gamma function and given by Γ ( ς ) = 0 z ς 1 e z d z .
Atangana and Baleanu originally suggested A-B fractinal operators in 2016 [21].
Definition 4.
Let ν o > μ o , ς [ 0 , 1 ] and ψ H 1 ( μ o , ν o ) , characterized by a non-local kernel, relates to the fractional integral of that function, expressed as
I y o ς μ o A B [ ψ ( y o ) ] = 1 ς B ( ς ) ψ ( y o ) + ς B ( ς ) Γ ( ς ) μ o y o ψ ( z ) ( y o z ) ς 1 d z ,
is termed as left A-B’s fractional operator. while the right A-B fractional operator takes the form
I y o ς ν o A B [ ψ ( y o ) ] = 1 ς B ( ς ) ψ ( y o ) + ς B ( ς ) Γ ( ς ) y o ν o ψ ( z ) ( z y o ) ς 1 d z .

2.3. Superquadraticity and Related Integral Inequalities

Definition 5.
A function ψ : 0 , , is said to be superquadratic, if ν o 0 and μ o 0 , a constant C ν o such that
ψ ( ν o ) ψ ( μ o ) + C μ o ( ν o μ o ) + ψ ( | ν o μ o | ) ,
In the same way that convexity and concavity of a function play a role in determining the next one, so do superquadraticity and subquadraticity. For instance, ψ yields a subquadratic function if ψ is superquadratic. The example that follows will show that superquadraticity and subquadraticity are independent of changing the function’s sign.
For each ν o 0 , the function
ψ ( ν o ) = ν o p
is superquadratic for p 2 and subquadratic for 0 p < 2 . C ν o = ψ ( ν o ) in this case. When p = 2 is taken into account in a function
ψ ( ν o ) = ν o p ,
the sign “≤” is replaced by “=” in (9).
The fundamental properties of superquadratic functions have been explained in [22] and [23] by the well known researchers Sinnamon, Jameson and Abramovich. In contrast to convexity, the characteristics and integral inequalities based on the condition (9) are more refined. Any arbitrary superquadratic function must satisfy the following requirements (1), (2), and (3):
1.
ψ ( 0 ) 0 .
2.
If ψ is differentiable at ν o > 0 as well as ψ ( 0 ) = 0 and ψ ( 0 ) = 0 then C ν o = ψ ( ν o ) .
3.
ψ is convex and ψ ( 0 ) = 0 as well as ψ ( 0 ) = 0 if ψ 0 .
Definition 6.
A superquadratic function is also one that satisfies the following conditions: (10) ∀ ν o , μ o 0 , and ς ( 0 , 1 ) .
ψ ( ( 1 ς ) ν o + ς μ o ) ( 1 ς ) ψ ( ν o ) + ς ψ ( μ o ) ς ψ ( ( 1 ς ) | ν o μ o | ) ( 1 ς ) ψ ( ς | ν o μ o | ) .
The condition (10) is known as the improvement of the Jensen inequality for nonnegative superquadratic functions. In the event that the inequality in (10) is inverted, the function ψ exhibits subquadratic behaviour.
To explore the notion of superquadratic functions within the framework of multiplicative calculus, we have a look at the following fundamental mathematics related to multiplicatively superquadratic or logarithmically functions. As we mentioned in the aforementioned paragraph that the idea of the multiplicatively superquadratic functions was introduced by Mario Karnić et al. in [26] is defined in the following way.
Definition 7.
The function ψ : [ 0 , ) ( 0 , ) is termed as multiplicatively superquadratic functions if
ψ ( ( 1 ς ) ν o + ς μ o ) ψ 1 ς ( ν o ) ψ ς ( μ o ) ψ ς ( ( 1 ς ) | ν o μ o | ) ψ 1 ς ( ς | ν o μ o | ) .
If a function ψ always takes positive values, we say that it is multiplicatively superquadratic when log ψ behaves like a superquadratic function. However, when log ψ is subquadratic, then ψ is described as logarithmically subquadratic.
Proposition 1.
If both ψ and Υ are multiplicatively superquadratic, then the functions ψ Υ and ψ g are also multiplicatively superquadratic.

2.4. Fundamental Properties of Multiplicative Calculus

We commence this section with a review of essential definitions, properties, and ideas pertaining to differentiation. Additionally, we examine key aspects of multiplicative integration, laying the groundwork for the subsequent development of our results.
Definition 8
([49]). Let the ψ : be a positive then the multiplicative derivative ψ * is defined as given below:
d * ψ d z = ψ * ( z ) = lim h 0 ψ ( z + h ) ψ ( z ) 1 h .
Remark 1.
For a positive differentiable function, the link between its multiplicative derivative and its ordinary derivative is given by:
ψ * = e ( ln ψ ( z ) ) = e ψ ( z ) ψ ( z ) .
Several fundamental properties can be attributed to the multiplicative derivative, including the following:
Proposition 2
([49]). Let ψ and Υ with c > 0 as an arbitrary constant be multiplicatively differentiable functions then c ψ , ψ + Υ , ψ Υ and ψ Υ are *differentiable functions
1.
( c ψ ) * ( z ) = ψ * ( z )
2.
( ψ Υ ) * ( z ) = ψ * ( z ) Υ * ( z )
3.
( ψ + Υ ) * ( z ) = ψ * ( z ) ψ ( z ) ψ ( z ) + Υ ( z ) Υ * ( z ) Υ ( z ) ψ ( z ) + Υ ( z )
4.
ψ Υ * ( z ) = ψ * ( z ) Υ * ( z )
5.
( ψ Υ ) * ( z ) = ψ * ( z ) Υ ( z ) ψ ( z ) Υ ( z )
The multiplicative integral, sometimes denoted as the *integral, is represented by the notation μ o ν o ( ψ ( z ) ) d z . This mathematical concept was introduced by Bashirov et al. in [49] as part of the foundational framework for multiplicative calculus. While the classical Riemann integral of a function ψ over an interval [ μ o , ν o ] is defined through the limit of a sum of rectangular areas, the multiplicative integral over the same interval is formulated using the limit of a product of function values, raised to powers that correspond to the subinterval lengths. In essence, the multiplicative integral replaces the additive structure of classical integration with a multiplicative structure, which is more natural in contexts involving exponential growth, proportional change, or scale-invariant systems. This shift in perspective makes it a valuable tool in various fields such as biomathematics, economics, and information theory, where multiplicative processes are prevalent.
The relationship between the multiplicative integral and the classical Riemann integral is given by the following expression, as established in [49]:
Proposition 3.
If the function ψ on the interval [ ν o , μ o ] is Riemann integrable then it is multiplicatively integrable over the same interval.
μ o ν o ( ψ ( z ) ) d z = e μ o ν o ln ( ψ ( z ) ) d z .
Additionally, Bashirov et al. [49] showed that a multiplicatively integrable function has the following characteristics and outcomes:
Proposition 4.
If ψ is Riemann integrable on [ μ o , ν o ] , then it is also multiplicatively integrable on that interval.
1.
μ o ν o ψ ( z ) p d z = μ o ν o ψ ( z ) d z p , p R .
2.
μ o ν o ψ ( z ) Υ ( z ) d z = μ o ν o ψ ( z ) d z · μ o ν o Υ ( z ) d z .
3.
μ o ν o ψ ( z ) Υ ( z ) d z = μ o ν o ψ ( z ) d z μ o ν o Υ ( z ) d z .
4.
μ o ν o ψ ( z ) d z = μ o c ψ ( z ) d z · c ν o ψ ( z ) d z , μ o c ν o .
5.
μ o μ o ψ ( z ) d z = 1 , μ o ν o ψ ( z ) d z = ν o μ o ψ ( z ) d z 1 .
The Theorem 2 provides the formula for multiplicative version of the integration by parts formula [49].
Theorem 2.
Let ψ and Υ be *differentiable and differentiable functions respectively and ψ Υ be a *integrable functions, then
μ o ν o ψ * ( z ) Υ ( z ) d z = ψ ( ν o ) Υ ( ν o ) ψ ( μ o ) Υ ( μ o ) × 1 μ o ν o ψ ( z ) Υ ( z ) d z .
Lemma 1
([23]). Let ψ and Υ be *differentiable and differentiable respectively, and ψ Υ is *integrable then
μ o ν o ψ * ( h ( z ) ) h ( z ) Υ ( z ) d z = ψ ( ν o ) Υ ( ν o ) ψ ( μ o ) Υ ( μ o ) × 1 μ o ν o ψ ( h ( z ) ) Υ ( z ) d z .
A noteworthy generalization of the classical R-L fractional integrals, known as the multiplicative R-L fractional integrals, was introduced in [48], marking a significant development in the field of multiplicative fractional calculus.
Definition 9.
The multiplicatively left R-L fractional operator I μ o * ς ψ ( z ) of order ς C , R e ( ς ) > 0 and z > μ o , is formulated as
I μ o * ς ψ ( z ) = e x p { I μ o + ς l n ψ ( z ) } = e x p 1 Γ ( ς ) μ o z ( z ς ) ς 1 l n ψ ( ς ) d ς ,
and the right-sided one I * ν o ς ψ ( z ) , where z < ν o is defined by
I * ν o ς ψ ( z ) = e x p { I ν o ς l n ψ ( z ) } = e x p 1 Γ ( ς ) z ν o ( ς z ) ς 1 l n ψ ( ς ) d ς ,
with Γ ( ς + 1 ) = ς Γ ( ς ) and Γ ( 1 ) = 1 .
Definition 10
([44]). The multiplicatively left A-B fractional operators I μ o A B * ς ψ ( z ) of order ς C , R e ( ς ) > 0 and z > μ o , is formulated as
I μ o A B * ς ψ ( z ) = e x p { A B I μ o + ς l n ψ ( z ) } = e x p 1 ς B ( ς ) l n ψ ( ) + ς B ( ς ) Γ ( ς ) μ o z ( z ) ς 1 l n ψ ( ) d ,
and the right-sided one I * A B ν o ς ψ ( z ) , where z < ν o is defined by
I * A B ν o ς ψ ( z ) = e x p { A B I ν o ς l n ψ ( z ) } = e x p 1 ς B ( ς ) l n ψ ( ) + ς B ( ς ) Γ ( ς ) z ν o ( z ) ς 1 l n ψ ( ) d ,
with Γ ( 1 ) = 1 , Γ ( z + 1 ) = z Γ ( z ) and B ( ς ) denotes the normalized function satisfying B ( 1 ) = B ( 0 ) = 1 .

3. Fractional Inequalities for Multiplicative Superquadratic Function via Multiplicatively A-B Fractional Operator

This section consists of integral inequalities of H-H’s type via multiplicative A-B fractional integrals for multiplicative superquadratic function.
Theorem 3.
Let ψ be a positive multiplicative superquadratic function on the interval [ μ o , ν o ] , then
ψ ( μ o ) ψ ( ν o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς μ o ν o ψ | μ o + ν o 2 z | ( z μ o ) ς 1 d z ς ( ν o μ o ) ς ψ μ o + ν o 2 I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) B ( ς ) Γ ( ς ) 2 ( ν o μ o ) ς ( G ( ( ψ ( μ o ) , ψ ( ν o ) ) ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 ,
holds ς C , R e ( ς ) > 0 . Here G(.,.) signifies the geometric mean.
Proof. 
Let the function ψ be a positive and multiplicative superquadratic; therefore, we have
ln ψ ( μ o + ν o 2 ) = ln ψ μ o + ( 1 ) ν o + ν o + ( 1 ) μ o 2 1 2 ln ψ ( μ o + ( 1 ) ν o ) + 1 2 ln ψ ( ν o + ( 1 ) μ o ) 1 2 ln ψ ( 1 2 ) ( μ o ν o ) 2 1 2 ln ψ ( 1 2 ) ( μ o ν o ) 2 .
Multiply both sides with ς 1 , then integrating the resulting inequality with respect to ℓ over [ 0 , 1 ] , we have
0 1 ς 1 ln ψ ( μ o + ν o 2 ) d 1 2 0 1 ς 1 ln ψ ( μ o + ( 1 ) ν o ) d + 1 2 0 1 ς 1 ln ψ ( ν o + ( 1 ) μ o ) d 0 1 ς 1 ln ψ ( 1 2 ) ( μ o ν o ) 2 d .
After carrying out elementary computations and introducing a change of variables, we attain
ln ψ ( μ o + ν o 2 ) Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς I μ o + ς A B ln ψ ( ν o ) + I ν o ς A B ln ψ ( μ o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς ln ( ψ ( μ o ) ψ ( ν o ) ) ς ( ν o μ o ) ς μ o ν o ( z μ o ) ς 1 ln ψ ( | μ o + ν o 2 z | ) d z .
Thus, we have
ψ ( μ o ) ψ ( ν o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς μ o ν o ψ | μ o + ν o 2 z | ( z μ o ) ς 1 d z ς ( ν o μ o ) ς ψ μ o + ν o 2 I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) B ( ς ) Γ ( ς ) 2 ( ν o μ o ) ς .
The preceding computation coincides with the first component of inequality (16). In order to derive the second component, we focus on the R.H.S of (17), leading to
I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) B ( ς ) Γ ( ς ) 2 ( ν o μ o ) ς = e x p Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς I μ o + ς A B ( ln ψ ) ( ν o ) + I ν o ς A B ( ln ψ ) ( μ o ) = e x p { ( 1 ς ) Γ ( ς ) ln ( ψ ( μ o ) ψ ( ν o ) ) 2 ( ν o μ o ) ς + ς 2 0 1 ς 1 ln ψ ( μ o ( 1 ) ν o ) d + 0 1 ς 1 ln ψ ( ν o ( 1 ) μ o ) d } exp { ( 1 ς ) Γ ( ς ) ln ( ψ ( μ o ) ψ ( ν o ) ) 2 ( ν o μ o ) ς + ς 2 [ ln ( ψ ( μ o ) ψ ( ν o ) ) ς 2 ( ν o μ o ) ς + 1 ( μ o ν o ln ψ ( | ν o z | ) ( z μ o ) ς d z + μ o ν o ln ψ ( | z μ o | ) ( ν o z ) ( z μ o ) ς 1 d z ) ] } = ( ψ ( μ o ) ψ ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 .
Upon merging (17) and (18), the desired result follows. □
Remark 2.
If we set ς = 1 in Theorem 3, we attain
ψ μ o + ν o 2 μ o ν o ψ | μ o + ν o 2 z | d z 1 ( ν o μ o ) μ o ν o ( ψ ( z ) ) d z 1 ν o μ o G ( ψ ( μ o ) , ψ ( ν o ) ) μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) · ( ψ ( | z μ o | ) ) ( ν o z ) d z 1 ( ν o μ o ) 2 .
Which is a new inequality and has not obtained before.
Example 1.
For ψ ( z ) = e x p { z k } , the validity of Theorem 3 is described in Figure 1.
Theorem 4.
If ψ and Υ are two positive and multiplicative superquadratic functions on [ μ o , ν o ] , then
G ( ψ ( μ o ) , ψ ( ν o ) ) G ( Υ ( μ o ) , Υ ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς ψ μ o + ν o 2 Υ μ o + ν o 2 × μ o ν o ψ Υ | μ o + ν o 2 z | ( z μ o ) ς 1 d z ς ( ν o μ o ) ς I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) · I * ς μ o A B ( Υ ) ( ν o ) · I ν o ς * A B ( Υ ) ( μ o ) Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς G ( ( ψ ( μ o ) , ψ ( ν o ) ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 × ( Υ ( μ o ) Υ ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( Υ ( | ν o z | ) ) ( z μ o ) ς · ( Υ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 .
Proof. 
Since ψ and Υ are positive and multiplicative superquadratic functions, note that
ln ψ μ o + ν o 2 Υ μ o + ν o 2 = ln ψ μ o + ν o 2 + ln Υ μ o + ν o 2 = ln ψ μ o + ( 1 ) ν o + ν o + ( 1 ) μ o 2 + ln Υ μ o + ( 1 ) ν o + ν o + ( 1 ) μ o 2 1 2 ln ψ ( μ o + ( 1 ) ν o ) + 1 2 ln ψ ( ν o + ( 1 ) μ o ) 1 2 ln ψ ( 1 2 ) ( μ o ν o ) 2 1 2 ln ψ ( 1 2 ) ( μ o ν o ) 2 + 1 2 ln Υ ( μ o + ( 1 ) ν o ) + 1 2 ln Υ ( ν o + ( 1 ) μ o ) 1 2 ln Υ ( 1 2 ) ( μ o ν o ) 2 1 2 ln Υ ( 1 2 ) ( μ o ν o ) 2 .
Multiplying (21) both sides with ς 1 , then integrating the resulting inequality with respect to ℓ over [ 0 , 1 ] , we have
0 1 ς 1 ln ψ μ o + ν o 2 Υ μ o + ν o 2 d 1 2 0 1 ς 1 ln ψ ( μ o + ( 1 ) ν o ) d + 1 2 0 1 ς 1 ln ψ ( ν o + ( 1 ) μ o ) d 0 1 ς 1 ln ψ ( 1 2 ) ( μ o ν o ) 2 d + 1 2 0 1 ς 1 ln Υ ( μ o + ( 1 ) ν o ) d + 1 2 0 1 ς 1 ln Υ ( ν o + ( 1 ) μ o ) d 0 1 ς 1 ln Υ ( 1 2 ) ( μ o ν o ) 2 d .
After carrying out elementary computations and introducing a change of variables, we obtain
ln ψ μ o + ν o 2 Υ μ o + ν o 2 Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς [ I μ o + ς A B ln ψ ( ν o ) + I ν o ς A B ln ψ ( μ o ) + I μ o + ς A B ln Υ ( ν o ) + I ν o ς A B ln Υ ( μ o ) ] ln ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς ς ( ν o μ o ) ς [ μ o ν o ( z μ o ) ς 1 ln ψ ( | μ o + ν o 2 z | ) d z + μ o ν o ( z μ o ) ς 1 ln Υ ( | μ o + ν o 2 z | ) d z ] .
It follows as,
ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς ψ μ o + ν o 2 Υ μ o + ν o 2 × μ o ν o ψ Υ | μ o + ν o 2 z | ( z μ o ) ς 1 d z ς ( ν o μ o ) ς I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) · I * ς μ o A B ( Υ ) ( ν o ) · I ν o ς * A B ( Υ ) ( μ o ) Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς .
The preceding computation coincides with the first component of inequality (20). In order to derive the second component, we focus on the R.H.S of (22), leading to
I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) · I * ς μ o A B ( Υ ) ( ν o ) · I ν o ς * A B ( Υ ) ( μ o ) Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς = e x p Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς I μ o + ς A B ln ψ ( ν o ) + I ν o ς A B ln ψ ( μ o ) + I μ o + ς A B ln Υ ( ν o ) + I ν o ς A B ln Υ ( μ o ) = e x p { ( 1 ς ) ( Γ ( ς ) ) ln ( ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ) 2 ( ν o μ o ) ς + ς 2 [ 0 1 ς 1 ln ψ ( μ o + ( 1 ) ν o ) d + 0 1 ς 1 ln ψ ( ν o + ( 1 ) μ o ) d + 0 1 ς 1 ln Υ ( μ o + ( 1 ) ν o ) d + 0 1 ς 1 ln Υ ( ν o + ( 1 ) μ o ) d ] } e x p { ( 1 ς ) ( Γ ( ς ) ) ln ( ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ) 2 ( ν o μ o ) ς ς ( ν o μ o ) ς + 1 μ o ν o ln ψ ( | ν o z | ) ( z μ o ) ς ψ ( | z μ o | ) ( ν o z ) ( z μ o ) ς 1 d z + 1 2 ln ψ ( μ o ) , ψ ( ν o ) + ln Υ ( μ o ) , Υ ( ν o ) ς ( ν o μ o ) ς + 1 μ o ν o ln Υ ( | ν o z | ) ( z μ o ) ς Υ ( | z μ o | ) ( ν o z ) ( z μ o ) ς 1 d z } .
Thus it follows as
= ( ψ ( μ o ) ψ ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 × ( Υ ( μ o ) Υ ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( Υ ( | ν o z | ) ) ( z μ o ) ς · ( Υ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 .
The combination of (22) and (23) yields the required result. □
Remark 3.
If ς = 1 is picked in Theorem 4, we get
ψ μ o + ν o 2 Υ μ o + ν o 2 μ o ν o ψ Υ | μ o + ν o 2 z | d z 1 ν o μ o μ o ν o ( ψ ( z ) ) d z μ o ν o ( Υ ( z ) ) d z 1 ν o μ o G ( ψ ( μ o ) , ψ ( ν o ) ) · G ( Υ ( μ o ) , Υ ( ν o ) ) μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) ( ψ ( | z μ o | ) ) ( ν o z ) ( Υ ( | ν o z | ) ) ( z μ o ) ( Υ ( | z μ o | ) ) ( ν o z ) d z 1 ( ν o μ o ) 2
which is a new inequality and has not obtained before.
Example 2.
The Figure 2 illustrates the authenticity of Theorem 4 for ψ ( z ) = e x p { z 3 } and Υ ( z ) = e x p { z 2 } .
Corollary 1.
Suppose ψ is positive and multiplicatively convex on [ μ o , ν o ] , and Υ is positive and multiplicatively superquadratic. Then
I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) · I * ς μ o A B ( Υ ) ( ν o ) · I ν o ς * A B ( Υ ) ( μ o ) Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς G ( ψ ( μ o ) , ψ ( ν o ) ) G ( Υ ( μ o ) , Υ ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( Υ ( | ν o z | ) ) ( z μ o ) ς ( Υ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 .
Theorem 5.
Let ψ be a multiplicatively superquadratic function and Υ be a multiplicatively subquadratic function on [ μ o , ν o ] , then
μ o ν o ( ψ ( | μ o + ν o 2 z | ) ) ( z μ o ) ς 1 d z μ o ν o ( Υ ( | μ o + ν o 2 z | ) ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς ψ ( μ o + ν o 2 ) Υ ( μ o + ν o 2 ) I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) I * ς μ o A B ( Υ ) ( ν o ) · I ν o ς * A B ( Υ ) ( μ o ) Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς
G ( ψ ( μ o ) , ψ ( ν o ) ) G ( Υ ( μ o ) , Υ ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 × μ o ν o ( Υ ( | ν o z | ) ) ( z μ o ) ς · ( Υ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 .
Proof. 
Since the functions ψ and Υ are multiplicatively superquadratic and subquadratic functions respectively; therefore, we have
ln ψ μ o + ν o 2 Υ μ o + ν o 2 = ln ψ μ o + ν o 2 ln Υ μ o + ν o 2 = ln ψ ( 1 ) μ o + ν o + ( 1 ) ν o + μ o 2 ln Υ ( 1 ) μ o + ν o + ( 1 ) ν o + μ o 2 1 2 ln ψ ( 1 ) μ o + ν o + 1 2 ln ψ ( 1 ) ν o + μ o ln ψ | ( 1 2 ) ( μ o ν o ) | 2 1 2 ln Υ ( 1 ) μ o + ν o 1 2 ln Υ ( 1 ) ν o + μ o + ln Υ | ( 1 2 ) ( μ o ν o ) | 2 .
Multiplying (26) through by ς 1 and integrating over [ 0 , 1 ] yields
0 1 ς 1 ln ψ ( μ o + ν o 2 ) Υ ( μ o + ν o 2 ) d 1 2 0 1 ς 1 ln ψ ( 1 ) μ o + ν o d + 1 2 0 1 ς 1 ln ψ ( 1 ) ν o + μ o d 0 1 ς 1 ln ψ | ( 1 2 ) ( μ o ν o ) | 2 d 1 2 0 1 ς 1 ln Υ ( 1 ) μ o + ν o d 1 2 0 1 ς 1 ln Υ ( 1 ) ν o + μ o d + 0 1 ς 1 ln Υ | ( 1 2 ) ( μ o ν o ) | 2 d .
Through elementary calculations and a change of variables, we get
ln ψ ( μ o + ν o 2 ) Υ ( μ o + ν o 2 ) Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς [ I μ o + ς A B ln ψ ( ν o ) + I ν o ς A B ln ψ ( μ o ) I μ o + ς A B ln Υ ( ν o ) I ν o ς A B ln Υ ( μ o ) ] + ς ( ν o μ o ) ς ( μ o ν o ( z μ o ) ς ln Υ ( | μ o + ν o 2 z | ) d z μ o ν o ( z μ o ) ς ln ψ ( | μ o + ν o 2 z | ) d z ) ln ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς .
Thus it follows as
μ o ν o ( ψ ( | μ o + ν o 2 z | ) ) ( z μ o ) ς 1 d z μ o ν o ( Υ ( | μ o + ν o 2 z | ) ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς ψ ( μ o + ν o 2 ) Υ ( μ o + ν o 2 )
I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) I * ς μ o A B ( Υ ) ( ν o ) · I ν o ς * A B ( Υ ) ( μ o ) Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς .
The preceding computation coincides with the first component of inequality (25). In order to derive the second component, we focus on the R.H.S of (27), leading to
I * ς μ o A B ( ψ ) ( ν o ) · I ν o ς * A B ( ψ ) ( μ o ) I * ς μ o A B ( Υ ) ( ν o ) · I ν o ς * A B ( Υ ) ( μ o ) Γ ( ς ) B ( ς ) 2 ( b a ) ς = e x p { Γ ( ς ) B ( ς ) 2 ( ν o μ o ) ς I μ o + ς A B ln ψ ( ν o ) + I ν o ς A B ln ψ ( μ o ) I μ o + ς A B ln Υ ( ν o ) I ν o ς A B ln Υ ( μ o ) } = e x p { ( 1 ς ) Γ ( ς ) ln ( ψ ( μ o ) ψ ( ν o ) ) 2 ( ν o μ o ) ς ( 1 ς ) Γ ( ς ) ln ( Υ ( μ o ) Υ ( ν o ) ) 2 ( ν o μ o ) ς + ς 2 [ 0 1 ς 1 ln ψ ( μ o + ( 1 ) ν o ) d + 0 1 ς 1 ln ψ ( ν o + ( 1 ) μ o ) d 0 1 ς 1 ln Υ ( μ o + ( 1 ) ν o ) d 0 1 ς 1 ln Υ ( ν o + ( 1 ) μ o ) d ] } e x p { ς ( ν o μ o ) ς + 1 [ μ o ν o ln Υ ( | ν o z | ) ( z μ o ) ς Υ ( | z μ o | ) ( ν o z ) ( z μ o ) ς 1 d z μ o ν o ln ψ ( | ν o z | ) ( z μ o ) ς ψ ( | z μ o | ) ( ν o z ) ( z μ o ) ς 1 d z ] + 1 2 ln ψ ( μ o ) , ψ ( ν o ) ln Υ ( μ o ) , Υ ( ν o ) + ( 1 ς ) Γ ( ς ) ln ( ψ ( μ o ) ψ ( ν o ) ) 2 ( ν o μ o ) ς ( 1 ς ) Γ ( ς ) ln ( Υ ( μ o ) Υ ( ν o ) ) 2 ( ν o μ o ) ς } .
Thus it follows as
= G ( ψ ( μ o ) , ψ ( ν o ) ) G ( Υ ( μ o ) , Υ ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 × μ o ν o ( Υ ( | ν o z | ) ) ( z μ o ) ς · ( Υ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1
The combination of (27) and (28) yields the required result. □
Remark 4.
If ς = 1 is picked in Theorem 5, we get
ψ ( μ o + ν o 2 ) Υ ( μ o + ν o 2 ) μ o ν o ψ ( | μ o + ν o 2 z | ) d z μ o ν o Υ ( | μ o + ν o 2 z | ) d z 1 ν o μ o μ o ν o ( ψ ( z ) ) d z μ o ν o ( Υ ( z ) ) d z 1 ν o μ o G ( ψ ( μ o ) , ψ ( ν o ) ) G ( Υ ( μ o ) , Υ ( ν o ) ) μ o ν o ( Υ ( | ν o z | ) ) ( z μ o ) · ( Υ ( | z μ o | ) ) ( ν o z ) d z μ o ν o ( ψ ( | ν o z | ) ) ( z μ o ) · ( ψ ( | z μ o | ) ) ( ν o z ) d z 1 ( ν o μ o ) 2 .
Which is a new inequality and has not obtained before.
Example 3.
A graphical representation given by Figure 3 confirming the authenticity of Theorem 5 is given below, for ψ ( z ) = e x p { z 3 } and Υ ( z ) = e x p { z 2 } .
Theorem 6.
Let ψ be a positive and multiplicatively superquadratic function on interval [ μ o , ν o ] ; then
ψ ( μ o ) ψ ( ν o ) 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς ψ μ o + ν o 2 μ o + ν o 2 ν o ( ψ ( | z μ o + ν o 2 | ) ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς I * ς ( ψ ) ( ν o ) · I μ o + ν o 2 ς * A B ( ψ ) ( μ o ) μ o + ν o 2 A B 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς G ( ( ψ ( μ o ) , ψ ( ν o ) ) ) 2 ς ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o + ν o 2 ν o ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς · 2 ς ( ν o μ o ) ς + 1 .
Proof. 
Since ψ be a positive and multiplicative superquadratic function. We have
ln ψ μ o + ν o 2 = ln ψ 1 2 2 μ o + 2 2 ν o + 2 ν o + 2 2 μ o 1 2 ln ψ ( 2 μ o + 2 2 ν o ) + 1 2 ln ψ ( 2 b + 2 2 μ o ) ln ψ ( 1 2 | ( 1 ) ( μ o ν o ) | ) .
Multiplying (31) through by ς 1 and integrating over [ 0 , 1 ] yields
0 1 ς 1 ln ψ ( μ o + ν o 2 ) d 1 2 0 1 ς 1 ln ψ ( 2 μ o + 2 2 ν o ) d + 1 2 0 1 ς 1 ln ψ ( 2 ν o + 2 2 μ o ) d 0 1 ς 1 ln ψ ( 1 2 | ( 1 ) ( μ o ν o ) | ) d
By changing of variables and after simple calculations, we get
ln ψ ( μ o + ν o 2 ) Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) 2 ς 1 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) ς · 2 ς ( ν o μ o ) ς μ o + ν o 2 ν o ( ν o z ) ς 1 ln ψ ( | z μ o + ν o 2 | ) d z .
Thus it follows as
e x p { ln ψ ( μ o + ν o 2 ) } e x p { Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) ς · 2 ς ( ν o μ o ) ς μ o + ν o 2 ν o ( ν o z ) ς 1 ln ψ ( | z μ o + ν o 2 | ) d z } .
Thus we have
ψ ( μ o ) ψ ( ν o ) 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς ψ μ o + ν o 2 μ o + ν o 2 ν o ( ψ ( | z μ o + ν o 2 | ) ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς I * ς μ o + ν o 2 A B ( ψ ) ( ν o ) · I μ o + ν o 2 ς * A B ( ψ ) ( ν o ) 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς
This completes the proof of first part of (30).
Proceeding with the proof of the second inequality, we now turn our attention to the right-hand side of (32), and obtain
I * ς μ o + ν o 2 A B ( ψ ) ( ν o ) · I μ o + ν o 2 ς * A B ( ψ ) ( ν o ) 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς = e x p Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) = e x p { 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) ς 2 [ 0 1 ς 1 ln ψ ( 2 μ o + 2 2 ν o ) d + 0 1 ς 1 ln ψ ( 2 ν o + 2 2 μ o ) d ] } e x p { 2 ς ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) 1 2 ln ( ψ ( μ o ) ψ ( ν o ) ) ς · 2 ς ( ν o μ o ) ς + 1 ( μ o + ν o 2 ν o ln ( ψ ( | z μ o | ) ) ( ν o z ) ς ( ψ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z ) } = ( G ( ψ ( μ o ) , ψ ( ν o ) ) ) 2 ς ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o + ν o 2 ν o ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 .
From (32) and (33) we have the desired result in (30). The proof is completed. □
Remark 5.
If ς = 1 is picked in Theorem 6, we get
ψ μ o + ν o 2 μ o + ν o 2 ν o ψ ( | z μ o + ν o 2 | ) d z 2 ν o μ o μ o + ν o 2 ν o ( ψ ( z ) ) d z 2 ν o μ o G ( ψ ( μ o ) , ψ ( ν o ) ) μ o + ν o 2 ν o ( ψ ( | z μ o | ) ) ( ν o z ) ( ψ ( | ν o z | ) ) ( z μ o ) d z 2 ( ν o μ o ) 2
which is a new inequality and has not been obtained before.
Example 4.
A graphical representation given Figure 4 supporting the conclusions of Theorem 6 is presented below, for ψ ( z ) = e x p { z 3 } .
Theorem 7.
If ψ and Υ are positive and multiplicatively superquadratic functions on the interval [ μ o , ν o ] ; then
ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς ψ μ o + ν o 2 Υ μ o + ν o 2 × μ o + ν o 2 ν o ( ψ Υ ( | z μ o + ν o 2 | ) ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς I * ς μ o + ν o 2 A B ( ψ ) ( ν o ) · I μ o + ν o 2 ς * A B ( ψ ) ( ν o ) · I * ς μ o + ν o 2 A B ( Y ) ( ν o ) · I μ o + ν o 2 ς * A B ( Y ) ( ν o ) 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς ( G ( ψ ( μ o ) , ψ ( ν o ) ) G ( Υ ( μ o ) , Υ ( ν o ) ) ) 2 ς ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 × [ μ o + ν o 2 ν o ( ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 × ( Υ ( | ν o z | ) ) ( z μ o ) ς · ( Υ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 ) d z ] ς · 2 ς ( ν o μ o ) ς + 1 .
Proof. 
Since ψ and Υ are positive and multiplicative superquadratic functions, We have
ln ψ μ o + ν o 2 Υ μ o + ν o 2 = ln ψ μ o + ν o 2 + ln Υ μ o + ν o 2 = ln ψ 1 2 2 μ o + 2 2 ν o + 2 ν o + 2 2 μ o + ln Υ ( 1 2 ( 2 μ o + 2 2 ν o + 2 ν o + 2 2 μ o ) ) 1 2 ln ψ ( 2 μ o + 2 2 ν o ) + 1 2 ln ψ ( 2 ν o + 2 2 μ o ) ln ψ ( 1 2 | ( 1 ) ( μ o ν o ) | ) + 1 2 ln Υ ( 2 μ o + 2 2 ν o ) + 1 2 ln Υ ( 2 ν o + 2 2 μ o ) ln Υ ( 1 2 | ( 1 ) ( μ o ν o ) | ) .
Multiplying (36) through by ς 1 and integrating over [ 0 , 1 ] yields
0 1 ς 1 ln ψ μ o + ν o 2 Υ μ o + ν o 2 d 1 2 0 1 ς 1 ln ψ ( 2 μ o + 2 2 ν o ) d + 1 2 0 1 ς 1 ln ψ ( 2 ν o + 2 2 μ o ) d 0 1 ς 1 ln ψ ( 1 2 | ( 1 ) ( μ o ν o ) | ) d + 1 2 0 1 ς 1 ln Υ ( 2 μ o + 2 2 ν o ) d + 1 2 0 1 ς 1 ln Υ ( 2 ν o + 2 2 μ o ) d 0 1 ς 1 ln Υ ( 1 2 | ( 1 ) ( μ o ν o ) | ) d .
By changing of variables and after simple calculations, we get
ln ψ μ o + ν o 2 Υ μ o + ν o 2 Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς [ I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) + I ( μ o + ν o 2 ) + ς A B ln Υ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln Υ ( μ o ) ] 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) + l n Υ ( μ o ) Υ ( ν o ) ς · 2 ς ( ν o μ o ) ς [ μ o + ν o 2 ν o ( ν o z ) ς 1 ln ψ ( | z μ o + ν o 2 | ) d z + μ o + ν o 2 ν o ( ν o z ) ς 1 ln Υ ( | z μ o + ν o 2 | ) d z ] .
Thus it follows as
e x p ln ψ μ o + ν o 2 Υ μ o + ν o 2 e x p { Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς [ I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) + I ( μ o + ν o 2 ) + ς A B ln Υ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln Υ ( μ o ) ] 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ς · 2 ς ( ν o μ o ) ς [ μ o + ν o 2 ν o ( ν o z ) ς 1 ln ψ ( | z μ o + ν o 2 | ) d z + μ o + ν o 2 ν o ( ν o z ) ς 1 ln Υ ( | z μ o + ν o 2 | ) d z ] } .
Hence
ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς ψ μ o + ν o 2 Υ μ o + ν o 2 × μ o + ν o 2 ν o ( ψ Υ ( | z μ o + ν o 2 | ) ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς I * ς μ o + ν o 2 A B ( ψ ) ( ν o ) · I μ o + ν o 2 ς * A B ( ψ ) ( μ o ) · I * ς μ o + ν o 2 A B ( Y ) ( ν o ) · I μ o + ν o 2 ς * A B ( Y ) ( μ o ) 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς .
This completes the proof of first part of (35).
The second part is established by analyzing the right-hand side of (37), leading to
I * ς μ o + ν o 2 A B ( ψ ) ( ν o ) · I μ o + ν o 2 ς * A B ( ψ ) ( μ o ) · I * ς μ o + ν o 2 A B ( Y ) ( ν o ) · I μ o + ν o 2 ς * A B ( Y ) ( μ o ) 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς = e x p { Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς [ I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) I ( μ o + ν o 2 ) + ς A B ln Υ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln Υ ( μ o ) ] } = e x p { 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ς 2 [ 0 1 ς 1 ln ψ ( 2 μ o + 2 2 ν o ) d + 0 1 ς 1 ln ψ ( 2 ν o + 2 2 μ o ) d + 0 1 ς 1 ln Υ ( 2 μ o + 2 2 ν o ) d + 0 1 ς 1 ln Υ ( 2 ν o + 2 2 μ o ) d ] } e x p { 2 ς ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) ς · 2 ς ( ν o μ o ) ς + 1 [ μ o + ν o 2 ν o ln ( ψ ( | z μ o | ) ) ( ν o z ) ς · ( ψ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z
+ μ o + ν o 2 ν o ln ( Υ ( | z μ o | ) ) ( ν o z ) ς · ( Υ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z ] + ln ( ψ ( μ o ) ψ ( ν o ) ) 2 + ln ( Υ ( μ o ) Υ ( ν o ) ) 2 } .
It follows as
= ( G ( ψ ( μ o ) , ψ ( ν o ) ) G ( Υ ( μ o ) , Υ ( ν o ) ) ) 2 ς ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 × [ μ o + ν o 2 ν o ( ( ψ ( | ν o z | ) ) ( z μ o ) ς · ( ψ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 × ( Υ ( | ν o z | ) ) ( z μ o ) ς · ( Υ ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 ) d z ] ς · 2 ς ( ν o μ o ) ς + 1 .
From (37) and (38), we get the desired result of (35). The proof is completed. □
Remark 6.
If ς = 1 is picked in Theorem 7, we attain
ψ μ o + ν o 2 μ o + ν o 2 ν o ψ Υ ( | z μ o + ν o 2 | ) d z 2 ν o μ o μ o + ν o 2 ν o ( ψ ( z ) ) d z μ o + ν o 2 ν o ( Υ ( z ) ) d z 1 ν o μ o G ( ψ ( μ o ) , ψ ( ν o ) ) μ o + ν o 2 ν o ( ψ ( | z μ o | ) ) ( ν o z ) ( ψ ( | ν o z | ) ) ( z μ o ) d z 2 ( ν o μ o ) 2 × G ( Υ ( μ o ) , Υ ( ν o ) ) μ o + ν o 2 ν o ( Υ ( | z μ o | ) ) ( ν o z ) ( Υ ( | ν o z | ) ) ( z μ o ) d z 2 ( ν o μ o ) 2 .
Which is a new inequality and has not obtained before.
Example 5.
The graph given by Figure 5 demonstrates the viability of Theorem 7 for ψ ( z ) = e x p { z 3 } and Υ ( z ) = e x p { z 2 } .
Theorem 8.
Suppose ψ is a positive function satisfying the multiplicative superquadratic condition on [ μ o , ν o ] and Υ is a positive multiplicatively subquadratic function. Then
ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) 2 ς 1 ( 1 ς ) Γ ( ς ) ( o o ) ς ψ μ o + ν o 2 Υ μ o + ν o 2 μ o + ν o 2 ν o ( ψ ( | z μ o + ν o 2 | ) ) ( ν o z ) ς 1 d z μ o + ν o 2 ν o ( Υ ( | z μ o + ν o 2 | ) ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς I * ς ( ψ ) ( ν o ) · I μ o + ν o 2 ς * A B ( ψ ) ( μ o ) μ o + ν o 2 A B I * ς ( Υ ) ( ν o ) · I μ o + ν o 2 ς * A B ( Υ ) ( μ o ) μ o + ν o 2 A B 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς ψ ( μ o ) , ψ ( ν o ) Υ ( μ o ) , Υ ( ν o ) 1 2 ( 2 ς ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 ) × μ o + ν o 2 ν o ( Υ ( | z μ o | ) ) ( ν o z ) ς · ( Υ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z μ o + ν o 2 ν o ( ψ ( | z μ o | ) ) ( ν o z ) ς · ( ψ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς + 1 .
Proof. 
Since ψ and Υ be multiplicative superquadratic and multiplicative subquadratic functions respectively, then we have
ln ψ μ o + ν o 2 Υ μ o + ν o 2 = ln ψ μ o + ν o 2 ln Υ μ o + ν o 2
= ln ψ 1 2 2 μ o + 2 2 ν o + 2 ν o + 2 2 μ o ln Υ 1 2 2 μ o + 2 2 ν o + 2 ν o + 2 2 μ o 1 2 ln ψ 2 μ o + 2 2 ν o + 1 2 ln ψ 2 ν o + 2 2 μ o ln ψ 1 2 | ( 1 ) ( μ o ν o ) | 1 2 ln Υ 2 μ o + 2 2 ν o 1 2 ln Υ 2 ν o + 2 2 μ o + ln Υ 1 2 | ( 1 ) ( μ o ν o ) | .
Multiply with ς 1 on both sides of (41) and integrating the resulting inequality with respect to ℓ over [ 0 , 1 ] , we get
0 1 ς 1 ln ψ μ o + ν o 2 Υ μ o + ν o 2 d 1 2 0 1 ς 1 ln ψ ( 2 μ o + 2 2 ν o ) d + 1 2 0 1 ς 1 ln ψ ( 2 ν o + 2 2 μ o ) d 0 1 ς 1 ln ψ ( 1 2 | ( 1 ) ( μ o ν o ) | ) d 1 2 0 1 ς 1 ln Υ ( 2 μ o + 2 2 ν o ) d 1 2 0 1 ς 1 ln Υ ( 2 ν o + 2 2 μ o ) d + 0 1 ς 1 ln Υ ( 1 2 | ( 1 ) ( μ o ν o ) | ) d .
After carrying out simple calculations and introducing a change of variables, we obtain
ln ψ μ o + ν o 2 Υ μ o + ν o 2 Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς [ I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) I ( μ o + ν o 2 ) + ς A B ln Υ ( ν o ) I ( μ o + ν o 2 ) ς A B ln Υ ( μ o ) ] + 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) l n Υ ( μ o ) Υ ( ν o ) ς · 2 ς ( ν o μ o ) ς [ μ o + ν o 2 ν o ( ν o z ) ς 1 ln ψ ( | z μ o + ν o 2 | ) d z μ o + ν o 2 ν o ( ν o z ) ς 1 ln Υ ( | z μ o + ν o 2 | ) d z ] .
That is
ln ψ μ o + ν o 2 Υ μ o + ν o 2 Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς [ I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) I ( μ o + ν o 2 ) + ς A B ln Υ ( ν o ) I ( μ o + ν o 2 ) ς A B ln Υ ( μ o ) ] + ς · 2 ς ( ν o μ o ) ς [ μ o + ν o 2 ν o ( ν o z ) ς 1 ln Υ ( | z μ o + ν o 2 | ) d z μ o + ν o 2 ν o ( ν o z ) ς 1 ln ψ ( | z μ o + ν o 2 | ) d z ] + 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) .
Thus it follows as
e x p ln ψ μ o + ν o 2 Υ μ o + ν o 2 e x p { Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς [ I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) I ( μ o + ν o 2 ) + ς A B ln Υ ( ν o ) I ( μ o + ν o 2 ) ς A B ln Υ ( μ o ) ] + ς · 2 ς ( ν o μ o ) ς [ μ o + ν o 2 ν o ( ν o z ) ς 1 ln Υ ( | z μ o + ν o 2 | ) d z μ o + ν o 2 ν o ( ν o z ) ς 1 ln ψ ( | z μ o + ν o 2 | ) d z ] + 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) } .
Thus we have
ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς ψ μ o + ν o 2 Υ μ o + ν o 2 μ o + ν o 2 ν o ( ψ ( | z μ o + ν o 2 | ) ) ( ν o z ) ς 1 d z μ o + ν o 2 ν o ( Υ ( | z μ o + ν o 2 | ) ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς I * ς ( ψ ) ( ν o ) · * I μ o + ν o 2 ς ( ψ ) ( μ o ) μ o + ν o 2 I * ς ( Υ ) ( ν o ) · * I μ o + ν o 2 ς ( Υ ) ( μ o ) μ o + ν o 2 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς .
This is the first part of inequality (40).
Proceeding to the second part of inequality (40), we now consider the R.H.S of (42).
I * ς ( ψ ) ( ν o ) · * I μ o + ν o 2 ς ( ψ ) ( μ o ) μ o + ν o 2 I * ς ( Υ ) ( ν o ) · * I μ o + ν o 2 ς ( Υ ) ( μ o ) μ o + ν o 2 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς = e x p { Γ ( ς ) B ( ς ) · 2 ς 1 ( ν o μ o ) ς [ I ( μ o + ν o 2 ) + ς A B ln ψ ( ν o ) + I ( μ o + ν o 2 ) ς A B ln ψ ( μ o ) I ( μ o + ν o 2 ) + ς A B ln Υ ( ν o ) I ( μ o + ν o 2 ) ς A B ln Υ ( μ o ) ] } = e x p { 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) + ς 2 [ 0 1 ς 1 ln ψ ( 2 μ o + 2 2 ν o ) d + 0 1 ς 1 ln ψ ( 2 ν o + 2 2 μ o ) d 0 1 ς 1 ln Υ ( 2 μ o + 2 2 ν o ) d 0 1 ς 1 ln Υ ( 2 ν o + 2 2 μ o ) d ] } e x p { 2 ς 1 ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς l n ψ ( μ o ) ψ ( ν o ) Υ ( μ o ) Υ ( ν o ) + ln ( ψ ( μ o ) ψ ( ν o ) ) 2 ln ( Υ ( μ o ) Υ ( ν o ) ) 2 + 2 ς + 1 ( ν o μ o ) ς + 1 [ μ o + ν o 2 ν o ln ( Υ ( | z μ o | ) ) ( ν o z ) ς · ( Υ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z μ o + ν o 2 ν o ln ( ψ ( | z μ o | ) ) ( ν o z ) ς · ( ψ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z ] } = G ψ ( μ o ) , ψ ( ν o ) G Υ ( μ o ) , Υ ( ν o ) 2 ς ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 × μ o + ν o 2 ν o ( Υ ( | z μ o | ) ) ( ν o z ) ς · ( Υ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z μ o + ν o 2 ν o ( ψ ( | z μ o | ) ) ( ν o z ) ς · ( ψ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς + 1 .
From (42) and (43), we have the desired result of inequality in (40). The proof is completed. □
Remark 7.
If we set ς = 1 in Theorem 8, we attain
ψ μ o + ν o 2 Υ μ o + ν o 2 μ o + ν o 2 ν o ψ ( | z μ o + ν o 2 | ) d z μ o + ν o 2 ν o Υ ( | z μ o + ν o 2 | ) d z 2 ν o μ o μ o + ν o 2 ν o ( f ( z ) ) d z μ o + ν o 2 ν o ( g ( z ) ) d z 1 ν o μ o G ψ ( μ o ) , ψ ( ν o ) G Υ ( μ o ) , Υ ( ν o ) μ o + ν o 2 ν o ( Υ ( | z μ o | ) ) ( ν o z ) ( Υ ( | ν o z | ) ) ( z μ o ) d z μ o + ν o 2 ν o ( ψ ( | z μ o | ) ) ( ν o z ) ( ψ ( | ν o z | ) ) ( z μ o ) d z 2 ( ν o μ o ) 2 .
Which is a new inequality and has not obtained before.
Example 6.
The graph given by Figure 6 demonstrates the viability of Theorem 7 for ψ ( z ) = e x p { z 4 } and Υ ( z ) = e x p { z 2 } .
Corollary 2.
Suppose that ψ is positive and satisfies the multiplicative superquadratic condition on [ μ o , ν o ] , and that Υ is a positive multiplicatively convex function. Then
I * ς ( ψ ) ( ν o ) · * I μ o + ν o 2 ς ( ψ ) ( μ o ) μ o + ν o 2 I * ς ( Υ ) ( ν o ) · * I μ o + ν o 2 ς ( Υ ) ( μ o ) μ o + ν o 2 2 ς 1 B ( ς ) Γ ( ς ) ( ν o μ o ) ς G ψ ( μ o ) , ψ ( ν o ) G Υ ( μ o ) , Υ ( ν o ) 2 ς ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 × μ o + ν o 2 ν o ( ψ ( | z μ o | ) ) ( ν o z ) ς · ( ψ ( | ν o z | ) ) ( z μ o ) ( ν o z ) ς 1 d z ς · 2 ς ( ν o μ o ) ς + 1 .
where G(.,.) is a geometric mean.

4. Applications

This section explains how the findings are applied to means, special functions, and moments. It also provides the implications of the findings.

4.1. Special Means

Let μ o ( μ o , o ) , W ( μ o , ν o , w 1 , w 2 ) and Υ ( μ o , ν o ) are the arithmetic, weighted arithmetic and geometric means defined by
μ o ( μ o , ν o ) = μ o + ν o 2 , W ( μ o , ν o , w 1 , w 2 ) = w 1 μ o + w 2 ν o w 1 + w 2 , Υ ( μ o , ν o ) = μ o ν o ,
respectively.
Proposition 5.
The inequality
e x p μ o p ( μ o , ν o ) + 2 1 + p μ o 1 + p ( μ o , ν o ) Γ ( 1 + ς ) Γ ( 1 + p ) Γ ( 1 + ς + p ) I * ς μ o A B ( ν o ) p · I ν o ς * A B ( μ o ) p B ( ς ) Γ ( ς ) 2 ( ν o μ o ) ς G ( μ o p , ν o p ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( | ν o z | ) p ( z μ o ) ς · ( | z μ o | ) p ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1 ,
holds for all [ μ o , ν o ] [ 1 , ] .
Proof. 
Theorem 3 can be utilised to demonstrate the outcome for a multiplicatively superquadratic function Ψ ( z ) = e x p { z p } for p 2 . □

4.2. Modified Bessel Functions

Taking into account the type-1 modified Bessel function J q [51].
J q ( z ) = n = 0 ( z 2 ) q + 2 n n ! Γ ( q + n + 1 ) , w h e r e z .
Suppose the mapping Ψ q : [ 0 , ) [ 0 , ) for every q 2 is defined by
Ψ q ( z ) = z q J q ( z ) .
Ψ q ( z ) = z q J q 1 ( z ) .
Ψ q ( z ) = z q 1 J q 1 ( z ) + z q J q 2 ( z ) .
Ψ q ( z ) = 3 z q 1 J q 2 ( z ) + z q J q 3 ( z ) .
Ψ q 4 ( z ) = 3 z q 2 J q 2 ( z ) + 6 z q 1 J q 3 ( z ) + z q J q 4 ( z ) .
Ψ q 5 ( z ) = 15 z q 2 J q 3 ( z ) + 10 z q 1 J q 4 ( z ) + z q J q 5 ( z ) .
As Ψ q 5 ( z ) > 0 , ∀ q 2 and z > 0 . It implies the convexity of Ψ q ( z ) on [ 0 , ) . It is evident that Ψ q ( 0 ) = 0 and Ψ q ( 0 ) = 0 .
So from Lemma (2.2) in [23], if Ψ q ( z ) is convex and Ψ q ( 0 ) = Ψ q ( 0 ) = 0 then Ψ q ( z ) is superquadratic. If log of a function is superquadratic then such a function is termed as a multiplicatively superquadratic. It is to be noted that Ψ q ( z ) may also be derived by employing log on e x p { Ψ q ( z ) } therefore the function e x p { Ψ q ( z ) } is a multiplicative superquadratic.
Proposition 6.
Let q 2 , μ o , ν o [ 1 , [ such that μ o < ν o then the inequality holds
Ψ q ( μ o ) Ψ q ( ν o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς μ o ν o Ψ q | μ o + ν o 2 z | ( z μ o ) ς 1 d z ς ( ν o μ o ) ς Ψ q μ o + ν o 2 I * ς μ o A B ( Ψ q ) ( ν o ) · I ν o ς * A B ( Ψ q ) ( μ o ) B ( ς ) Γ ( ς ) 2 ( ν o μ o ) ς ( Ψ q ( μ o ) Ψ q ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( Ψ q ( | ν o z | ) ) ( z μ o ) ς · ( Ψ q ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1
Proof. 
The result (46) is proved by considering the Theorem 3 for the function e x p { Ψ q ( z ) } , where
Ψ q ( z ) = z q 1 J q 1 ( z ) + z q J q 2 ( z ) .

4.3. Moment of Random Variables

The distribution of probabilities associated with a random variable is described by its distribution and density functions, but these alone often fail to distinguish similar distributions. Key statistical measures like mean, variance, skewness, and kurtosis, along with moments, provide deeper insights by capturing the distribution’s shape and spread. Moments also indicate the existence and bounds of distributions under certain conditions. Recent studies have applied mathematical inequalities to estimate moments, enhancing distribution analysis and comparison [52,53].
In probability theory, let M k ( z ) denote the kth moment of a random variable χ , where z serves as an arbitrary location parameter relative to χ . The expression for the kth moment M k ( z ) , for any k 1 , is given by:
M k ( z ) = μ o ν o Ψ ( z ) k ( ) d , k = 1 , 2 , .
In (47), the probability function is assumed to be Ψ which in this case is the probability density function (pdf). Specifically, the assumption is that the pdf is uniform on the range [ μ o , ν o ] . This means that any value in this range has an equal probability of being the random variable. The uniform distribution is commonly used when no particular outcome within the specified range is preferred.
Considering (47), we arrive to the following assertion.
Proposition 7.
Suppose that Υ : I [ 0 , ) + is a p d f on I = [ μ o , ν o ] for χ, being a random variable then k 1 , we acquire
M k ( μ o ) M k ( ν o ) ( 1 ς ) Γ ( ς ) 2 ( ν o μ o ) ς μ o ν o M k | μ o + ν o 2 z | ( z μ o ) ς 1 d z ς ( ν o μ o ) ς M k μ o + ν o 2 I * ς μ o A B ( Ψ q ) ( ν o ) · I ν o ς * A B ( M k ) ( μ o ) B ( ς ) Γ ( ς ) 2 ( ν o μ o ) ς ( M k ( μ o ) M k ( ν o ) ) ( 1 ς ) Γ ( ς ) ( ν o μ o ) ς + 1 μ o ν o ( M k ( | ν o z | ) ) ( z μ o ) ς · ( M k ( | z μ o | ) ) ( ν o z ) ( z μ o ) ς 1 d z ς ( ν o μ o ) ς + 1
Proof. 
The inequality (48) can be developed by utilizing the Theorem 3 for the function M k ( z ) , which is clearly possessing multiplicatively subquadraticity, in which Υ is picked as p d f on the interval of real numbers I . □

4.4. Implications

The main objective of these results is to extend classical H-H type inequalities to the context of multiplicative fractional calculus through the A-B operators. By means of multiplicatively superquadratic and subquadratic functions, the inequalities provide sharper and more refined estimates than in the classical convexity-based results. This extends the analytical toolkit for studying nonlinear and nonlocal dynamics of mathematical models. The results also merge and generalize several familiar inequalities, and develop new fractional recurrence relations, thereby contributing to theory and applications.
The obtained results have a wide range of potential applications across both theoretical and applied domains. In optimization theory, the sharper bounds derived from multiplicatively superquadratic functions provide more accurate estimates that can be employed in designing efficient algorithms and improving stability analyses. In information theory, the refined inequalities offer tools for deriving stronger bounds in problems related to entropy measures and coding theory. The results also extend naturally to inequalities involving special means and special functions, yielding new insights and recurrence relations in the multiplicative fractional setting. Furthermore, by incorporating moments of random variables, the developed inequalities contribute to probability and statistics, enabling a more precise characterization of random phenomena. Beyond pure mathematics, the nonlinear and nonlocal structure captured through multiplicative A-B fractional operators makes these results relevant for modeling complex dynamical systems in physics, biology, engineering, and economics, where memory effects and multiplicative processes play a crucial role.

5. Conclusions

In this work, we established novel H-H type inequalities in the framework of multiplicative A-B fractional calculus, focusing particularly on multiplicatively superquadratic and subquadratic functions. These results significantly extend classical H-H inequalities by embedding nonlinear and nonlocal dynamics through the use of multiplicative A-B fractional operators. The enhanced sharpness of the bounds derived from the superquadratic nature of the functions leads to more precise estimates, with promising implications in optimization, information theory, and the analysis of complex dynamical systems.
The theoretical contributions have been validated through illustrative graphical plots and numerical computations, demonstrating both the accuracy and the practical relevance of the proposed inequalities. Moreover, we explored applications involving special means, special functions, and moments of random variables, resulting in the formulation of new fractional recurrence relations within the multiplicative setting.
Looking ahead, this study opens up several avenues for future exploration. One promising direction is the development of analogous inequalities using broader classes of multiplicative fractional integral operators. These include, among others, the multiplicative forms of conformable fractional integrals, ψ -Riemann–Liouville integrals, generalized Riemann–Liouville integrals, Katugampola fractional integrals, k- and (k-p)-fractional integrals, generalized proportional integrals, and Sarikaya fractional integrals. Extending the current framework to incorporate these operators may yield deeper insights and more versatile tools for modeling and analyzing nonlinear phenomena in fields such as physics, biology, and engineering.

Author Contributions

Conceptualization, G.J., S.I.B. and Y.S.; methodology, G.J. and S.I.B.; software, D.K.; validation, G.J., S.I.B., D.K. and Y.S.; formal analysis, G.J. and D.K.; investigation, G.J. and S.I.B.; writing—original draft preparation, D.K.; writing—review and editing, G.J., S.I.B. and D.K.; visualization, Y.S.; supervision, S.I.B.; project administration, Y.S.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research recieved no external funding.

Data Availability Statement

Not applicable.

Acknowledgments

This research was supported by Dong-A University, Busan 49315, Republic of Korea.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Visual illustration of Theorem 3 for μ o [ 0 , 0.4 ] , ν o [ 1 , 1.25 ] , k = 3 and ς = 0.5 .
Figure 1. Visual illustration of Theorem 3 for μ o [ 0 , 0.4 ] , ν o [ 1 , 1.25 ] , k = 3 and ς = 0.5 .
Fractalfract 09 00617 g001
Figure 2. Graphical illustration for μ o [ 0 , 0.3 ] , ν o [ 1.0 , 1.30 ] and ς = 0.5 .
Figure 2. Graphical illustration for μ o [ 0 , 0.3 ] , ν o [ 1.0 , 1.30 ] and ς = 0.5 .
Fractalfract 09 00617 g002
Figure 3. Visual illustration for μ o [ 0 , 0.4 ] , ν o [ 1.0 , 1.3 ] and ς = 0.5 .
Figure 3. Visual illustration for μ o [ 0 , 0.4 ] , ν o [ 1.0 , 1.3 ] and ς = 0.5 .
Fractalfract 09 00617 g003
Figure 4. Graphical Interpretation of Theorem 6 for μ o [ 0 , 0.35 ] , ν o [ 0.7 , 1.0 ] and ς = 0.5 .
Figure 4. Graphical Interpretation of Theorem 6 for μ o [ 0 , 0.35 ] , ν o [ 0.7 , 1.0 ] and ς = 0.5 .
Fractalfract 09 00617 g004
Figure 5. Visual illustration for μ o [ 0 , 0.3 ] , ν o [ 0.6 , 0.9 ] and ς = 0.5 .
Figure 5. Visual illustration for μ o [ 0 , 0.3 ] , ν o [ 0.6 , 0.9 ] and ς = 0.5 .
Fractalfract 09 00617 g005
Figure 6. Visual illustration for μ o [ 0 , 0.35 ] , ν o [ 0.7 , 1 ] and ς = 0.5 .
Figure 6. Visual illustration for μ o [ 0 , 0.35 ] , ν o [ 0.7 , 1 ] and ς = 0.5 .
Fractalfract 09 00617 g006
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MDPI and ACS Style

Jallani, G.; Butt, S.I.; Khan, D.; Seol, Y. Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures. Fractal Fract. 2025, 9, 617. https://doi.org/10.3390/fractalfract9090617

AMA Style

Jallani G, Butt SI, Khan D, Seol Y. Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures. Fractal and Fractional. 2025; 9(9):617. https://doi.org/10.3390/fractalfract9090617

Chicago/Turabian Style

Jallani, Ghulam, Saad Ihsan Butt, Dawood Khan, and Youngsoo Seol. 2025. "Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures" Fractal and Fractional 9, no. 9: 617. https://doi.org/10.3390/fractalfract9090617

APA Style

Jallani, G., Butt, S. I., Khan, D., & Seol, Y. (2025). Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures. Fractal and Fractional, 9(9), 617. https://doi.org/10.3390/fractalfract9090617

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