Inequalities of Ostrowski Type for Functions Whose Derivative Module Is Relatively Convex on Time Scales

Abstract

In this article, we discuss several novel generalized Ostrowski-type inequalities for functions whose derivative module is relatively convex in time scales calculus. Our core findings are proved by using the integration by parts technique, Hölder’s inequality, and the chain rule on time scales. These derived inequalities expand the existing literature, enriching specific integral inequalities within this domain.

1. Introduction

The Ostrowski inequality, established in classical literature since 1938 [1], provides an upper bound for the approximation of the integral average by the value of the function’s derivative at a point within the interval of integration. Mathematically, it is expressed as follows: suppose that $\Phi :I\subset [0,+\infty )\to \mathbb{R}$ is a differentiable function in $I°$ (the interior of the interval I), such that $\Phi \prime \in \pounds [{\theta }_0,\theta ]$, where ${\theta }_0,\theta \in I$, and ${\theta }_0<\theta .$ If $|\Phi \prime \left(\varkappa \right)|\le M$, then

$$\left|\Phi \left(\varkappa \right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)du\right|\le \frac{M}{\theta -{\theta }_0}\left[\frac{{(\varkappa -{\theta }_0)}^2+{(\theta -\varkappa )}^2}{2}\right].$$

(1)

In [2], Alomari et al. evidenced that if $\Phi :I\subset {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a differentiable function in $I°$ such that $\Phi \prime \in \pounds [{\theta }_0,\theta ]$ where ${\theta }_0,\theta \in I$ with ${\theta }_0<\theta .$ Additionally, they assumed that if $|\Phi \prime |$ is s-convex in the second sense in $[{\theta }_0,\theta ]$ for some fixed $s\in (0,1],$ $|\Phi \prime \left(\varkappa \right)|\le M$, and $\varkappa \in [{\theta }_0,\theta ],$ then

$$\left|\Phi \left(\varkappa \right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)du\right|\le \frac{M}{\theta -{\theta }_0}\left[\frac{{(\varkappa -{\theta }_0)}^2+{(\theta -\varkappa )}^2}{s+1}\right],$$

(2)

is satisfied for each $\varkappa \in [{\theta }_0,\theta ]$. The function $\Phi$ as s-convex in the second sense or $s_2$-convex if

$$\Phi (\lambda \tau +\mu \upsilon )\le {\lambda }^s\Phi \left(\tau \right)+{\mu }^s\Phi \left(\upsilon \right),$$

$\forall \tau ,\upsilon \in [0,+\infty ),$ $\lambda ,\mu \in (0,1)$, and $\lambda +\mu =1.$ Moreover, we denominate the function $\Phi$ as s-convex in the first sense or $s_1$-convex if

$$\Phi (\lambda \tau +\mu \upsilon )\le {\lambda }^s\Phi \left(\tau \right)+{\mu }^s\Phi \left(\upsilon \right),$$

$\forall \tau ,\upsilon \in [0,+\infty ),$ $\lambda ,\mu \in (0,1)$, and ${\lambda }^s+{\mu }^s=1.$

In [3], Alomari et al. evidenced that if $\Phi :I\subset {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a differentiable function on $I°$ such that ${\Phi }^{\prime }\in \pounds [{\theta }_0,\theta ],$ where ${\theta }_0,\theta \in I,$ $\theta >{\theta }_0.$ Moreover, if $|{\Phi }^{\prime }{|}^{\eta }$ is relatively s-convex in the second sense with respect to a function $\varpi :\mathbb{R}\to \mathbb{R}$ for some fixed $s\in (0,1],$ $\gamma ,\eta >1,$ $\frac{1}{\gamma }+\frac{1}{\eta }=1$, and $|{\Phi }^{\prime }\left(\varkappa \right)|\le M$ for $\varkappa \in [{\theta }_0,\theta ],$ then

$$\left|\Phi \left(\varkappa \right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)du\right|\le \frac{M}{{(1+\gamma )}^{\frac{1}{\gamma }}}{\left(\frac{2}{s+1}\right)}^{\frac{1}{\eta }}\left[\frac{{(\varkappa -{\theta }_0)}^2+{(\theta -\varkappa )}^2}{\theta -{\theta }_0}\right],$$

(3)

is satisfied $\forall \varkappa \in [{\theta }_0,\theta ].$ A function $\Phi :K_{\varpi }\to [0,+\infty )$ is said to be relatively s-convex in the second sense with respect to a function $\varpi :H\to H,$ where $s\in (0,1],$ if

$$\Phi (\tau \varpi \left(\varkappa \right)+(1-\tau )y)\le {\tau }^s\Phi \left(\varpi \left(\varkappa \right)\right)+{(1-\tau )}^s\Phi \left(y\right),$$

is satisfied $\forall \varkappa ,y\in [0,+\infty ),$ $\varpi \left(\varkappa \right), y\in K_{\varpi }$, and $\tau \in [0,1],$ where $K_{\varpi }$ is a subset of H and is said to be relatively convex with respect to a function $\varpi :H\to H$ if

$$\tau \varpi \left(\nu \right)+(1-\tau )u\in K_{\varpi },$$

$\forall u,\nu \in H,$ $u, \varpi \left(\nu \right)\in K_{\varpi }$, and $\tau \in [0,1].$

In [4], Cortez et al. evidenced that if $\Phi :I\subset {\mathbb{R}}_{+}\to \mathbb{R}$ is a differentiable function on $I°$ such that ${\Phi }^{\prime }\in \pounds [{\theta }_0,\theta ],$ where ${\theta }_0,\theta \in I$ with $\theta >{\theta }_0.$ Moreover, if $|{\Phi }^{\prime }|$ is relatively convex with respect to a function $\varpi :\mathbb{R}\to \mathbb{R}$ in $[{\theta }_0,\theta ]$ and $|{\Phi }^{\prime }\left(\varkappa \right)|\le M,$ then

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)du\right|\le \frac{M}{\theta -{\theta }_0}\left[\frac{{(\varpi \left(\varkappa \right)-{\theta }_0)}^2+{(\varpi \left(\varkappa \right)-\theta )}^2}{2}\right],$$

(4)

is satisfied $\forall \varkappa \in {\varpi }^{-1}\left(I\right).$

On the other hand, in this article, we establish some new generalizations of Ostrowski-type inequalities on a general domain called a time scale $\mathbb{T}$, which enable us to avoid establishing the dynamic inequalities twice (once in continuous calculus and another in discrete calculus). The dynamic inequalities on time scales have been developed by many authors. For a comprehensive overview of the dynamic inequalities on time scales, see the papers [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19].

Our objective in this article is to establish the time-scale version of the inequality (4) and explore some implications for relatively convex functions in both the first and second senses.

The structure of the article is as follows: In Section 2, we present a few preliminaries on time scales and prove the essential lemmas needed in Section 3 to establish the main results.

2. Preliminaries

A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the real numbers $\mathbb{R}$. The forward jump operator is defined as $\sigma \left(\tau \right):=inf\{r\in \mathbb{T}:r>\tau \}.$ The set of all such rd-continuo functions is denoted by ${\mathrm{C}}_{rd}(\mathbb{T},\mathbb{R})$ and for any function $\Phi :\mathbb{T}\to \mathbb{R}$, the notation ${\Phi }^{\sigma }\left(\tau \right)$ denotes $\Phi \left(\sigma \right(\tau \left)\right).$

The derivatives of $\Phi \varpi$ and $\Phi /\varpi$ (where $\varpi {\varpi }^{\sigma }\ne 0$) of two differentiable functions $\Phi$ and $\varpi$ are given by

$${(\Phi \varpi )}^{\Delta }={\Phi }^{\Delta }\varpi +{\Phi }^{\sigma }{\varpi }^{\Delta }=\Phi {\varpi }^{\Delta }+{\Phi }^{\Delta }{\varpi }^{\sigma },{\left(\frac{\Phi }{\varpi }\right)}^{\Delta }=\frac{{\Phi }^{\Delta }\varpi -\Phi {\varpi }^{\Delta }}{\varpi {\varpi }^{\sigma }}.$$

(5)

In this article, we refer to the (delta) integral, which is defined as follows: if $G^{\Delta }\left(\tau \right)=\varpi \left(\tau \right)$, then ${\int }_{{\tau }_0}^{\tau }\varpi \left(s\right)\Delta s:=G\left(\tau \right)-G\left({\tau }_0\right).$ It can be shown (see [20]) that if $\varpi \in C_{rd}(\mathbb{T},\mathbb{R}),$ then the Cauchy integral $G\left(\tau \right):={\int }_{{\tau }_0}^{\tau }\varpi \left(s\right)\Delta s$ exists, where ${\tau }_0\in \mathbb{T}$. The integration by parts formula on $\mathbb{T}$ is given by

$${\int }_{{\upsilon }_0}^{\upsilon }\lambda \left(\tau \right){\phi }^{\Delta }\left(\tau \right)\Delta \tau ={\left[\lambda \left(\tau \right)\phi \left(\tau \right)\right]}_{{\upsilon }_0}^{\upsilon }-{\int }_{{\upsilon }_0}^{\upsilon }{\lambda }^{\Delta }\left(\tau \right){\phi }^{\sigma }\left(\tau \right)\Delta \tau .$$

(6)

The time-scale chain rule (see ([20], Theorem 1.87)) is given by

$${(\varpi \circ \delta )}^{\Delta }\left(\tau \right)=\varpi \prime \left(\delta \left(\zeta \right)\right){\delta }^{\Delta }\left(\tau \right),\mathrm{where}\zeta \in \left[\tau ,\sigma \left(\tau \right)\right],$$

(7)

where it is assumed that $\varpi :\mathbb{R}\to \mathbb{R}$ is continuously differentiable and $\delta :\mathbb{T}\to \mathbb{R}$ is delta differentiable.

The Hölder inequality (see ([20], Theorem 6.13)) on $\mathbb{T}$ is given by

$${\int }_{{\zeta }_0}^{\zeta }|\Phi \left(\tau \right)\varpi \left(\tau \right)|\Delta \tau \le {\left[{\int }_{{\zeta }_0}^{\zeta }{|\Phi \left(\tau \right)|}^{\gamma }\Delta \tau \right]}^{\frac{1}{\gamma }}{\left[{\int }_{{\zeta }_0}^{\zeta }{\left|\varpi \left(\tau \right)\right|}^{\nu }\Delta \tau \right]}^{\frac{1}{\nu }},$$

(8)

where ${\zeta }_0,\zeta \in \mathbb{T},$ $\Phi ,\varpi \in {\mathrm{C}}_{rd}(\mathbb{I},\mathbb{R}),$ $\gamma >1$, and $1/\gamma +1/\nu =1.$

Definition 1

([21]). Suppose I is an interval in $\mathbb{R}$. A function $\Phi :I\to \mathbb{R}$ is said to be convex, if $\forall \alpha ,\beta \in I$ and $\tau \in (0,1),$ the inequality

$$\Phi (\tau \alpha +(1-\tau \left)\beta \right)\le \tau \Phi \left(\alpha \right)+(1-\tau )\Phi \left(\beta \right),$$

is satisfied. If this inequality holds in the opposite sense, then we say that Φ is concave.

Definition 2

([21]). Suppose $0<s\le 1.$ A function $\Phi :[0,+\infty )\to$ $\mathbb{R}$ is s-convex in the first sense or $s_1-$ convex if

$$\Phi (\lambda \varkappa +\mu y)\le {\lambda }^s\Phi \left(\varkappa \right)+{\mu }^s\Phi \left(y\right),$$

is satisfied $\forall \varkappa ,y\in [0,+\infty ),$ $\lambda ,\mu \in (0,1)$, and ${\lambda }^s+{\mu }^s=1.$ If this inequality holds in the opposite sense, then we say that Φ is $s-$concave in the first sense.

Definition 3

([21]). The function Φ is s-convex in the second sense or $s_2$-convex if

$$\Phi (\lambda \varkappa +\mu y)\le {\lambda }^s\Phi \left(\varkappa \right)+{\mu }^s\Phi \left(y\right),$$

is satisfied $\forall \varkappa ,y\in [0,+\infty ),$ $\lambda ,\mu \in (0,1)$, and $\lambda +\mu =1.$

Definition 4.

Suppose $K_{\varpi }$ is a subset of $H.$ $K_{\varpi }$ is said to be relatively convex with respect to a function $\varpi :H\to H$ if

$$t\varpi \left(\nu \right)+(1-t)u\in K_{\varpi },$$

$\forall u,\nu \in H,$$u,\varpi \left(\nu \right)\in K_{\varpi }$, and $t\in [0,1].$

Definition 5.

Suppose I is an interval in $\mathbb{R}$. A function $\Phi :K_{\varpi }\subseteq$ $\mathbb{R}$ $\to \mathbb{R}$ is said to be relatively convex with respect to a function $\varpi :\mathbb{R}\to \mathbb{R}$ if

$$\Phi \left(\tau \varpi \right(\alpha )+(1-\tau \left)\beta \right)\le \tau \Phi \left(\varpi \right(\alpha \left)\right)+(1-\tau )\Phi \left(\beta \right),$$

is satisfied $\forall \beta ,\varpi \left(\alpha \right)\in K_{\varpi },$ $\alpha ,\beta \in \mathbb{R}$, and $\tau \in [0,1].$ If this inequality holds in the opposite sense, then we say that f is relatively concave.

Definition 6.

A function $\Phi :K_{\varpi }\to [0,+\infty )$ is said to be relatively s-convex in the first sense with respect to a function $\varpi :H\to H,$ where $s\in (0,1],$ if

$$\Phi (\tau \varpi \left(\alpha \right)+(1-\tau )\beta )\le {\tau }^s\Phi \left(\varpi \left(\alpha \right)\right)+{(1-\tau )}^s\Phi \left(\beta \right),$$

is satisfied $\forall \alpha ,\beta \in [0,+\infty ),$ $\varpi \left(\alpha \right), \beta \in K_{\varpi }$, and ${\tau }^s+{(1-\tau )}^s=1$ for $\tau \in [0,1].$

Definition 7.

A function $\Phi :K_{\varpi }\to [0,+\infty )$ is said to be relatively s-convex in the second sense with respect to a function $\varpi :H\to H,$ where $s\in (0,1],$ if

$$\Phi (\tau \varpi \left(\varkappa \right)+(1-\tau )y)\le {\tau }^s\Phi \left(\varpi \left(\alpha \right)\right)+{(1-\tau )}^s\Phi \left(\beta \right),$$

is satisfied $\forall \alpha ,\beta \in [0,+\infty ),$ $\varpi \left(\alpha \right),\beta \in K_{\varpi }$, and $\tau \in$ $[0,1].$

Theorem 1.

Assume that $\nu :\mathbb{T}\to \mathbb{R}$ is strictly increasing, $I\subset \mathbb{T}$ and $J\subset$ $\tilde{\mathbb{T}}$ where $\tilde{\mathbb{T}}=\nu \left(\mathbb{T}\right)$ is a time scale and $J=\nu \left(I\right)$. If $f:\mathbb{T}\to \mathbb{R}$ is an rd-continuous function and ν is differentiable with rd-continuous derivative, then

$${\int }_{I}g\left(t\right){\nu }^{\Delta }\left(t\right)\Delta t={\int }_J\left(g\circ {\nu }^{-1}\right)\left(s\right)\tilde{\Delta }s,forI\subset \mathbb{T}.$$

Throughout the article, we will suppose that the functions (without mentioning) are nonnegative rd-continuous functions and the integrals considered are assumed to exist (finite i.e., convergent). The following lemma is needed to prove our essential results.

Lemma 1.

Let $\Phi :{[{\theta }_0,\theta ]}_{\mathbb{T}}\to \mathbb{R}$ be a $\Delta$–differentiable function and $\varpi :\mathbb{R}\to \mathbb{R}$ be a function. Then

$$\begin{array}{ccc}& & \Phi \left(\varpi \left(\zeta \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(s\right)\tilde{\Delta }s\hfill \\ & =& \frac{\varpi \left(\zeta \right)-{\theta }_0}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & & -\frac{\varpi \left(\zeta \right)-\theta }{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\zeta \right)+(1-\tau )\theta )\Delta \tau ,\hfill \end{array}$$

(9)

is satisfied $\forall \zeta \in \mathbb{R}.$

Proof. 

Utilizing (6) with $u^{\Delta }\left(\tau \right)={\Phi }^{\Delta }(\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)$ and $v^{\sigma }\left(\tau \right)=\sigma \left(\tau \right)$ (note $v\left(\tau \right)=\tau$), we acquire

$$\begin{array}{ccc}& & {\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & =& {\left.\tau \Phi (\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\right|}_0^1-{\int }_0^1\Phi (\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & =& \Phi \left(\varpi \left(\zeta \right)\right)-{\int }_0^1\Phi (\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\Delta \tau .\hfill \end{array}$$

(10)

Applicability Theorem 1 with $\varpi =\Phi \circ \nu$ such that $\nu \left(\tau \right)=\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0,$ we see (note ${\nu }^{\Delta }\left(\tau \right)=\varpi \left(\zeta \right)-{\theta }_0,$ $\nu \left(0\right)={\theta }_0$ and $\nu \left(1\right)=\varpi \left(\zeta \right)$) that

$${\int }_0^1\Phi (\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0){\nu }^{\Delta }\left(\tau \right)\Delta \tau ={\int }_{{\theta }_0}^{\varpi \left(\zeta \right)}\Phi \left(s\right)\tilde{\Delta }s,$$

and then

$$\left(\varpi \left(\zeta \right)-{\theta }_0\right){\int }_0^1\Phi (\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\Delta \tau ={\int }_{{\theta }_0}^{\varpi \left(\zeta \right)}\Phi \left(s\right)\tilde{\Delta }s,$$

thus

$${\int }_0^1\Phi (\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\Delta \tau =\frac{1}{\varpi \left(\zeta \right)-{\theta }_0}{\int }_{{\theta }_0}^{\varpi \left(\zeta \right)}\Phi \left(s\right)\tilde{\Delta }s.$$

Therefore, we acquire from (10) that

$$\begin{array}{ccc}& & \left(\varpi \left(\zeta \right)-{\theta }_0\right){\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & =& \left(\varpi \left(\zeta \right)-{\theta }_0\right)\Phi \left(\varpi \left(\zeta \right)\right)-{\int }_{{\theta }_0}^{\varpi \left(\zeta \right)}\Phi \left(s\right)\tilde{\Delta }s,\hfill \end{array}$$

(11)

in the same way, by replacing ${\theta }_0$ with $\theta$ in (11), we obtain

$$\begin{array}{ccc}& & \left(\varpi \left(\zeta \right)-\theta \right){\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\zeta \right)+(1-\tau )\theta )\Delta \tau \hfill \\ & =& \left(\varpi \left(\zeta \right)-\theta \right)\Phi \left(\varpi \left(\zeta \right)\right)-{\int }_{\theta }^{\varpi \left(\zeta \right)}\Phi \left(s\right)\tilde{\Delta }s.\hfill \end{array}$$

(12)

By subtracting (11) and (12), we get (where $\theta >{\theta }_0$) that

$$\begin{array}{ccc}\hfill \Phi \left(\varpi \left(\zeta \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(s\right)\tilde{\Delta }s& =& \frac{\left(\varpi \left(\zeta \right)-{\theta }_0\right)}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\zeta \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & & -\frac{\left(\varpi \left(\zeta \right)-\theta \right)}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\zeta \right)+(1-\tau )\theta )\Delta \tau ,\hfill \end{array}$$

which is (9). □

3. Main Results

In this section, we state and prove our essential results.

Theorem 2.

Suppose $\Phi :I_{\mathbb{T}}=I\cap \mathbb{T}\to \mathbb{R}$ is a differentiable function and ${\theta }_0,\theta \in I_{\mathbb{T}}$ such that ${\theta }_0<\theta$. If $\left|{\Phi }^{\prime }\right|$ is relatively convex with respect to a function $\varpi :\mathbb{R}\to \mathbb{R},$ $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M$ and $\sigma \left(\tau \right)\le m\tau$ for $m\ge 1,$ then the inequality

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\le \frac{mM}{2}\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right],$$

(13)

is satisfied $\forall \varkappa \in {\varpi }^{-1}\left(I_{\mathbb{T}}\right).$

Proof. 

Using Lemma 1, we have

$$\begin{array}{ccc}& & \Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\hfill \\ & =& \frac{\varpi \left(\varkappa \right)-{\theta }_0}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & & -\frac{\varpi \left(\varkappa \right)-\theta }{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )\Delta \tau ,\hfill \end{array}$$

(14)

Applying the chain rule formula (7) on ${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)$ with $v\left(\tau \right)=\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0$, then

$${\Phi }^{\Delta }\left(v\left(\tau \right)\right)={\Phi }^{\prime }\left(v\left(\zeta \right)\right)v^{\Delta }\left(\tau \right),\zeta \in [\tau ,\sigma \left(\tau \right)],$$

which implies

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)=(\varpi \left(\varkappa \right)-{\theta }_0){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0),\zeta \in [\tau ,\sigma \left(\tau \right)].$$

(15)

In the same way, we get

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )=(\varpi \left(\varkappa \right)-\theta ){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta ),\lambda \in [\tau ,\sigma \left(\tau \right)].$$

(16)

Substituting (15) and (16) into (14), we get

$$\begin{array}{ccc}\hfill \Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y& =& \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\Delta \tau \hfill \\ & & -\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\Delta \tau ,\hfill \end{array}$$

Using the triangle inequality, we obtain

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & \le & \left|\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\Delta \tau \right|\hfill \\ & & +\left|\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\Delta \tau \right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\right|\Delta \tau \hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\right|\Delta \tau ,\hfill \end{array}$$

and then by utilizing the assumptions $\sigma \left(\tau \right)\le m\tau$ and that $\left|{\Phi }^{\prime }\right|$ is relatively convex such that $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M,$ we obtain

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left(\zeta \left|{\Phi }^{\prime }(\varpi \left(\varkappa \right)\right|+(1-\zeta )\left|{\Phi }^{\prime }\left({\theta }_0\right)\right|\right)\Delta \tau \hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left(\lambda \left|{\Phi }^{\prime }(\varpi \left(\varkappa \right)\right|+(1-\lambda )\left|{\Phi }^{\prime }\left(\theta \right)\right|\right)\Delta \tau \hfill \\ & \le & M\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right]{\int }_0^1\sigma \left(\tau \right)\Delta \tau \hfill \\ & \le & mM\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right]{\int }_0^1\tau \Delta \tau \hfill \\ & =& mM\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right]\left(\frac{1}{2}{\int }_0^1(\tau +\tau )\Delta \tau \right)\hfill \\ & \le & mM\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right]\left(\frac{1}{2}{\int }_0^1\left(\tau +\sigma \left(\tau \right)\right)\Delta \tau \right)\hfill \\ & =& mM\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right]\left(\frac{1}{2}{\int }_0^1{\left({\tau }^2\right)}^{\Delta }\Delta \tau \right)\hfill \\ & =& \frac{mM}{2}\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right],\hfill \end{array}$$

which is (13). □

Remark 1.

As a special case of Theorem 2 when $\mathbb{T}=\mathbb{R},$ we acquire the inequality proved by Cortez et al. [4].

Remark 2.

If $\varpi \left(\varkappa \right)=\varkappa ,$ then we have the classical ostrowski inequality (1).

In the following, we prove the same inequality in Theorem 2, by taking $\left|{\Phi }^{\prime }\right|$ as relatively s-convex in the first sense.

Theorem 3.

Suppose that $\Phi :I_{\mathbb{T}}=I\cap \mathbb{T}\to \mathbb{R}$ is a differentiable function and ${\theta }_0,\theta \in I_{\mathbb{T}}$ such that ${\theta }_0<\theta$. If $\left|{\Phi }^{\prime }\right|$ is relatively s-convex in the first sense with respect to a function $\varpi :\mathbb{R}\to \mathbb{R}$ for some fixed $0<s\le 1,$ $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M$, and $\sigma \left(\tau \right)\le m\tau$ for $m\ge 1,$ then the inequality

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\le \frac{mM}{2}\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right),$$

(17)

is satisfied $\forall \varkappa \in {\varpi }^{-1}\left(I_{\mathbb{T}}\right).$

Proof. 

From Lemma 1, we observe that

$$\begin{array}{ccc}& & \Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\hfill \\ & =& \frac{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & & -\frac{\left(\varpi \left(\varkappa \right)-\theta \right)}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )\Delta \tau .\hfill \end{array}$$

(18)

Applying (7) to ${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)$ with $v\left(\tau \right)=\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0$, then

$${\Phi }^{\Delta }\left(v\left(\tau \right)\right)={\Phi }^{\prime }\left(v\left(\zeta \right)\right)v^{\Delta }\left(\tau \right),\zeta \in [\tau ,\sigma \left(\tau \right)],$$

and hence

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)=(\varpi \left(\varkappa \right)-{\theta }_0){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0),\zeta \in [\tau ,\sigma \left(\tau \right)].$$

(19)

In the same way, we get

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )=(\varpi \left(\varkappa \right)-\theta ){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta ),\lambda \in [\tau ,\sigma \left(\tau \right)].$$

(20)

Substituting (19) and (20) into (18), we have

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & =& \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}\left|{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\Delta \tau \right|\hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\left|{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\Delta \tau \right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\right|\Delta \tau \hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\right|\Delta \tau .\hfill \end{array}$$

Utilizing the fact that $(\left|{\Phi }^{\prime }\right|$ is relatively s-convex in the first sense) such that $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M$, we have

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left[{\zeta }^s\left|{\Phi }^{\prime }\left(\varpi \left(\varkappa \right)\right)\right|+{(1-\zeta )}^s\left|{\Phi }^{\prime }\left({\theta }_0\right)\right|\right]\Delta \tau \hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left[{\lambda }^s\left|{\Phi }^{\prime }\left(\varpi \left(\varkappa \right)\right)\right|+{(1-\lambda )}^s\left|{\Phi }^{\prime }\left(\theta \right)\right|\right]\Delta \tau \hfill \\ & \le & \frac{M{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left[{\zeta }^s+{(1-\zeta )}^s\right]\Delta \tau \hfill \\ & & +\frac{M{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left[{\lambda }^s+{(1-\lambda )}^s\right]\Delta \tau \hfill \\ & =& \frac{M{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\Delta \tau +\frac{M{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\Delta \tau \hfill \\ & =& M\left({\int }_0^1\sigma \left(\tau \right)\Delta \tau \right)\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right).\hfill \end{array}$$

Since $\sigma \left(\tau \right)\le m\tau ,$ then

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)\tilde{\Delta }u\right|\hfill \\ & \le & mM\left({\int }_0^1\tau \Delta \tau \right)\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right)\hfill \\ & \le & mM\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right)\left(\frac{1}{2}{\int }_0^1\left(\tau +\sigma \left(\tau \right)\right)\Delta \tau \right)\hfill \\ & =& mM\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right)\left(\frac{1}{2}{\int }_0^1{\left({\tau }^2\right)}^{\Delta }\Delta \tau \right)\hfill \\ & \le & \frac{mM}{2}\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right),\hfill \end{array}$$

which is (17). □

The following theorem corresponds to those functions whose derivatives in the modulus (i.e., $\left|{\Phi }^{\prime }\right|)$ are relatively s-convex in the second sense.

Theorem 4.

Suppose that $\Phi :I_{\mathbb{T}}=I\cap \mathbb{T}\to \mathbb{R}$ is a differentiable function and ${\theta }_0,\theta \in I_{\mathbb{T}}$ such that ${\theta }_0<\theta$. If $\left|{\Phi }^{\prime }\right|$ is relatively s-convex of the second sense with respect to a function $\varpi :\mathbb{R}\to \mathbb{R}$ for some fixed $0<s\le 1,$ $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M$, and $\sigma \left(\tau \right)\le m\tau$ for $m\ge 1,$ then the inequality

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\le 2^{-s}mM\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right],$$

(21)

is satisfied $\forall \varkappa \in {\varpi }^{-1}\left(I_{\mathbb{T}}\right).$

Proof. 

From Lemma 1, we observe that

$$\begin{array}{ccc}& & \Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)\tilde{\Delta }u\hfill \\ & =& \frac{\varpi \left(\varkappa \right)-{\theta }_0}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & & -\frac{\varpi \left(\varkappa \right)-\theta }{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )\Delta \tau .\hfill \end{array}$$

(22)

By applying the chain rule formula on ${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)$ with $v\left(\tau \right)=\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0$, then

$${\Phi }^{\Delta }\left(v\left(\tau \right)\right)={\Phi }^{\prime }\left(v\left(\zeta \right)\right)v^{\Delta }\left(\tau \right),\zeta \in [\tau ,\sigma \left(\tau \right)],$$

and then

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)=(\varpi \left(\varkappa \right)-{\theta }_0){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0),\zeta \in [\tau ,\sigma \left(\tau \right)].$$

(23)

In the same way, we get

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )=(\varpi \left(\varkappa \right)-\theta ){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta ),\lambda \in [\tau ,\sigma \left(\tau \right)].$$

(24)

Substituting (23) and (24) into (22) and utilizing the triangle inequality, we get

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)\tilde{\Delta }u\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\right|\Delta \tau \hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\right|\Delta \tau ,\hfill \end{array}$$

by using the fact that $(\left|{\Phi }^{\prime }\right|$ is relatively s-convex of second sense) and $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M$, we acquire

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)\tilde{\Delta }u\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left[{\zeta }^s\left|{\Phi }^{\prime }\left(\varpi \left(\varkappa \right)\right)\right|+{(1-\zeta )}^s\left|{\Phi }^{\prime }\left({\theta }_0\right)\right|\right]\Delta \tau \hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left[{\lambda }^s\left|{\Phi }^{\prime }\left(\varpi \left(\varkappa \right)\right)\right|+{(1-\lambda )}^s\left|{\Phi }^{\prime }\left(\theta \right)\right|\right]\Delta \tau \hfill \\ & \le & \frac{M{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left[{\zeta }^s+{(1-\zeta )}^s\right]\Delta \tau \hfill \\ & & +\frac{M{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left[{\lambda }^s+{(1-\lambda )}^s\right]\Delta \tau .\hfill \end{array}$$

Since ${\zeta }^s+{(1-\zeta )}^s\le 2^{1-s}{\left[\zeta +(1-\zeta )\right]}^s,$ for $0<s\le 1,$ and also ${\lambda }^s+{(1-\lambda )}^s\le$ $2^{1-s}{\left[\lambda +(1-\lambda )\right]}^s,$ for $0<s\le 1,$ then we have for $\sigma \left(\tau \right)\le m\tau$ that

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(u\right)\tilde{\Delta }u\right|\hfill \\ & \le & \frac{M{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{2}^{1-s}{\int }_0^1\sigma \left(\tau \right){\left[\zeta +(1-\zeta )\right]}^s\Delta \tau \hfill \\ & & +\frac{M{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{2}^{1-s}{\int }_0^1\sigma \left(\tau \right){\left[\lambda +(1-\lambda )\right]}^s\Delta \tau \hfill \\ & =& \frac{M{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{2}^{1-s}{\int }_0^1\sigma \left(\tau \right)\Delta \tau +\frac{M{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{2}^{1-s}{\int }_0^1\sigma \left(\tau \right)\Delta \tau \hfill \\ & \le & \frac{mM{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{2}^{1-s}{\int }_0^1\tau \Delta \tau +\frac{mM{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{2}^{1-s}{\int }_0^1\tau \Delta \tau \hfill \\ & =& 2^{1-s}mM\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right]{\int }_0^1\tau \Delta \tau \hfill \\ & \le & 2^{-s}mM\left[\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right],\hfill \end{array}$$

which is (21). □

Next, we give the following extension of (13) in Theorem 2.

Theorem 5.

Suppose that $\Phi :I_{\mathbb{T}}=I\cap \mathbb{T}\to \mathbb{R}$ is a differentiable function and ${\theta }_0,\theta \in I_{\mathbb{T}}$ such that ${\theta }_0<\theta$. If $\left|{\Phi }^{\prime }\right|$ is relatively convex with respect to a function $\varpi :\mathbb{R}\to \mathbb{R},$ $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M$, and $\sigma \left(\tau \right)\le m\tau$ for $m\ge 1,$ then the inequality

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\le {\left(\frac{m^{\gamma }{M}^{\gamma }}{\gamma +1}\right)}^{\frac{1}{\gamma }}\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right),$$

(25)

is satisfied $\forall \gamma >1$ and $\varkappa \in {\varpi }^{-1}\left(I_{\mathbb{T}}\right).$

Proof. 

By Lemma 1, we get

$$\begin{array}{ccc}& & \Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\hfill \\ & =& \frac{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & & -\frac{\left(\varpi \left(\varkappa \right)-\theta \right)}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )\Delta \tau .\hfill \end{array}$$

(26)

By applying the chain rule formula on ${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)$ with $v\left(\tau \right)=\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0$, then

$${\Phi }^{\Delta }\left(v\left(\tau \right)\right)={\Phi }^{\prime }\left(v\left(\zeta \right)\right)v^{\Delta }\left(\tau \right),\zeta \in [\tau ,\sigma \left(\tau \right)],$$

and then

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)=(\varpi \left(\varkappa \right)-{\theta }_0){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0),\zeta \in [\tau ,\sigma \left(\tau \right)].$$

(27)

In the same way, we show that

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )=(\varpi \left(\varkappa \right)-\theta ){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta ),\lambda \in [\tau ,\sigma \left(\tau \right)].$$

(28)

By substituting (27) and (28) into (26) and utilizing the triangle inequality, we notice that

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}\left|{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\Delta \tau \right|\hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\left|{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\Delta \tau \right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\right|\Delta \tau \hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\right|\Delta \tau .\hfill \end{array}$$

(29)

By appying the Hölder inequality on (29) with $\gamma >1$ and $\gamma /(\gamma -1),$ we get

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\left({\int }_0^1{\sigma }^{\gamma }\left(\tau \right)\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left|{\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\right|}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}\hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\left({\int }_0^1{\sigma }^{\gamma }\left(\tau \right)\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left|{\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\right|}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}.\hfill \end{array}$$

(30)

Utilizing the assumption $\sigma \left(\tau \right)\le m\tau$ and the fact that $\left|{\Phi }^{\prime }\right|$ is relatively convex with $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M,$ the inequality (30) becomes

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\left({\int }_0^1{\sigma }^{\gamma }\left(\tau \right)\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left(\zeta \left|{\Phi }^{\prime }\left(\varpi \left(\varkappa \right)\right)\right|+(1-\zeta )\left|{\Phi }^{\prime }\left({\theta }_0\right)\right|\right)}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}\hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\left({\int }_0^1{\sigma }^{\gamma }\left(\tau \right)\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left(\lambda \left|{\Phi }^{\prime }\left(\varpi \left(\varkappa \right)\right)\right|+(1-\lambda )\left|{\Phi }^{\prime }\left(\theta \right)\right|\right)}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}\hfill \\ & \le & M\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\left({\int }_0^1{\sigma }^{\gamma }\left(\tau \right)\Delta \tau \right)}^{\frac{1}{\gamma }}+M\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\left({\int }_0^1{\sigma }^{\gamma }\left(\tau \right)\Delta \tau \right)}^{\frac{1}{\gamma }}\hfill \\ & \le & mM{\left({\int }_0^1{\tau }^{\gamma }\Delta \tau \right)}^{\frac{1}{\gamma }}\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right).\hfill \end{array}$$

(31)

By applying the chain rule formula on the term ${\tau }^{\gamma +1},$ we obtain

$${\left({\tau }^{\gamma +1}\right)}^{\Delta }=(\gamma +1){\zeta }^{\gamma },\zeta \in [\tau ,\sigma \left(\tau \right)],$$

and then we have for $\zeta \ge \tau$ that ${\zeta }^{\gamma }\ge {\tau }^{\gamma }$; therefore, ${\tau }^{\gamma }\le \left(1/\left(\gamma +1\right)\right){\left({\tau }^{\gamma +1}\right)}^{\Delta },$ thus

$${\int }_0^1{\tau }^{\gamma }\Delta \tau \le \frac{1}{\gamma +1}{\int }_0^1{\left({\tau }^{\gamma +1}\right)}^{\Delta }\Delta \tau =\frac{1}{\gamma +1},$$

(32)

By substituting (32) into (31), we observe that

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\le {\left(\frac{m^{\gamma }{M}^{\gamma }}{\gamma +1}\right)}^{\frac{1}{\gamma }}\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right),$$

which is (25). □

Remark 3.

Clearly, for $\gamma =1,$ (25) in Theorem 5 reduces to (13) in Theorem 2.

By using the same method in Theorem 5, we have the following corollary.

Corollary 1.

Suppose that $\Phi :I_{\mathbb{T}}=I\cap T\to R$ is a differentiable function and ${\theta }_0,\theta \in I_{\mathbb{T}}$ such that ${\theta }_0<\theta$. If $\left|{\Phi }^{\prime }\right|$ is relatively s-convex of first sense with respect to a function $\varpi :\mathbb{R}\to \mathbb{R}$ for some fixed $0<s\le 1$ such that $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M$ and $\sigma \left(\tau \right)\le m\tau$ for $m\ge 1,$ then the inequality

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\le {\left(\frac{m^{\gamma }{M}^{\gamma }}{\gamma +1}\right)}^{\frac{1}{\gamma }}\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right),$$

is satisfied $\forall \gamma >1$ and $\varkappa \in {\varpi }^{-1}\left(I_{\mathbb{T}}\right).$

The following theorem is a generalization of Theorem 5.

Theorem 6.

Suppose that $\Phi :I_{\mathbb{T}}=I\cap \mathbb{T}\to \mathbb{R}$ is a differentiable function and ${\theta }_0,\theta \in I_{\mathbb{T}}$ such that ${\theta }_0<\theta$. If $\left|{\Phi }^{\prime }\right|$ is relatively s-convex in the second sense with respect to a function $\varpi :\mathbb{R}\to \mathbb{R}$ for some fixed $0<s\le 1,$ $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M$ and $\sigma \left(\tau \right)\le m\tau$ for $m\ge 1,$ then the inequality

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\le {\left(\frac{2^{\gamma \left(1-s\right)}{m}^{\gamma }{M}^{\gamma }}{\gamma + 1}\right)}^{\frac{1}{\gamma }}\left(\frac{{\left(\varpi \left(\varkappa \right) - {\theta }_0\right)}^2 + {\left(\varpi \left(\varkappa \right) - \theta \right)}^2}{\theta - {\theta }_0}\right),$$

(33)

is satisfied $\forall \gamma >1$ and $\varkappa \in {\varpi }^{-1}\left(I_{\mathbb{T}}\right).$

Proof. 

By Lemma 1, we get

$$\begin{array}{ccc}& & \Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\hfill \\ & =& \frac{\varpi \left(\varkappa \right)-{\theta }_0}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)\Delta \tau \hfill \\ & & -\frac{\varpi \left(\varkappa \right)-\theta }{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right){\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )\Delta \tau .\hfill \end{array}$$

(34)

By applying the chain rule formula on ${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)$ with $v\left(\tau \right)=\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0$, then

$${\Phi }^{\Delta }\left(v\left(\tau \right)\right)={\Phi }^{\prime }\left(v\left(\zeta \right)\right)v^{\Delta }\left(\tau \right),\zeta \in [\tau ,\sigma \left(\tau \right)],$$

and then

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau ){\theta }_0)=(\varpi \left(\varkappa \right)-{\theta }_0){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0),\zeta \in [\tau ,\sigma \left(\tau \right)].$$

(35)

In the same way, we get

$${\Phi }^{\Delta }(\tau \varpi \left(\varkappa \right)+(1-\tau )\theta )=(\varpi \left(\varkappa \right)-\theta ){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta ),\lambda \in [\tau ,\sigma \left(\tau \right)].$$

(36)

By substituting (35) and (36) into (34) and utilizing the triangle inequality, we notice that

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}\left|{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\Delta \tau \right|\hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\left|{\int }_0^1\sigma \left(\tau \right){\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\Delta \tau \right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\right|\Delta \tau \hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}{\int }_0^1\sigma \left(\tau \right)\left|{\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\right|\Delta \tau .\hfill \end{array}$$

(37)

By applying Hölder inequality on (37) with $\gamma >1$ and $\gamma /(\gamma -1),$ we get that

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(s\right)\tilde{\Delta }s\right|\hfill \\ & \le & \frac{{\left(\varpi \left(\varkappa \right) - {\theta }_0\right)}^2}{\theta - {\theta }_0}{\left({\int }_0^1{\sigma }^{\gamma }\left(\tau \right)\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left|{\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\right|}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}\hfill \\ & & +\frac{{\left(\varpi \left(\varkappa \right) - \theta \right)}^2}{\theta - {\theta }_0}{\left({\int }_0^1{\sigma }^{\gamma }\left(\tau \right)\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left|{\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\right|}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}.\hfill \end{array}$$

(38)

Utilizing the assumption $\sigma \left(\tau \right)\le m\tau$ and the fact that $\left|{\Phi }^{\prime }\right|\le M,$ the inequality (38) becomes

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\hfill \\ & \le & \frac{m{\left(\varpi \left(\varkappa \right) - {\theta }_0\right)}^2}{\theta - {\theta }_0}{\left({\int }_0^1{\tau }^{\gamma }\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left|{\Phi }^{\prime }(\zeta \varpi \left(\varkappa \right)+(1-\zeta ){\theta }_0)\right|}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}\hfill \\ & & +\frac{m{\left(\varpi \left(\varkappa \right) - \theta \right)}^2}{\theta - {\theta }_0}{\left({\int }_0^1{\tau }^{\gamma }\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left|{\Phi }^{\prime }(\lambda \varpi \left(\varkappa \right)+(1-\lambda )\theta )\right|}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}.\hfill \end{array}$$

(39)

Since $\left|{\Phi }^{\prime }\right|$ is relatively s-convex in the second sense, $0<s\le 1$ such that $\left|{\Phi }^{\prime }\left(\varkappa \right)\right|\le M,$ then the inequality (39) becomes

$$\begin{array}{ccc}& & \Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }sy\hfill \\ & \le & \frac{m{\left(\varpi \left(\varkappa \right) - {\theta }_0\right)}^2}{\theta - {\theta }_0}{\left({\int }_0^1{\tau }^{\gamma }\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left({\zeta }^s\left|{\Phi }^{\prime }\left(\varpi \left(\varkappa \right)\right)\right|+{(1-\zeta )}^s\left|{\Phi }^{\prime }\left({\theta }_0\right)\right|\right)}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}\hfill \\ & & +\frac{m{\left(\varpi \left(\varkappa \right) - \theta \right)}^2}{\theta - {\theta }_0}{\left({\int }_0^1{\tau }^{\gamma }\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left({\lambda }^s\left|{\Phi }^{\prime }\left(\varpi \left(\varkappa \right)\right)\right|+{(1-\lambda )}^s\left|{\Phi }^{\prime }\left(\theta \right)\right|\right)}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}\hfill \\ & \le & \frac{mM{\left(\varpi \left(\varkappa \right) - {\theta }_0\right)}^2}{\theta - {\theta }_0}{\left({\int }_0^1{\tau }^{\gamma }\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left({\zeta }^s+{(1-\zeta )}^s\right)}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}\hfill \\ & & +\frac{mM{\left(\varpi \left(\varkappa \right) - \theta \right)}^2}{\theta - {\theta }_0}{\left({\int }_0^1{\tau }^{\gamma }\Delta \tau \right)}^{\frac{1}{\gamma }}{\left({\int }_0^1{\left({\lambda }^s+{(1-\lambda )}^s\right)}^{\frac{\gamma }{\gamma -1}}\Delta \tau \right)}^{\frac{\gamma -1}{\gamma }}.\hfill \end{array}$$

(40)

Since $0<s<1,$ then

$${\zeta }^s+{(1-\zeta )}^s\le 2^{1-s}{\left[\zeta +(1-\zeta )\right]}^s=2^{1-s}.$$

Also, we have

$${\lambda }^s+{(1-\lambda )}^s\le 2^{1-s}{\left[\lambda +(1-\lambda )\right]}^s=2^{1-s}.$$

Therefore, we get from (40) that

$$\begin{array}{ccc}& & \left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(s\right)\tilde{\Delta }s\right|\hfill \\ & \le & 2^{1-s}mM\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right){\left({\int }_0^1{\tau }^{\gamma }\Delta \tau \right)}^{\frac{1}{\gamma }}.\hfill \end{array}$$

(41)

By applying the chain rule formula on the term ${\tau }^{\gamma +1},$ we get

$${\left({\tau }^{\gamma +1}\right)}^{\Delta }=(\gamma +1){\zeta }^{\gamma },\zeta \in [\tau ,\sigma \left(\tau \right)],$$

and then we have for $\zeta \ge \tau$ that ${\zeta }^{\gamma }\ge {\tau }^{\gamma }$; therefore, ${\tau }^{\gamma }\le \left(1/\left(\gamma +1\right)\right){\left({\tau }^{\gamma +1}\right)}^{\Delta },$ thus

$${\int }_0^1{\tau }^{\gamma }\Delta \tau \le \frac{1}{\gamma +1}{\int }_0^1{\left({\tau }^{\gamma +1}\right)}^{\Delta }\Delta \tau =\frac{1}{\gamma +1}.$$

(42)

Substituting (42) into (41), we notice that

$$\left|\Phi \left(\varpi \left(\varkappa \right)\right)-\frac{1}{\theta -{\theta }_0}{\int }_{{\theta }_0}^{\theta }\Phi \left(y\right)\tilde{\Delta }y\right|\le {\left(\frac{2^{\gamma \left(1-s\right)}{m}^{\gamma }{M}^{\gamma }}{\gamma +1}\right)}^{\frac{1}{\gamma }}\left(\frac{{\left(\varpi \left(\varkappa \right)-{\theta }_0\right)}^2+{\left(\varpi \left(\varkappa \right)-\theta \right)}^2}{\theta -{\theta }_0}\right),$$

which is (33). □

Remark 4.

Clearly, for $\gamma =1,$ (33) in Theorem 6 reduces to (25) in Theorem 5.

4. Conclusions

In this manuscript, we discussed different types of Ostrowski inequalities for functions whose derivative module is relatively convex. Our obtained results for ($\mathbb{T}=\mathbb{R}$) generalize the inequalities of Vivas-Cortez, Garcia, and Hernández [4]. In future work, we will continue to generalize more fractional dynamic inequalities by using Specht’s ratio and Kantorovich’s ratio.