Some novel inequalities for Caputo Fabrizio fractional integrals involving (α,s)-convex functions with applications

ABSTRACT

Fractional calculus is extremely important and should not be undervalued due to its critical role in the theory of inequalities. In this article, different generalized Hermite-Hadamard type inequalities for functions whose modulus of first derivatives are $(\alpha ,s)$-convex are presented, via Caputo-Fabrizio integrals. Graphical justifications of main results are presented. Graphs enable us to support our conclusions and show the reliability of our findings. Additionally, some applications to probability theory and numerical integration are also established. As special cases, certain established outcomes from different articles are recaptured. This study acts as a stimulant for future studies, inspiring researchers to investigate more thorough results by utilizing generalized fractional operators and broadening the idea of convexity.

KEYWORDS:

• Convex function
• Caputo-Fabrizio integrals
• power-mean inequality

1 Introduction

In recent monographs and symposium proceedings, the use of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics has been highlighted. The fractional operators precisely describe natural phenomena that occur in such common engineering issues as heat transfer, electrode/electrolyte behaviour, and sub-threshold nerve propagation. Many issues in biomedical research can be resolved using methodologies of fractional calculus when defined as a Laplace or Fourier convolution product. See [1,2,]. In [3], Adam Loverro discussed the history, definitions and applications of fractional calculus for the Engineers and demonstrated how these can be used to resolve a few current issues.

Over the last few decades, many research have been drawn to fractional calculus, see [4–10]. Numerous academician in the field of fractional analysis have concentrated on creating novel operators, modelling and solving issues based on their characteristics. The characteristics which distinguish the operators include singularity and locality, whereas the kernel expression of the operator can be chosen as the power function, the exponential function or a Mittag-Leffler function, see [11,12].

The Caputo-Fabrizio operator is distinguished by its non-singular kernel. In [5], new investigations on several modelling and realistic scenarios have been conducted with the assistance of the Caputo-Fabrizio operator. This is because the Caputo-Fabrizio operator is particularly effective in better explaining heterogeneousness and systems with multiple scales with memory effects [4,13]. The basic characteristic of the Caputo-Fabrizio operator is a real power converted to an integer through the Laplace transformation. As a result, precise solutions to many difficulties are quickly identified.

In 2015, Caputo and Fabrizio proposed the following operator:

Caputo-Fabrizio integral operator [5,7]:

Let $H^1(\omega ,\zeta )$ be the Sobolev space of order one defined as

$H^1(\omega ,\zeta )=\{g\in L^2(\omega ,\zeta ):g{ }^{\mathrm{\prime }}\in L^2(\omega ,\zeta )\},$

where

$L^2(\omega ,\zeta )=\{g(z):(\underset{\omega }{\overset{\zeta }{\int }}{g}^2(z)dz{)}^{\frac{1}{2}}<\mathrm{\infty }\}.$

Let $\delta \in H^1(\omega ,\zeta ),\omega <\zeta$ and $\varsigma \in [0,1]$, then left derivative in the sense of Caputo-Fabrizio is defined as

${(}_{\omega }^{CFD}{D}^{\varsigma }\delta )(z)=\frac{T(\varsigma )}{1-\varsigma }\underset{\omega }{\overset{z}{\int }}\delta { }^{\mathrm{\prime }}(l)e^{\frac{-\varsigma {(z-l)}^{\varsigma }}{1-\varsigma }}dl,$

$z>\varsigma$ and the associated integral operator is

(1.1) ${(}_{\omega }^{CF}{I}^{\varsigma }\delta )(z)=\frac{1-\varsigma }{T(\varsigma )}\delta (z)+\frac{\varsigma }{T(\varsigma )}\underset{\omega }{\overset{z}{\int }}\delta (v)dv,$(1.1)

where $T(\varsigma )>0$ is normalization function and T(0)=T(1) = 1. For $\varsigma =0,\varsigma =1$ the left derivative is defined as follows, respectively

${(}_{\omega }^{CFD}{D}^0\delta )(z)=\delta { }^{\mathrm{\prime }}(z),$

${(}_{\omega }^{CFD}{I}^1\delta )(z)=\delta (z)-\delta (\omega ).$

For the right derivative operator

${(}_{\zeta }^{CFD}{D}^{\varsigma }\delta )(z)=\frac{-T(\varsigma )}{1-\varsigma }\underset{z}{\overset{\zeta }{\int }}\delta { }^{\mathrm{\prime }}(l)e^{\frac{-\varsigma {(l-z)}^{\varsigma }}{1-\varsigma }}dl,$

$z<\zeta$ and associated integral operator is

(1.2) ${(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(z)=\frac{1-\varsigma }{T(\varsigma )}\delta (z)+\frac{\varsigma }{T(\varsigma )}\underset{z}{\overset{\zeta }{\int }}\delta (v)dv,$(1.2)

where $T(\varsigma )>0$ is a normalization function and $T(0)=T(1)=1$.

Convex functions have a rich history. The origin may be traced back to the late nineteenth century. Its foundations may be traced in the seminal contributions of Hölder (1889), Stolz (1893), and Hadamard (1894). At the beginning of the last century, Jensen first realized the importance and undertook a systematic study of convex functions. Following this study, the theory of convex functions emerged as an independent topic of mathematical analysis. Convex functions are of great interest, see [11,14,15].

Convex function [16]:

A real valued function $\delta$ is said to be convex on close interval $I$, if

(1.3) $\delta (\omega \varrho +(1-\varrho )\zeta )\le \varrho \delta (\omega )+(1-\varrho )\delta (\zeta )$(1.3)

holds, for all $\omega ,\zeta$ in $I$ and $\varrho \in [0,1]$.

$s$-convex function [17]:

A function $\delta :[0,\mathrm{\infty })\to \mathbb{R}$ is supposed to be $s$-convex in second sense if

(1.4) $\delta (\omega \varrho +(1-\varrho )\zeta )\le {\varrho }^s\delta (\omega )+(1-\varrho {)}^s\delta (\zeta )$(1.4)

holds, provided that all $\omega ,\zeta \in [0,\mathrm{\infty })$, $s\in (0,1]$ and $\varrho \in [0,1]$.

In [18], Xi et al. introduced the concept of $(\alpha ,s)$-convex function as:

$(\alpha ,s)$-convex function:

A function $\delta :I\subseteq \mathbb{R}\to \mathbb{R}$ is said to be $(\alpha ,s)$-convex, if

(1.5) $\delta (\omega \varrho +(1-\varrho )\zeta )\le {\varrho }^{\alpha s}\delta (\omega )+(1-{\varrho }^{\alpha }{)}^s\delta (\zeta )$(1.5)

holds, provided that all $\omega ,\zeta \in I,s\in [-1,1],\alpha \in (0,1]$ and $\varrho \in (0,1)$. $(\alpha ,s)$-convex function is generalization of classical convex function for $\alpha =s=1$.

Inequalities have been very effective and exciting subject of research in recent years and they play a vital role in all disciplines of mathematics. Inequalities have played a significant influence in the realm of classical differential and integral equations [19,20].

There are many inequalities for convex functions that have been set up by various authors, such as the Hardy type inequality, Ostrowski type inequality, Olsen type inequality and Gagliardo-Nirenberg type inequality, but the Hermite-Hadamard inequality is among the most eminent inequalities. It has a straightforward geometrical presentation and a diverse variety of use. In 1893, Charles Hermite and Jacques Hadamard derived the following Hermite-Hadamard inequality for convex function [21,22].

Theorem 1.

Let $\delta :I\to \mathbb{R}$ be convex function, then

(1.6) $\delta \left(\frac{b+u}{2}\right)\le \frac{1}{u-b}\underset{b}{\overset{u}{\int }}\delta (y)dy\le \frac{\delta (b)+\delta (u)}{2}$(1.6)

hold, where $b,u\in I,b<u$ and $I$ is a closed interval in $\mathbb{R}$.

Theorem 2.

The Hermite-Hadamard inequality for $s$-convex function $\delta :I\to \mathbb{R}$ is defined by:

(1.7) $2^{s-1}\delta \left(\frac{\omega +\zeta }{2}\right)\le \frac{1}{\zeta -\omega }\underset{\omega }{\overset{\zeta }{\int }}\delta (y)dy\le \frac{\delta (\omega )+\delta (\zeta )}{s+1},$(1.7)

where $s\in (0,1]$, $\omega ,\zeta \in I$ and $\omega <\zeta$.

Hermite-Hadamard inequality have undergone significant generalization and improvement. Since then, several researchers have extensively applied concepts of Hadamard inequalities and obtained several new generalizations via convex functions and their refinements.

In [23], Nosheen et al. established the Hermite-Hadamard type inequalities involving Caputo fractional derivatives and Caputo-Fabrizio integrals for class of $(s,m)$- convex functions. They established the following Hermite-Hadamard inequality for the class of $(\alpha ,s)$- convex functions [24]:

Theorem 3.

If a function $\delta :[0,v]\to \mathbb{R}$ is $(\alpha ,s)$-convex and integrable on $[\omega ,\zeta ]$, then

(1.8) $2^{\alpha s-1}\delta \left(\frac{\omega +\zeta }{2}\right)\le \frac{1}{\zeta -\omega }\underset{\omega }{\overset{\zeta }{\int }}\delta (v)dv\le \frac{\delta (\omega )+\delta (\zeta )}{\alpha s+1},$(1.8)

for all $\omega ,\zeta \in [0,v]$, $\omega <\zeta$ and $s\in (0,1]$.

Theorem 4.

(Hölder’s inequality [25]) Let $p>1$. If $\xi$ and $\delta$ are real functions defined on $[\omega ,\zeta ]$ and $|\xi {|}^p,|\delta {|}^q$are integrable on $[\omega ,\zeta ]$, then

(1.9) $\underset{\omega }{\overset{\zeta }{\int }}|\xi (v)\delta (v)|dv\le (\underset{\omega }{\overset{\zeta }{\int }}|\xi (v){|}^{p}dv{)}^{\frac{1}{p}}(\underset{\omega }{\overset{\zeta }{\int }}|\delta (v){|}^{q}dv{)}^{\frac{1}{q}}$(1.9)

holds, where $\frac{1}{p}+\frac{1}{q}=1$.

Theorem 5.

(Power-mean inequality [25]) If $\xi$ and $\delta$ are defined on $[\omega ,\zeta ]$ and are real functions. If $|\xi |,|\xi ||\delta {|}^q$are integrable on $[\omega ,\zeta ]$, then

(1.10) $\underset{\omega }{\overset{\zeta }{\int }}|\xi (v)\delta (v)|dv\le (\underset{\omega }{\overset{\zeta }{\int }}|\xi (v)|dv{)}^{1-\frac{1}{q}}(\underset{\omega }{\overset{\zeta }{\int }}|\xi (v)||\delta (v){|}^{q}dv{)}^{\frac{1}{q}}$(1.10)

holds, where $q>1$.

In [26] Kavurmaci et al. established the following identity.

Lemma 1.1

Let $\delta :[\omega ,\zeta ]\to \mathbb{R}$ be a differentiable function on $(\omega ,\zeta ),\omega <\zeta$. If $\delta { }^{\mathrm{\prime }}$ is integrable on $[\omega ,\zeta ]$ and $x\in [\omega ,\zeta ]$, then

(1.11) $\begin{array}{cc}\hspace{0.17em}& \frac{(x-\omega )\delta (\omega )+(\zeta -x)\delta (\zeta )}{(\zeta -\omega )}-\frac{1}{\zeta -\omega }\underset{\omega }{\overset{\zeta }{\int }}\delta (v)dv\\ \hspace{0.17em}& =\frac{{(x-\omega )}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(\varrho -1)\delta { }^{\mathrm{\prime }}(\varrho x+(1-\varrho )\omega )d\varrho +\frac{{(\zeta -x)}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )\delta { }^{\mathrm{\prime }}(\varrho x+(1-\varrho )\zeta )d\varrho .\end{array}$(1.11)

In [27] Hermite-Hadamard type inequalities for uniformly $p$-convex function and uniformly $q$-convex function are presented. Wang et al. [28] provided inequalities for modified $h$-convex functions. Li et al. gave inequalities for strongly convex functions using Caputo-Fabrizio integrals [29]. In [26,30], authors provide inequalities of the Hermite-Hadamard type for function with absolute value of the derivatives is convex and $s$-convex respectively.

Some refinements of Hermite-Hadamard inequality using $k$ fractional Caputo derivatives were established in [31]. Rahman et al. applied a new approach to establish bounds of generalized proportional fractional integrals in general form via convex functions [32]. Tassaddiq et al. established fractional conformable inequalities for the weighted and extended Chebyshev functionals [33]. Abd-Allah Hyder presented new fractional inequalities through convex functions and comprehensive RiemannLiouville integrals in the year 2023 [34]. In the same year, Breaz et al. demonstrated several inequalities for the products of two modified h-convex functions [35] and Nosheen et al. acquired various Hermite-Hadamard inequalities for convex functions and for $s$-convex functions with the use of q-symmetric calculus [36]. Vivas-Cortez et al. explored the Hermite-Hadamard-Type integral inequalities for twice differentiable $h$-convex functions [37]. New extensions of Hermite-Hadamard inequality using $k$ fractional Caputo derivatives established in [38].

The purpose of this article is to accomplish some bounds for Caputo-Fabrizio integrals involving $(\alpha ,s)$-convex functions. In Section 2, we generalize Lemma 1.1 and apply Hölder and Power-mean inequalities to obtain these bounds. In Section 3, error estimates for trapezoidal formula are calculated, and Section 4 presents applications to probability theory.

2 Main results

Following lemma is established for the class of $(\alpha ,s)$-convex functions.

Lemma 2.1.

If $\delta :I\subseteq \mathbb{R}\to \mathbb{R}$ be $(\alpha ,s)$-convex function, then

(2.1) $\delta \left(\omega h+(1-h)\zeta \right)\le h^{\alpha s}\delta (\omega )+(1-h{)}^{\alpha s}\delta (\zeta )$(2.1)

holds, for all $\omega ,\zeta$ in $I$, $\alpha \in (0,1]$, $s\in [0,1]$ and $h\in (0,1)$.

Proof. Since $h^{\alpha }\ge h$ for all $\alpha \in (0,1]$ and $h\in (0,1)$, therefore

$1-h^{\alpha }\le 1-h\Rightarrow 1-h^{\alpha }\le (1-h{)}^{\alpha }\Rightarrow (1-h^{\alpha }{)}^s\le (1-h{)}^{\alpha s}.$

$\delta (\cdot )$ is $(\alpha ,s)$-convex, thus (1.5) takes the form:

$\delta \left(\omega h+(1-h)\zeta \right)\le h^{\alpha s}\delta (\omega )+(1-h^{\alpha }{)}^s\delta (\zeta )$

$\le h^{\alpha s}\delta (\omega )+(1-h{)}^{\alpha s}\delta (\zeta ).$

Lemma 1.1 is generalized first in the following lemma, which involves the Caputo-Fabrizio integrals. Then fractional Hadamard inequalities will be proposed using the lemma.

Lemma 2.2.

Let $\delta :[\omega ,\zeta ]\to \mathbb{R}$ be a differentiable function on $(\omega ,\zeta ),\omega <\zeta$. If $\delta { }^{\mathrm{\prime }}$ is integrable on $[\omega ,\zeta ]$, $k\in [\omega ,\zeta ]$ and $\varsigma \in [0,1]$, then (2.2) holds

(2.2) $\begin{array}{cc}\hspace{0.17em}& \frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{T(\varsigma )}{\varsigma (\zeta -\omega )}[{(}_{\omega }^{CF}{I}^{\varsigma }\delta )(k)+{(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(k)]+\frac{2(1-\varsigma )}{(\zeta -\omega )\varsigma }\delta (k)\\ \hspace{0.17em}& =\frac{{(k-\omega )}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(\varrho -1)\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\omega )d\varrho +\frac{{(\zeta -k)}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\zeta )d\varrho .\end{array}$(2.2)

Proof. Let

(2.3) $\begin{array}{c}F=\frac{{(k-\omega )}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(\varrho -1)\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\omega )d\varrho +\frac{{(\zeta -k)}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\zeta )d\varrho .\end{array}$(2.3)

Apply integration by parts on left side of (2.3)

(2.4) $\begin{array}{c}F=\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{1}{\zeta -\omega }\underset{\omega }{\overset{\zeta }{\int }}\delta (l)dl.\end{array}$(2.4)

Multiply both sides of (2.4) by $\frac{\varsigma }{T(\varsigma )}$ and subtract $\frac{2(1-\varsigma )\delta (k)}{T(\varsigma )(\zeta -\omega )}$ to get,

(2.5) $\begin{array}{cc}\hspace{0.17em}& \frac{\varsigma }{T(\varsigma )}F-\frac{2(1-\varsigma )\delta (k)}{T(\varsigma )(\zeta -\omega )}\\ \hspace{0.17em}& =\frac{\varsigma }{T(\varsigma )}[\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{1}{\zeta -\omega }\underset{\omega }{\overset{\zeta }{\int }}\delta (l)dl]-\frac{2(1-\varsigma )\delta (k)}{T(\varsigma )(\zeta -\omega )}\\ \hspace{0.17em}& =\frac{\varsigma }{T(\varsigma )}\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{1}{(\zeta -\omega )}[{(}_{\omega }^{CF}{I}^{\varsigma }\delta )(k)+{(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(k)].\end{array}$(2.5)

Further simplification of (2.5) leads towards (2.2).

Outcome of Lemma 2.2 is in the following remark.

Remark 1 Put $\varsigma =1$ in Lemma 2.2 to obtain (1.11).

Consider

$|L|=|\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{T(\varsigma )}{\varsigma (\zeta -\omega )}[({(}_{\omega }^{CF}{I}^{\varsigma }\delta )(k)+{(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(k))]$

$+\frac{2(1-\varsigma )}{\varsigma (\zeta -\omega )}\delta (k)|.$

and

$|J\left|=\right|\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{1}{(\zeta -\omega )}\underset{\omega }{\overset{\zeta }{\int }}\delta (c)dc|$

throughout the paper.

Fractional Hadamard inequalities for the function whose absolute value of first derivative is $(\alpha ,s)$-convex is established in the undermentioned theorem.

Theorem 6.

Let $\delta :[\omega ,\zeta ]\to \mathbb{R}$ be a differentiable function on $(\omega ,\zeta ),\omega <\zeta$. If $|\delta { }^{\mathrm{\prime }}|$ is $(\alpha ,s)$-convex function and integrable on $[\omega ,\zeta ]$, then:

(2.6) $\begin{array}{cc}\hspace{0.17em}& |\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{T(\varsigma )}{\varsigma (\zeta -\omega )}[{(}_{\omega }^{CF}{I}^{\varsigma }\delta )(k)+{(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(k)]+\frac{2(1-\varsigma )}{(\zeta -\omega )\varsigma }\delta (k)|\\ \hspace{0.17em}& \le \frac{{(k-\omega )}^2}{\zeta -\omega }[\frac{|\delta { }^{\mathrm{\prime }}(k)|}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\omega )|}{\alpha s+2}]+\frac{{(\zeta -k)}^2}{\zeta -\omega }[\frac{|\delta { }^{\mathrm{\prime }}(k)|}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\zeta )|}{\alpha s+2}]\end{array}$(2.6)

holds for $k\in [\omega ,\zeta ]$ and $s\in [0,1]$.

Proof. Use Lemma 2.2 and property of modulus

(2.7) $\begin{array}{cc}|L|& \le \frac{{(k-\omega )}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )|\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\omega )|d\varrho \\ \hspace{0.17em}& +\frac{{(\zeta -k)}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )|\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\zeta )|d\varrho .\end{array}$(2.7)

$|\delta { }^{\mathrm{\prime }}(\cdot )|$ is $(\alpha ,s)$-convex, use Lemma 2.1 in (2.7)

$|L|\le \frac{{(k-\omega )}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )[{\varrho }^{\alpha s}|\delta { }^{\mathrm{\prime }}(k)|+(1-\varrho {)}^{\alpha s}|\delta { }^{\mathrm{\prime }}(\omega )|]d\varrho$

$+\frac{{(\zeta -k)}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )[{\varrho }^{\alpha s}|\delta { }^{\mathrm{\prime }}(k)|+(1-\varrho {)}^{\alpha s}|\delta { }^{\mathrm{\prime }}(\zeta )|]d\varrho$

$\le \frac{{(k-\omega )}^2}{\zeta -\omega }[\frac{|\delta { }^{\mathrm{\prime }}(k)|}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\omega )|}{\alpha s+2}]+\frac{{(\zeta -k)}^2}{\zeta -\omega }[\frac{|\delta { }^{\mathrm{\prime }}(k)|}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\zeta )|}{\alpha s+2}].$

Following remark summarizes the implications of Theorem 6.

Remark 2 Put $\varsigma =1$ in Theorem 6 to get

(2.8) $\begin{array}{cc}\hspace{0.17em}& |J|=|\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{1}{(\zeta -\omega )}\underset{\omega }{\overset{\zeta }{\int }}\delta (c)dc|\\ \hspace{0.17em}& \le \frac{{(k-\omega )}^2}{\zeta -\omega }[\frac{|\delta { }^{\mathrm{\prime }}(k)|}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\omega )|}{\alpha s+2}]+\frac{{(\zeta -k)}^2}{\zeta -\omega }[\frac{|\delta { }^{\mathrm{\prime }}(k)|}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\zeta )|}{\alpha s+2}].\end{array}$(2.8)

1. Substitute $\alpha =1$ in (2.8), to obtain [30, Theorem 5].

2. Substitute $\alpha =s=1$ in (2.8), to obtain [29, Theorem 4].

3. Put $\alpha =s=1$ and choose $k=\frac{\omega +\zeta }{2}$ in (19), to get [29, Corollary 2].

Obtained results are graphically elaborated in the following examples to help readers comprehend properly.

Example 2.1.

$\delta (x)=e^x$ is $(\alpha ,s)$-convex function for $\alpha =\frac{1}{3}$ and $s=\frac{1}{7}$ on $[0,1]$ because substituting $\omega =0$, $\zeta =1$, $\alpha =\frac{1}{3}$ and $s=\frac{1}{7}$ in the definition of $(\alpha ,s)$-convex function, (1.5) holds true. Graphically it is shown in figure (a).

$\delta (0\varrho +(1-\varrho )1)\le {\varrho }^{\frac{1}{21}}\delta (0)+(1-{\varrho }^{\frac{1}{3}}{)}^{\frac{1}{7}}\delta (1)$

(2.9) $u(\varrho )=e^{1-\varrho }\le {\varrho }^{\frac{1}{21}}+(1-{\varrho }^{\frac{1}{3}}{)}^{\frac{1}{7}}e=v(\varrho ).$(2.9)

Same substitution in Theorem 6 gives:

(2.10) $\begin{array}{cc}\hspace{0.17em}& w(k)=|k-ke+1|\le k^2[\frac{441e^k}{946}+\frac{21}{43}]+{(1-k)}^2[\frac{441e^k}{946}+\frac{21e}{43}]=r(k),\end{array}$(2.10)

where $k\in [0,1]$. (2.10) is graphically represented in figure(b).

Example 2.2.

For $\alpha =\frac{1}{2}$ and $s=\frac{1}{2}$, $\delta (x)=e^x$ is $(\alpha ,s)$-convex function on $[1,2]$. Substitute $\omega =1$, $\zeta =2$, $\alpha =\frac{1}{2}$ and $s=\frac{1}{2}$ in (1.5) to get

(2.11) $a(\varrho )=e^{2-\varrho }\le {\varrho }^{\frac{1}{4}}e+(1-{\varrho }^{\frac{1}{2}}{)}^{\frac{1}{2}}{e}^2=b(\varrho ).$(2.11)

Figure(c) shows that (2.11) holds true. Same substitution in Theorem 6 gives:

(2.12) $\begin{array}{cc}\hspace{0.17em}& z(k)=|ke+(1-k)e^2|\le {(k-1)}^2[\frac{16e^k}{45}+\frac{4e}{9}]+{(2-k)}^2[\frac{16e^k}{45}+\frac{9e^2}{4}]=l(k).\end{array}$(2.12)

(2.12) is represented graphically in figure (d).

Another approach to the Hadamard inequality for Caputo-Fabrizio integrals of $(\alpha ,s)$-convex function is discussed in forthcoming results.

Theorem 7.

Let $\delta :[\omega ,\zeta ]\to \mathbb{R}$ be a differentiable function on $(\omega ,\zeta ),\omega <\zeta$. If $|\delta { }^{\mathrm{\prime }}{|}^q$, $q>1$, is $(\alpha ,s)$-convex function and integrable on $[\omega ,\zeta ]$, then

(2.13) $\begin{array}{cc}\hspace{0.17em}& |\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{T(\varsigma )}{\varsigma (\zeta -\omega )}[{(}_{\omega }^{CF}{I}^{\varsigma }\delta )(k)+{(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(k)]+\frac{2(1-\varsigma )}{(\zeta -\omega )\varsigma }\delta (k)|\\ \hspace{0.17em}& \le {(\frac{1}{p+1})}^{\frac{1}{p}}{(\frac{1}{\alpha s+1})}^{\frac{1}{q}}[\frac{{(k-\omega )}^2{(|\delta { }^{\mathrm{\prime }}(\omega {)|}^q+|\delta { }^{\mathrm{\prime }}(k{)|}^q)}^{\frac{1}{q}}+{(\zeta -k)}^2{(|\delta { }^{\mathrm{\prime }}(\zeta {)|}^q+|\delta { }^{\mathrm{\prime }}(k{)|}^q)}^{\frac{1}{q}}}{(\zeta -\omega )}]\end{array}$(2.13)

holds, for $s\in [0,1]$, $k\in (\omega ,\zeta )$ and $p^{-1}=1-q^{-1}$.

Proof. Use Lemma 2.2 and (1.9)

$|\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{T(\varsigma )}{\varsigma (\zeta -\omega )}[({(}_{\omega }^{CF}{I}^{\varsigma }\delta )(k)+{(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(k))]+\frac{2(1-\varsigma )}{\varsigma (\zeta -\omega )}\delta (k)|$

$\le \frac{{(k-\omega )}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )|\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\omega )|d\varrho +\frac{{(\zeta -k)}^2}{\zeta -\omega }\underset{0}{\overset{1}{\int }}(1-\varrho )|\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\zeta )|d\varrho$

$\le \frac{{(k-\omega )}^2}{\zeta -\omega }(\underset{0}{\overset{1}{\int }}(1-\varrho {)}^{p}d\varrho {)}^{\frac{1}{p}}(\underset{0}{\overset{1}{\int }}|\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\omega ){|}^{q}d\varrho {)}^{\frac{1}{q}}$

$+\frac{{(\zeta -k)}^2}{\zeta -\omega }(\underset{0}{\overset{1}{\int }}(1-\varrho {)}^{p}d\varrho {)}^{\frac{1}{p}}(\underset{0}{\overset{1}{\int }}|\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\zeta ){|}^{q}d\varrho {)}^{\frac{1}{q}}.$

Since $|\delta { }^{\mathrm{\prime }}{|}^q$ is $(\alpha ,s)$-convex function, therefore apply Theorem 3

$\begin{array}{cc}\hspace{0.17em}& |\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{T(\varsigma )}{\varsigma (\zeta -\omega )}[{(}_{\omega }^{CF}{I}^{\varsigma }\delta )(k)+{(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(k)]+\frac{2(1-\varsigma )}{\varsigma (\zeta -\omega )}\delta (k)|\\ \hspace{0.17em}& \le {(\frac{1}{p+1})}^{\frac{1}{p}}{(\frac{1}{\alpha s+1})}^{\frac{1}{q}}[\frac{{(k-\omega )}^2{(|\delta { }^{\mathrm{\prime }}(\omega {)|}^q+|\delta { }^{\mathrm{\prime }}(k{)|}^q)}^{\frac{1}{q}}+{(\zeta -k)}^2{(|\delta { }^{\mathrm{\prime }}(\zeta {)|}^q+|\delta { }^{\mathrm{\prime }}(k{)|}^q)}^{\frac{1}{q}}}{(\zeta -\omega )}].\end{array}$

Remark 3. Put $\varsigma =1$ in Theorem 7 to obtain

(2.14) $\begin{array}{cc}|J|\le & {(\frac{1}{p+1})}^{\frac{1}{p}}{(\frac{1}{\alpha s+1})}^{\frac{1}{q}}\\ \hspace{0.17em}& [\frac{{(k-\omega )}^2{(|\delta { }^{\mathrm{\prime }}(\omega {)|}^q+|\delta { }^{\mathrm{\prime }}(k{)|}^q)}^{\frac{1}{q}}+{(\zeta -k)}^2{(|\delta { }^{\mathrm{\prime }}(\zeta {)|}^q+|\delta { }^{\mathrm{\prime }}(k{)|}^q)}^{\frac{1}{q}}}{(\zeta -\omega )}].\end{array}$(2.14)

1. Choose $\alpha =1$ in (2.14) to obtain [30, Theorem 6].

2. Put $\alpha =1$ and choose $k=\frac{\omega +\zeta }{2}$ in (2.14), to get [30, Corollary 4].

3. Choose $\alpha =s=1$ in (2.14) to obtain [29, Theorem 5].

4. Put $\alpha =s=1$ and choose $k=\frac{\omega +\zeta }{2}$ in (2.14), to get [29, Corollary 3].

Theorem 8.

Let $\delta :[\omega ,\zeta ]\to \mathbb{R}$ be a differentiable function on $(\omega ,\zeta )$. If $|\delta { }^{\mathrm{\prime }}{|}^q$, $q>1$, is $(\alpha ,s)$-convex function and integrable on $[\omega ,\zeta ]$, then

(2.15) $\begin{array}{cc}\hspace{0.17em}& |\frac{(k-\omega )\delta (\omega )+(\zeta -k)\delta (\zeta )}{(\zeta -\omega )}-\frac{T(\varsigma )}{\varsigma (\zeta -\omega )}[{(}_{\omega }^{CF}{I}^{\varsigma }\delta )(k)+{(}^{CF}{I}_{\zeta }^{\varsigma }\delta )(k)]+\frac{2(1-\varsigma )}{(\zeta -\omega )\varsigma }\delta (k)|\\ \hspace{0.17em}& \le \frac{{(k-\omega )}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{[\frac{|\delta { }^{\mathrm{\prime }}(k{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\omega {)|}^q}{\alpha s+2}]}^{\frac{1}{q}}\\ \hspace{0.17em}& +\frac{{(\zeta -k)}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{[\frac{|\delta { }^{\mathrm{\prime }}(k{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\zeta {)|}^q}{\alpha s+2}]}^{\frac{1}{q}}\end{array}$(2.15)

holds for $s\in [0,1]$, $\omega <\zeta$ and $k\in [\omega ,\zeta ]$.

Proof. Use Lemma 2.2 and (1.10) to get

(2.16) $\begin{array}{cc}\hspace{0.17em}& |L|\le \frac{{(k-\omega )}^2}{\zeta -\omega }{(\underset{0}{\overset{1}{\int }}(1-\varrho )d\varrho )}^{1-\frac{1}{q}}{(\underset{0}{\overset{1}{\int }}(1-\varrho )|\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\omega {)|}^{q}d\varrho )}^{\frac{1}{q}}\\ \hspace{0.17em}& +\frac{{(\zeta -k)}^2}{\zeta -\omega }{(\underset{0}{\overset{1}{\int }}(1-\varrho )d\varrho )}^{1-\frac{1}{q}}{(\underset{0}{\overset{1}{\int }}(1-\varrho )|\delta { }^{\mathrm{\prime }}(\varrho k+(1-\varrho )\zeta {)|}^{q}d\varrho )}^{\frac{1}{q}}.\end{array}$(2.16)

Since $|\delta { }^{\mathrm{\prime }}{|}^q$ is $(\alpha ,s)$-convex, therefore use Lemma 2.1 in (2.16):

$\begin{array}{cc}\hspace{0.17em}& |L|\le \frac{{(k-\omega )}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{(\underset{0}{\overset{1}{\int }}(1-\varrho )[{\varrho }^{\alpha s}|\delta { }^{\mathrm{\prime }}(k{)|}^q+{(1-\varrho )}^{\alpha s}|\delta { }^{\mathrm{\prime }}(\omega {)|}^q]d\varrho )}^{\frac{1}{q}}\\ \hspace{0.17em}& +\frac{{(\zeta -k)}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{(\underset{0}{\overset{1}{\int }}(1-\varrho )[{\varrho }^{\alpha s}|\delta { }^{\mathrm{\prime }}(k{)|}^q+{(1-\varrho )}^{\alpha s}|\delta { }^{\mathrm{\prime }}(\zeta {)|}^q]d\varrho )}^{\frac{1}{q}}\\ \hspace{0.17em}& \le \frac{{(k-\omega )}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{[\frac{|\delta { }^{\mathrm{\prime }}(k{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\omega {)|}^q}{\alpha s+2}]}^{\frac{1}{q}}\\ \hspace{0.17em}& +\frac{{(\zeta -k)}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{[\frac{|\delta { }^{\mathrm{\prime }}(k{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\zeta {)|}^q}{\alpha s+2}]}^{\frac{1}{q}}.\end{array}$

The following remark states the conclusions of Theorem 8.

Remark 4. Put $\varsigma =1$ in Theorem 8,

(2.17) $\begin{array}{cc}|J|\le & \frac{{(k-\omega )}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{[\frac{|\delta { }^{\mathrm{\prime }}(k{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\omega {)|}^q}{\alpha s+2}]}^{\frac{1}{q}}\\ \hspace{0.17em}& +\frac{{(\zeta -k)}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{[\frac{|\delta { }^{\mathrm{\prime }}(k{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(\zeta {)|}^q}{\alpha s+2}]}^{\frac{1}{q}}.\end{array}$(2.17)

1. Choose $\alpha =1$ in (2.17) to obtain [30, Theorem 7].

2. Choose $\alpha =s=1$ in (2.17) to obtain [29, Theorem 7].

3. Put $\alpha =s=1$ and choose $k=\frac{\omega +\zeta }{2}$ in (2.17), to get [29, Corollary 4].

3 Application to numerical integration

In this section, trapezoidal error are estimated by applying the substitution in our obtained results.

The Trapezoidal Formula [26]: Suppose $b$ is from set of natural numbers, $B:\omega =r_0<r_1<r_2<\dots <r_b=\zeta$ is partition of interval $[\omega ,\zeta ]$ and consider the quadrature formula

(3.1) $\underset{\omega }{\overset{\zeta }{\int }}\delta (q)dq=Y(\delta ,B)+S(\delta ,B),$(3.1)

where

$Y(\delta ,B)=\sum _{a=0}^{b-1}\left(\frac{\delta (r_a)+\delta (r_{a+1})}{2}\right)(r_{a+1}-r_a)$

for the area of trapeziums and $S(\delta ,B)$ denotes the associated approximation error.

Proposition 3.1.

Consider the assumptions of Theorem 6, for each partition $B$ of $[\omega ,\zeta ]$, the trapezoidal error estimate satisfies

(3.2) $\left|S(\delta ,B)\right|\le \sum _{a=0}^{b-1}\frac{{(r_{a+1}-r_a)}^2}{4}\left[\frac{|\delta { }^{\mathrm{\prime }}(r_a)|+|\delta { }^{\mathrm{\prime }}(r_{a+1}|)}{\alpha s+2}+\frac{2|\delta { }^{\mathrm{\prime }}(\frac{r_a+r_{a+1}}{2})|}{(\alpha s+1)(\alpha s+2)}\right].$(3.2)

Proof. In (2.8) choose

$k=\frac{\omega +\zeta }{2}$ to get

(3.3) $\begin{array}{cc}\hspace{0.17em}& \left|\frac{\delta (\omega )+\delta (\zeta )}{2}-\frac{1}{(\zeta -\omega )}\underset{\omega }{\overset{\zeta }{\int }}\delta (c)dc\right|\\ \hspace{0.17em}& \le \frac{(\zeta -\omega )}{4}\left[\frac{|\delta { }^{\mathrm{\prime }}(\zeta )|+|\delta { }^{\mathrm{\prime }}(\omega |)}{\alpha s+2}+\frac{2|\delta { }^{\mathrm{\prime }}(\frac{\zeta +\omega }{2})|}{(\alpha s+1)(\alpha s+2)}\right].\end{array}$(3.3)

Consider (3.3) for $[r_a,r_{a+1}]$ $(a=0,1,\dots ,b-1)$

$\begin{array}{cc}\hspace{0.17em}& \left|\frac{\delta (r_a)+\delta (r_{a+1})}{2}-\frac{1}{(r_{a+1}-r_a)}\underset{r_a}{\overset{r_{a+1}}{\int }}\delta (c)dc\right|\\ \hspace{0.17em}& \le \frac{(r_{a+1}-r_a)}{4}\left[\frac{|\delta { }^{\mathrm{\prime }}(r_{a+1})|+|\delta { }^{\mathrm{\prime }}(r_a)|}{\alpha s+2}+\frac{2|\delta { }^{\mathrm{\prime }}(\frac{r_{a+1}+r_a}{2})|}{(\alpha s+1)(\alpha s+2)}\right].\end{array}$

Hence in (3.1) we have

(3.4) $\begin{array}{cc}\hspace{0.17em}& \left|\underset{\omega }{\overset{\zeta }{\int }}\delta (c)dc-Y(\delta ,B)\right|\\ \hspace{0.17em}& =\left|\sum _{a=0}^{b-1}\underset{r_a}{\overset{r_{a+1}}{\int }}\delta (c)dc-\sum _{a=0}^{b-1}\frac{\delta (r_a)+\delta (r_{a+1})}{2}(r_{a+1}-r_a)\right|\\ \hspace{0.17em}& \le \sum _{a=0}^{b-1}\left|\underset{r_a}{\overset{r_{a+1}}{\int }}\delta (c)dc-\frac{\delta (r_a)+\delta (r_{a+1})}{2}(r_{a+1}-r_a)\right|.\end{array}$(3.4)

$|S(\delta ,B)|\le \sum _{a=0}^{b-1}\frac{{(r_{a+1}-r_a)}^2}{4}\left[\frac{|\delta { }^{\mathrm{\prime }}(r_{a+1})|+|\delta { }^{\mathrm{\prime }}(r_a|)}{\alpha s+2}+\frac{2|\delta { }^{\mathrm{\prime }}(\frac{r_{a+1}+r_a}{2})|}{(\alpha s+1)(\alpha s+2)}\right].$

Remark 5. For each partition $B$ of $[\omega ,\zeta ]$ the following error estimates are satisfied:

(I) If we follow the procedure in the proof of Proposition 3.1 for (2.17), then we obtain:

$\left|S(\delta ,B)\right|\le \sum _{a=0}^{b-1}\frac{{(r_{a+1}-r_a)}^2}{4}(\frac{1}{2}{)}^{1-\frac{1}{q}}$

$\left({[\frac{|\delta { }^{\mathrm{\prime }}(\frac{r_{a+1}+r_a}{2}{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(r_a{)|}^q}{\alpha s+2}]}^{\frac{1}{q}}+{[\frac{|\delta { }^{\mathrm{\prime }}(\frac{r_{a+1}+r_a}{2}{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta { }^{\mathrm{\prime }}(r_{a+1}{)|}^q}{\alpha s+2}]}^{\frac{1}{q}}\right).$

(II) In (2.14) choose

$k=\frac{\omega +\zeta }{2}:$

(3.5) $\begin{array}{cc}\hspace{0.17em}& \left|\frac{\delta (\omega )+\delta (\zeta )}{2}-\frac{1}{(\zeta -\omega )}\underset{\omega }{\overset{\zeta }{\int }}\delta (c)dc\right|\\ \hspace{0.17em}& \le \frac{\zeta -\omega }{4}{(\frac{1}{p+1})}^{\frac{1}{p}}{(\frac{1}{\alpha s+1})}^{\frac{1}{q}}[(|\delta { }^{\mathrm{\prime }}(\omega {)|}^q+|\delta { }^{\mathrm{\prime }}(\frac{\omega +\zeta }{2}{)|}^q{)}^{\frac{1}{q}}+{(|\delta { }^{\mathrm{\prime }}(\zeta {)|}^q+|\delta { }^{\mathrm{\prime }}(\frac{\omega +\zeta }{2}{)|}^q)}^{\frac{1}{q}}].\end{array}$(3.5)

Analogous to [29, Corollary 3], for $s\in [0,1)$, $p>1$ and $0<1-\frac{1}{p}<1$

$\sum _{a=0}^b({\omega }_a+{\zeta }_a{)}^s\le \sum _{a=0}^b({\omega }_a{)}^s+\sum _{a=0}^b({\zeta }_a{)}^s,$

where ${\omega }_1,{\omega }_2,{\omega }_3,\dots ,{\omega }_a>0$ and ${\zeta }_1,{\zeta }_2,{\zeta }_3,\dots ,{\zeta }_a>0$, (3.5) becomes:

(3.6) $\begin{array}{cc}\hspace{0.17em}& \left|\frac{\delta (\omega )+\delta (\zeta )}{2}-\frac{1}{(\zeta -\omega )}\underset{\omega }{\overset{\zeta }{\int }}\delta (c)dc\right|\\ \hspace{0.17em}& \le \frac{\zeta -\omega }{2}{(\frac{1}{p+1})}^{\frac{1}{p}}{(\frac{1}{\alpha s+1})}^{\frac{1}{q}}[|\delta { }^{\mathrm{\prime }}(\omega )|+|\delta { }^{\mathrm{\prime }}(\zeta )|].\end{array}$(3.6)

Follow the procedure in the proof of Proposition 3.1 and use (3.4) to get

(3.7) $\left|S(\delta ,B)\right|\le (\frac{1}{\alpha s+1}{)}^{\frac{1}{q}}(\frac{1}{p+1}{)}^{\frac{1}{p}}\sum _{a=0}^{b-1}\frac{{(r_{a+1}-r_a)}^2}{2}\left(|\delta { }^{\mathrm{\prime }}(r_a)|+|\delta { }^{\mathrm{\prime }}(r_{a+1})|\right).$(3.7)

(III) For $\alpha =s=1$ (3.7) reduces to [29, Proposition 3].

4 Application to probability theory

In this section bounds for expectation value of random variable are obtained.

Suppose $Z$ is a continuous random variable having values in finite interval $[\omega ,\zeta ]$ with the probability density function $\delta :[\omega ,\zeta ]\to [0,1]$ and with the cumulative distribution function

(4.1) $N(z)=P(Z\le z)=\underset{\omega }{\overset{z}{\int }}\delta (r)dr,$(4.1)

where $z\in [\omega ,\zeta ]$, $N(\omega )=0$ and $N(\zeta )=1$. The expectation of Z is:

(4.2) $E(Z)=\underset{\omega }{\overset{\zeta }{\int }}r\delta (r)dr=\zeta -\underset{\omega }{\overset{\zeta }{\int }}N(r)dr.$(4.2)

Theorem 9.

Consider the assumptions of Theorem 6 to get:

(4.3) $\begin{array}{cc}\left|\frac{E(Z)-k}{\zeta -\omega }\right|& \le \frac{{(k-\omega )}^2}{\zeta -\omega }[\frac{|\delta (k)|}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta (\omega )|}{\alpha s+2}]+\\ \hspace{0.17em}& \frac{{(\zeta -k)}^2}{\zeta -\omega }[\frac{|\delta (k)|}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta (\zeta )|}{\alpha s+2}].\end{array}$(4.3)

Proof. Choose $\delta =N$ in (2.8) and apply (4.2) to get (4.3).

Theorem 10.

Consider the assumptions of Theorem 7 to get:

(4.4) $\begin{array}{cc}\left|\frac{E(Z)-k}{\zeta -\omega }\right|& \le {(\frac{1}{p+1})}^{\frac{1}{p}}{(\frac{1}{\alpha s+1})}^{\frac{1}{q}}\\ \hspace{0.17em}& [\frac{{(k-\omega )}^2{(|\delta (\omega {)|}^q+|\delta (k{)|}^q)}^{\frac{1}{q}}+{(\zeta -k)}^2{(|\delta (\zeta {)|}^q+|\delta (k{)|}^q)}^{\frac{1}{q}}}{(\zeta -\omega )}].\end{array}$(4.4)

Proof. Choose $\delta =N$ in (2.14) and apply (4.2) to obtain (4.4).

Theorem 11

Consider the assumptions of Theorem 8:

(4.5) $\begin{array}{cc}\left|\frac{E(Z)-k}{\zeta -\omega }\right|& \le \frac{{(k-\omega )}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{[\frac{|\delta (k{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta (\omega {)|}^q}{\alpha s+2}]}^{\frac{1}{q}}\\ \hspace{0.17em}& +\frac{{(\zeta -k)}^2}{\zeta -\omega }{(\frac{1}{2})}^{1-\frac{1}{q}}{[\frac{|\delta (k{)|}^q}{(\alpha s+1)(\alpha s+2)}+\frac{|\delta (\zeta {)|}^q}{\alpha s+2}]}^{\frac{1}{q}}.\end{array}$(4.5)

Proof. Choose $\delta =N$ in (2.17), then applying (4.2), we obtain (4.5).

5 Conclusion

The paper has shown a vital use of a generalized identity that helps to prove new integral inequalities involving Caputo-Fabrizio fractional integrals for which absolute values of the first derivative is $(\alpha ,s)$-convex function.

Obtained results are graphically elaborated in the examples to strengthen the significance of our study and to help readers comprehend properly. Our results are generalization of the results of [26,30]. Error estimates for the trapezoidal formula are calculated. In addition, some applications of the results to theory of probability are also indulged.

The results of this article could encourage researchers in this subject to conduct more research for other classes of convex functions as well as for other fractional operators.

Acknowledgments

The researchers would like to acknowledge Deanship of Scientific Research, Taif University for funding this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Postdoctoral Fellowship of Asfand Fahad at Zhejiang Normal University, China [YS304023966].