Trapezium-Type Inequalities for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.79534 12.533" width="8.79534pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg>-</span>Fractional Integral via New Exponential-Type Convexity and Their Applications

Kashuri, Artion; Iqbal, Sajid; Butt, Saad Ihsan; Nasir, Jamshed; Nisar, Kottakkaran Sooppy; Abdeljawad, Thabet

doi:https://doi.org/10.1155/2020/8672710

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Volume 2020 | Article ID 8672710 | https://doi.org/10.1155/2020/8672710

Trapezium-Type Inequalities for -Fractional Integral via New Exponential-Type Convexity and Their Applications

Artion Kashuri,¹Sajid Iqbal,²Saad Ihsan Butt,³Jamshed Nasir,⁴Kottakkaran Sooppy Nisar,⁵and Thabet Abdeljawad^6,7,8

Academic Editor: Basil K. Papadopoulos

Received27 May 2020

Accepted24 Aug 2020

Published16 Oct 2020

Abstract

In this paper, the authors investigated the concept of -exponential-type convex functions and their algebraic properties. New generalizations of Hermite–Hadamard-type inequality for the -exponential-type convex function and for the products of two -exponential-type convex functions and are proved. Many refinements of the (H–H) inequality via -exponential-type convex are obtained. Finally, several new bounds for special means and new error estimates for the trapezoidal and midpoint formula are provided as well. The ideas and techniques of this paper may stimulate further research in different areas of pure and applied sciences.

1. Introduction

Theory of convexity also played a significant role in the development of theory of inequalities. Many famously known results in inequalities theory can be obtained using the convexity property of the functions. Hermite–Hadamard’s double inequality is one of the most intensively studied results involving convex functions. This result provides us necessary and sufficient condition for a function to be convex. It is also known as classical equation of H–H inequality.

The Hermite–Hadamard inequality assert that if a function is convex in for and , then

Interested readers can refer to [1–29].

Definition 1 (see [30]). A function is said to be -convex in the second sense for a real number or belongs to the class of , ifholds for all and .
A -convex function was introduced in Breckner’s article in [30], and a number of properties and connections with -convexity in the first sense are discussed in [10]. Usually, convexity means for -convexity when . Dragomir et al. proved a variant of Hadamard’s inequality in [5], which holds for -convex functions in the second sense. In the last decade, many mathematicians added the rich literature in the field of mathematical inequalities involving fractional calculus (see [8, 9, 13, 15, 22, 25, 28, 31]).
Toader introduced the class of -convex functions in [26].

Definition 2 (see [26]). A function , , is said to be -convex, where , ifholds and . Otherwise, is -concave if is -convex.
In a recent paper, Eftekhari [6] defined the class of -convex functions in the second sense as follows.

Definition 3. A function is said to be -convex for some fixed real numbers ifholds and .
Fractional integral inequalities are useful to find the uniqueness of solutions for certain fractional partial differential equations (see [19, 32]).
Let . Then, Riemann–Liouville fractional integrals of order with are defined as follows:For further details, one may see [14, 16, 18].
In [7, 17], there is given definition of fractional Riemann–Liouville integrals.
Let . Then, fractional integrals of order with are defined as follows:where is the gamma function defined asWe can notice thatBy choosing , the above fractional integrals yield Riemann–Liouville integrals.
Also that, the incomplete gamma function is defined for and by integralThe gamma function is defined for by integralMotivated by above results and literatures, we will give first in Section 2 the concept of -exponential-type convex function, and we will study some of their algebraic properties. In Section 3, we will prove new generalizations of Hermite–Hadamard-type inequality for the -exponential-type convex function and for the products of two -exponential-type convex functions and . In Section 4, we will obtain some refinements of the (H–H) inequality for functions whose first derivative in absolute value at certain power is -exponential-type convex. In Section 5, some new bounds for special means and error estimates for the trapezoidal and midpoint formula will be provided. In Section 6, a briefly conclusion will be given as well.

2. Some Algebraic Properties of -Exponential-Type Convex Functions

In this section, we will introduce a new definition, called -exponential-type convex function, and we will study some basic algebraic properties of it.

Definition 4. A nonnegative function is said -exponential-type convex for some fixed ifholds for all and .

Remark 1. For , we get exponential- type convexity given by İşcan in [11].

Remark 2. -exponential-type convex functions for some fixed and have the range .

Proof. Let be arbitrary for some fixed and . Using the Definition 4 for , we have

Lemma 1. For all and for some fixed and , the following inequalities and hold.

Proof. The proof is evident.

Proposition 1. Every nonnegative -convex function is -exponential-type convex function for some fixed and .

Proof. By using Lemma 1, for some fixed and , we have

Theorem 1. Let . If and are -exponential-type convex functions for some fixed , then the following holds:(1) is -exponential-type convex function.(2)For nonnegative real number , is -exponential-type convex function.

Proof. By Definition 4, for some fixed , the proof is obvious.

Theorem 2. Let be -convex function for and some fixed and is nondecreasing and -exponential-type convex function for some fixed . Then, for the same fixed numbers , the function is -exponential-type convex.

Proof. For all and and for some fixed numbers , we have

Theorem 3. Let be an arbitrary family of -exponential-type convex functions for the same fixed and let . If , then is an interval and is -exponential-type convex function on .

Proof. For all and and for the same fixed numbers , we haveThis means simultaneously that is an interval, and is -exponential-type convex function on .

Theorem 4. If the function is -exponential-type convex for some fixed , then is bounded on .

Proof. Let and be an arbitrary point for some fixed . Then, there exists such that . Thus, since and for some fixed , we haveWe have shown that is bounded above from real number . Interested reader can also prove the fact that is bounded below using the same idea as in Theorem 4 in [11].

3. New Generalizations of (H–H)-Type Inequality

In this section, we will establish some new generalizations of Hermite–Hadamard-type inequality for the -exponential-type convex function and for the products of two -exponential-type convex functions and .

Theorem 5. Let be -exponential-type convex function for some fixed and . If , then

Proof. Let denote, respectively,Using -exponential-type convexity of , we haveNow, integrating on both sides in the last inequality with respect to over , we getwhich completes the left-side inequality. For the right-side inequality, using -exponential-type convexity of , we obtainwhich gives the right-side inequality.

Corollary 1. By taking in Theorem 5, we get (Theorem 5, [11]).

Theorem 6. Assume that are, respectively, and -exponential-type convex functions for the same fixed and for some fixed , where and . If are synchronous functions and , thenwhere

Proof. Let us denote for all . Using the property of the and -exponential-type convex functions and , respectively, we haveMultiplying above inequalities on both sides, we getApplying Chebyshev integral inequality (see [23]), we obtainChanging the variable of integration, we getwhich completes the left-side inequality. For the right-side inequality, integrating on both sides of the inequality (25) with respect to over , we havewhich give the right-side inequality.

In this section, we will obtain some refinements of the (H–H) inequality via -exponential-type convex functions.

Lemma 2. Suppose and a mapping is differentiable on with . If , then the following equality for fractional integral holds:

where and is the Euler Gamma function.

Proof. Using the integrating by parts, we havewhich completes the proof.

Lemma 3. Suppose and a mapping is differentiable on with . If then the following equality for fractional integral holds:

Proof. Using the integrating by parts, we havewhereCombining equations (33) and (34) in (32) and multiplying by , we get (31), which completes the proof.

Theorem 7. Suppose and a mapping is differentiable on with If is -exponential-type convex on for and , then for some fixed , then the following inequality for fractional integral holds:where

Proof. From Lemma 2, Hölder’s inequality and -exponential-type convexity of , we havewhich completes the proof.

Theorem 8. Suppose and a mapping is differentiable on with If is -exponential-type convex on for , then for some fixed , then the following inequality for fractional integral holds:

Proof. From Lemma 2, power mean inequality and -exponential-type convexity of , we havewhich completes the proof.

Theorem 9. Suppose and a mapping is differentiable on with If is -exponential-type convex on for and , then for some fixed , then the following inequality for fractional integral holds:

Proof. From Lemma 3, Hölder’s inequality and -exponential-type convexity of , we havewhich completes the proof.

Theorem 10. Suppose and a mapping is differentiable on with If is -exponential-type convex on for , then for some fixed , then the following inequality for fractional integral holds:

Proof. From Lemma 3, power mean inequality and -exponential-type convexity of , we havewhich completes the proof.

5. Applications

Let consider the following two special means for different positive-real numbers .(1)The arithmetic mean:(2)The generalized -mean:

Dragomir et al. [5] have proved that, for , where , the function is -convex function. Then, from Proposition 1, it is also -exponential convex function for some fixed .

Using Section 4, we have the following interesting results:

Proposition 2. Let , , and such that . Then, for some fixed , where , we have

Proof. Consider the -exponential convex function , and using Theorem 7 for , we obtain the required result.

Proposition 3. Let , and . Then, for some fixed , where , we get

Proof. Consider the -exponential convex function , and using Theorem 8 for , the result is obvious.

Proposition 4. Let , and , such that . Then, for some fixed , where , we obtain

Proof. Consider the -exponential convex function , and using Theorem 9 for , the result is evident.

Proposition 5. Let , and . Then, for some fixed , where , we have

Proof. Consider the -exponential convex function , and using Theorem 10 for , we obtain the required result.
At the end, let consider some applications of the integral inequalities obtained above, to find new bounds for the trapezoidal and midpoint formula.
For , let is a partition of .
We denotewhere is the remainder term and for .
Using above notations, we are in position to prove the following error estimations.

Proposition 6. Suppose a mapping is differentiable on with If is -exponential-type convex on for and , then for some fixed , the remainder term satisfies the following error estimation:

Proof. Using Theorem 7 on subinterval of closed interval , for all and , we getSumming inequality (52) over from 0 to and using the property of modulus, we obtain the desired inequality (51).

Proposition 7. Suppose a mapping is differentiable on with If is -exponential-type convex on for , then for some fixed , the remainder term satisfies the following error estimation:

Proof. Apply the same technique as in Proposition 6 but using Theorem 8.

6. Conclusion

In this article, the authors showed new generalizations of trapezium-type inequality for the new class of functions, the so-called -exponential-type convex function and for the products of two -exponential-type convex functions and . We have obtained refinements of the (H–H) inequality for functions using -exponential-type convex and founded new bounds for special means and for the error estimates for the trapezoidal and midpoint formula. We hope that current work will attract the attention of researchers working in mathematical analysis, fractional calculus, quantum calculus, postquantum calculus, and other related fields.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2020 Artion Kashuri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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