Inequalities for a Unified Integral Operator for Strongly <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M1" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 42.2278 15.2545" width="42.2278pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g><g transform="matrix(.017,0,0,-0.017,15.686,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,22.474,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g><g transform="matrix(.017,0,0,-0.017,36.015,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Convex Function and Related Results in Fractional Calculus

Jung, Chahn Yong; Farid, Ghulam; Mahreen, Kahkashan; Shim, Soo Hak

doi:https://doi.org/10.1155/2021/6610836

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Volume 2021 | Article ID 6610836 | https://doi.org/10.1155/2021/6610836

Inequalities for a Unified Integral Operator for Strongly -Convex Function and Related Results in Fractional Calculus

Chahn Yong Jung,¹Ghulam Farid,²Kahkashan Mahreen,²and Soo Hak Shim³

Academic Editor: Shuangping Tao

Received06 Dec 2020

Accepted11 Feb 2021

Published16 Mar 2021

Abstract

In this paper, we study integral inequalities which will provide refinements of bounds of unified integral operators established for convex and -convex functions. A new definition of function, namely, strongly -convex function is applied in different forms and an extended Mittag-Leffler function is utilized to get the required results. Moreover, the obtained results in special cases give refinements of fractional integral inequalities published in this decade.

1. Introduction

Fractional integral operators are very useful and are extensively utilized in mathematics, physics, engineering, and many other subjects. The researchers have introduced variety of fractional integral operators, most of them generalize classical Riemann-Liouville integrals. In recent study of mathematical inequalities, fractional integral operators are playing an important role. Many classical inequalities have been studied for fractional integral operators of different kinds. For example, one can see recent articles dealing with fractional inequalities in [1–15] and in the references therein. Our aim in this article is to study an integral operator for a newly defined strongly -convex function, which has direct consequences in fractional integral operators. Next, we give some definitions of generalized fractional integral operators.

Definition 1 (see [16]). Let be an integrable function. Also let be an increasing and positive function on , having a continuous derivative on . The left-sided and right-sided fractional integrals of a function with respect to another function on of order where are defined bywhere is the gamma function.

A -analogue of the above definition is defined as follows.

Definition 2 (see [17]). Let be an integrable function. Also let be an increasing and positive function on , having a continuous derivative on . The left-sided and right-sided fractional integrals of a function with respect to another function on of order are defined bywhere is given by

A well-known function named Mittag-Leffler function is defined by [18]where and

One can see the references [19–22] to study the Mittag-Leffler function and its generalizations.

Fractional integral operator containing an extended Mittag-Leffler function is defined as follows.

Definition 3 (see [1]). Let , , with , and . Let and Then, the generalized fractional integral operators and are defined bywhereis the extended generalized Mittag-Leffler function.

Recently, Farid defined the following unified integral operator which unifies several kinds of fractional integrals. Also, new kinds of fractional integrals can be generated from it.

Definition 4 (see [23]). Let, , be the functions such thatbe positive and, andbe differentiable and strictly increasing. Also letbe an increasing function onand , , , and. Then, for, the left and right integral operators are defined bywhere .

The following property of the kernel used in the unified integral operator will be used in sequel.

P: Let and be increasing functions. Then, for , , the kernel satisfies the following inequality:

This can be obtained from following two straightforward inequalities:The reverse of inequality (9) holds when and are of opposite monotonicity. For suitable settings of functions , and certain values of parameters included in the Mittag-Leffler function (6), many well-known fractional integral operators can be reproduced, see ([11], Remarks 6 and 7).

The objective of this paper is to obtain bounds of unified integral operators using strongly -convexity. It is defined as follows [24].

Definition 5. A function is said to be strongly -convex, where ifholds for all and

Remark 6. (i) If we put , then (11) gives the definition of -convex functions
(ii) If we put , then (11) gives the definition of strongly -convex functions
(iii) If we put , then (11) gives the definition of -convex function
(iv) If we put , then (11) gives the definition of convex function
(v) If we put , then (11) gives the definition of star-shaped function
(vi) If we put and , then (11) gives the definition of convex function

In the upcoming section, bounds of unified integral operators are established by using strongly -convexity. These bounds provide general formulas to get bounds of many fractional integral operators along with described in [11], Remarks 6 and 7]. Many mathematicians worked on new types of Hadamard inequalities using convex functions, see [9, 25–27]. We also established general Hadamard type inequality by applying Lemma 10 which further produces various inequalities of Hadamard type for fractional integrals.

2. Main Results

Theorem 7. Let be a positive integrable strongly -convex function with , . Let be differentiable and strictly increasing function, also let be an increasing function on . Then, unified integral operators (7) and (8) satisfy the following inequality:where are Riemann-Liouville fractional integrals.

Proof. According to property P, the following inequalities hold:

Strongly -convex function satisfies the following inequalities:

Multiplying (13) with (15) and integrating over , one can obtain

From the above inequality, one can obtain

Now, adopting the same procedure as we did for (13) and (15), the following inequality can be obtained from (14) and (16):

From inequalities (18) and (19), inequality (12) can be obtained.

Corollary 8. If we consider and in (12), then the following inequality holds for -convex functions:

Remark 9. (i) If we consider in (12), ([28], Theorem 7) is obtained, otherwise the refinement is obtained
(ii) If we consider in (12), ([12], Theorem 4]) is obtained
(iii) If we consider and , in (12), then ([24], Theorem 4]) is obtained
(iv) If we consider , in (12), ([11], Theorem 8]) is obtained
(v) If we consider , and in (12), then ([10], Theorem 7]) can be obtained
(vi) If we consider in the result of (v), then ([10], Corollary 8]) can be obtained
(vii) If we consider , , , and in (12), ([6], Theorem 7]) is obtained
(viii) If we consider in the result of (vi), ([6], Corollary 8]) is obtained
(ix) If we consider , , , , and , then ([4], Theorem 7]) can be obtained
(x) If we consider in the result of (xi), then ([4], Corollary 8]) can be obtained
(xi) If we consider , , , and and in (12), then ([5], Theorem 7]) is obtained
(xii) If we consider in the result of (xi), ([5], Corollary 8]) can be obtained
(xiii) If we consider and or in the result of (xii), ([5], Corollary 12]) can be obtained
(xiv) If we consider and in the result of (xii), ([5], Corollary 15]) can be obtained

To prove the next result, we need the following lemma.

Lemma 10 (see [24]). Letbe strongly-convex function,. Ifandwith, then the following inequality holds:

The following result provides upper and lower bounds of sum of operators (7) and (8) in the form of a Hadamard inequality.

Theorem 11. With the assumptions of Theorem 7 in addition if with , then we have

Proof. According to P, the following inequalities hold:Using the definition of strongly -convex function , the following inequality holds:

Multiplying (23) and (25) and integrating the resulting inequality over , one can obtain

From the above inequality, one can obtain

Adopting the same pattern of simplification as we did for (23) and (25), the following inequality can be obtained from (25) and (24)

From (27) and (28), the following inequality can be obtained:

Multiplying both sides of (21) by , and integrating over , we have

From the above inequality, one can obtain

Similarly multiplying both sides of (21) by , and integrating over , we have

From inequalities (29), (31), and (32), inequality (22) can be obtained.

Corollary 12. If we consider and in (22), then the following inequality holds for -convex function:

Remark 13. (i) If we consider in (22), ([29], Theorem 5]) is obtained
(ii) If we consider and , in (22), then ([24], Theorem 6]) is obtained
(iii) If we consider , in (22), ([11], Theorem 22]) is obtained
(iv) If we consider and in (22), , and in (22), ([6], Theorem 14]) is obtained
(v) If we consider in the result of (iv), ([6], Corollary 15]) is obtained
(vi) If we consider , , =(1,1), and in (22), then ([4], Theorem 14]) can be obtained
(vii) If we consider in the result of (vi), then ([4], Corollary 6]) can be obtained
(viii) If we consider , , , and in (22), ([5], Theorem 14]) can be obtained
(ix) If we consider in the result of (viii), ([5], Corollary 6]) can be obtained

Theorem 14. Let be a differentiable function. If is strongly -convex with and be differentiable and strictly increasing function, also let be an increasing function on . Then, for unified integral operators, the following inequality holds:where

Proof. Using the definition of strongly -convexity for , the following inequality is valid:

The inequality (36) can be written as follows:

We consider the second inequality of (37)

Multiplying (13) and (38) and integrating over , we can obtain

From the above inequality, one can obtain

If we consider the left hand side from inequality (37) and adopt the same pattern as did for the right hand side inequality, then

From (40) and (41), the following inequality is obtained:

Now, using again the definition of strongly -convexity for , the following inequality is valid:

On the same procedure as we did for (13) and (36), one can obtain following inequality from (14) and (43):

From (42) and (44), inequality (34) can be obtained.

Corollary 15. If we consider and in (34), then the following inequality holds for the -convex function:

Remark 16. (i) If we consider in (34), ([28], Theorem 14]) is obtained
(ii) If we consider in (34), ([29], Theorem 6]) is obtained
(iii) If we consider and , in (34), then ([24], Theorem 5]) is obtained
(iv) If we consider , in (34), then ([11], Theorem 25]) is obtained
(v) If we consider , and in (34), then ([10], Theorem 11]) can be obtained
(vi) If we consider in the result of (v), then ([10], Corollary 12]) can be obtained
(vii) If we consider , and in (34), then ([6], Theorem 11]) is obtained
(viii) If we consider in the result of (vii), then ([6], Corollary 12]) is obtained
(ix) If we consider , , =(1,1), , and in (34), then ([4], Theorem 11]) can be obtained
(x) If we consider in the result of (ix), then ([4], Corollary 4]) can be obtained
(xi) If we consider and , in the result of (ix), then ([4], Corollary 5]) can be obtained
(xii) If we consider , , , and and in (34), then ([5], Theorem 11]) is obtained
(xiii) If we consider in the result of (xii), then ([5], Corollary 5]) can be obtained

3. Concluding Remarks

In this paper, integral inequalities for strongly -convex functions are given by utilizing unified integral operators. Results for -convex functions are presented in particular in the form of corollaries. In the form of remarks, many published results are highlighted as consequences of presented results. The reader can get integral and fractional integral inequalities for convex functions, strongly convex functions, -convex functions, -convex functions, and strongly -convex functions.

Data Availability

There is no any data required for this paper

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Chahn Yong Jung 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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