Further on Inequalities for <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 68.8967 15.2545" width="68.8967pt"><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="g113-105" d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"/></g><g transform="matrix(.017,0,0,-0.017,35.165,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,49.072,0)"><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,-0.017,62.613,0)"><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 Functions via <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="M2" 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 Operators

Yan, Tao; Farid, Ghulam; Demirel, Ayşe Kübra; Nonlaopon, Kamsing

doi:https://doi.org/10.1155/2022/9135608

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Volume 2022 | Article ID 9135608 | https://doi.org/10.1155/2022/9135608

Further on Inequalities for -Convex Functions via -Fractional Integral Operators

Tao Yan,¹Ghulam Farid,²Ayşe Kübra Demirel,³and Kamsing Nonlaopon⁴

Academic Editor: Sun Young Cho

Received18 Dec 2021

Accepted17 Feb 2022

Published06 May 2022

Abstract

The purpose of this article is to demonstrate new generalized -fractional Hadamard and Fejér–Hadamard integral inequalities for -convex functions. To prove these inequalities, -fractional integral operators including the generalization of the Mittag–Leffler function are used. The results presented in this article can be considered an important advancement of previously published inequalities.

1. Introduction

Theory of convexity offers an effective and charming area of research and is also a theory that featured prominently and surprisingly in distinct disciplines such as mathematical analysis, optimization, economics, finance, engineering and game theory. Convexity theory is very closely related with the theory of inequalities. Many inequalities well known in the literature are direct applications of the properties of convex functions. The usage of fractional integral operators for getting the generalized types of classic inequalities has become an important method in advanced mathematical studies of inequalities.

One of the convexity theory studies in the literature belongs to Gao et al. [1]. They presented a new type of functions called -polynomial harmonically exponential type convex, and specified some of their algebraic features. Mehrez and Agarwal [2] established new type of integral inequalities for convex functions and indicated new inequalities for some special and -special functions. Tariq [3] defined the concept of –harmonic exponential type convex functions. Also, they investigated some integral inequalities in the form of applications for some means. Another study on convexity theory and inequalities was presented by Butt et al. [4]. They presented the notion of –polynomial -harmonic exponential type convex functions and demonstrated various new integral inequalities. Srivastava et al. [5] obtained a new class of the bi-close-to-convex functions described in the open unit disk by using the Borel distribution series of the Mittag–Leffler type. Also, the authors demonstrated the Fekete–Szego type inequalities via the bi-close-to-convex function class.

Fractional calculus, which is the study of integrals and derivatives of fractional order, has expanded significantly over the late nineteenth century. It ranges from chemical, viscoelasticity, and statistical physics to electrical and mechanical engineering. The fundamental working doctrine of fractional analysis is to present new fractional derivative and integral operators, and to analyze the benefits of these operators through the instrument of modeling studies, and collations. Integral operators, which form a significant part of fractional calculus, are now resources of many fields such as inequality theory, engineering, statistics, mathematical biology, and modeling, which take advantage of fractional analysis. Many inequalities have been generalized through the instrument of fractional integral operators and provide construction of new approximations.

One of the fractional calculus studies in the literature belongs to Abdeljawad et al. [6]. They obtained generalized Hermite–Hadamard type inequalities and generalized Simpson type inequalities for -convex functions with the help of local fractional integration. Akdemir et al. [7] used generalized fractional integral operators. By using these operators, they proved new and general variants of Chebyshev’s inequality. Butt et al. [8] established a general integral identity to acquire new integral inequalities of several Hadamard types. For this purpose, they used a new version of the Atangana–Baleanu integral operator. Khan et al. [9] explored two fractional integral operators related to Fox -function owing to Saxena and Kumbhat. They proved series expansion of the images of the -series with the help of these fractional operators. Another study to -fractional integrals was presented by Qi et al. [10]. They constructed some generalized fractional integral inequalities of the Hermite–Hadamard type via -convex functions. Also, they demonstrated that one can get and expand some Riemann–Liouville fractional integral inequalities and classical integral inequalities of Hermite–Hadamard’s type. Tunc et al. [11] presented the generalized -fractional integrals of a function with respect to the another function that generalizes many several types of fractional integrals. Also, they studied trapezoid inequalities for the functions whose derivatives in absolute value are convex. Önalan et al. [12] proved many Hermite–Hadamard type integral inequalities for functions whose absolute values of the second derivatives are -convex and -concave using fractional integral operators with the Mittag–Leffler kernel. Zhu et al. [13] explored a weighted integral identity of Simpson-like type. Relying on this identity, they obtained some estimation-type results connected with the weighted Simpson-like type integral inequalities for the first order differentiable functions. Srivastava et al. [14] established the homogeneous -shift operator and the homogeneous -difference operator. Based on these operators, they searched generalized Cauchy and Hahn polynomials.

2. Preliminaries

Now let us define some important functions.

Definition 1 (see [15]). A function is called a convex function, ifholds for all and .

Definition 2 (see [16]). The function ,, is called the-convex function, ifholds for all and .

Definition 3 (see [17]). A functionis said to be-convex, ifholds for all , and .

Definition 4 (see [18]). A function is said to be-convex in the second sense, ifholds for all , and .

Definition 5 (see [19]). Let be an interval includingand letbe a nonnegative function. Then the functionis called the-convex function, ifholds for all , and .

Definition 6 (see [20]). Let be an interval includingand letbe a nonnegative function. Then the functionis called the-convex function, ifholds for all , and .

Remark 1. (i)By takingandin (6), we obtain the definition of convex function (1).(ii)By takingin (6), we obtain the definition of-convex function (2).(iii)By takingandin (6), we obtain the definition of-convex function (3).(iv)By takingandin (6), we obtain the definition of-convex function in the second sense (4).(v)By takingin(6), we obtain the definition of-convex function (5).(vi)By takingin (6), we obtain the definition of-function described by Dragomir et al. in [21].

Now let us represent some definitions of fractional integral operators that will form the basis for this article.

Definition 7 (see [22]). Let with,, and. Let,. In that case, the generalized fractional operators are defined bywhereis generalized extended Mittag–Leffler function, and is the expansion of beta function described as below:where .

Definition 8 (see [23]). Let with , be the functions, be positive, and be differentiable and strictly increasing. Let be an increasing on , with , and . In that case, for , the fractional operators are described by

Definition 9 (see [23]). Let with , be the functions such that be positive and and be differentiable and strictly increasing. Let , , and . In that case, for , the united operators are described by

Recently, Yue et al. defined generalized -fractional operators including a further extension of Mittag–Leffler function in [24] as noted below:

Definition 10. Letwith; be the functions such thatbe positive andandbe differentiable and strictly increasing. Letand,with,and. In that case, for, the right-left generalized-fractional operatorsandare defined by

The following inequality is the admitted Hadamard inequality.

Theorem 1. Letwith, be a convex function. In that case, the below inequality occurs:

Theorem 2. Letbe a convex andbe nonnegative and symmetric in respect ofand integrable. In that case, the below inequality occurs:

This inequality in [25] presented by Fejér is known as a weighted type of Hadamard’s inequality.

Many authors have been established several refinements and extensions of the Hadamard and the Fejér–Hadamard inequalities for various fractional integral operators (for details see, [2, 7, 11, 16, 17, 19–21, 26–34] and references therein). This article aims to derive the Hadamard and Fejér–Hadamard inequalities about generalized -fractional integrals involving Mittag–Leffler functions via -convex functions. In the upcoming section, we will utilize -fractional integral operators and -convexity to prove the two versions of the Hadamard inequality and the Fejér–Hadamard inequality.

3. The -Fractional Inequalities of Hadamard and Fejér–Hadamard Type

In this section, we first describe the below generalized -fractional Hadamard’s inequality.

Theorem 3. Letis nonnegative, nonzero and integrable function and,, be the functions such thatandbe positive andbe differentiable and strictly increasing. Ifis-convex, the below inequalities for-fractional operators (12) and (13) occur:where for all .

Proof. Since is -convex on , for all , we haveSetting and in above inequality, we haveMultiplying both sides of (18) by , then integrating over , we haveBy specifying and in (19), we haveBy usage -fractional operators (12) and (13), the first side of (16) is achieved.
To evidence the second side of (16), once again -convexity of over , for , we achieveMultiplying both sides of (21) by , next integrating over , we achieveSetting and in (22), in that case by utilizing -fractional operators (12) and (13), the second side of (16) is achieved.

Corollary 1. By usage (16), anymore-fractional inequalities are offered as noted below:(i)By choosingand, we obtain(ii)By choosingand, we obtain(iii)By settingand, we obtain(iv)By choosingand, we obtain(v)By settingand, we get(vi)By setting,and, we get

Remark 2. The above-fractional inequalities are farther in line with already known conclusions as noted below: (i) By choosingin Corollary 1(v), an inequality for extended generalized fractional integrals is acquired. (ii) By choosingandin Corollary 1(v), Theorem 2.1 of [28] is acquired. (iii) By choosing, andin Corollary 1(v), Theorem 2.1 of [27] is acquired. (iv) By choosingin Corollary 1(v), Theorem 2.1 of [20] is acquired.

Remark 3. (i) By choosingandin Remark 1(iii), an inequality for extended generalized fractional integrals is acquired. (ii) By choosingandin Remark 1(iii), Theorem 2 of [29] is acquired. (iii) By choosingin Remark 1(iv), Corollary 2.2 of [20] is acquired.

The below lemma is beneficial to offer the Fejér–Hadamard’s inequality for generalized -fractional integrals.

Lemma 1. Letwith, be the functions such thatandpositive andbe differentiable and strictly increasing. If, in that case for generalized-fractional operators (11) and (12), we getfor all .

Proof. By description of generalized -fractional operators (12) and (13), we getSetting in the above equation and using , we haveThis impliesBy adding on both sides of (32), we haveFrom equations (32) and (33), the result can be obtained.

The first type of Fejér–Hadamard inequality is endued through generalized -fractional integrals as noted below:

Theorem 4. Letbe nonnegative, nonzero, and integrable function and,, be the functions such thatandbe positive andbe differentiable and strictly increasing,is a nonnegative and integrable function. Ifis-convex and, in that case the below inequalities for generalized-fractional operators (12) and (13) occur:where for all .

Proof. We demonstrate the claim as follows:
Multiplying both sides of (18) by and then integrating over , we haveBy specifying and , that is , in (35), then using , we haveThis impliesUsing Lemma 1 in the above inequality, we have the first side of (34).
To demonstrate second side of (34), multiplying both parts of (21) by and then integrating over , we haveSetting and , then using in (38), we haveBy usage Lemma 1 in the above inequality, we have the second side of (34).

Corollary 2. By using (34), some more-fractional inequalities are offered as noted below:(i)By choosingand, we obtain(ii)By choosingand, we obtain(iii)By choosingand, we obtain(iv)By choosingand, we obtain(v)By choosingand, we obtain(vi)By choosing,and, we obtain(vii)By choosingand, we obtain

Remark 4. The above-fractional inequalities are farther in line with foreknown conclusions as noted below: (i) By choosingin Corollary 2(vi), Theorem 2.2 of [27] is acquired. (ii) By choosingin in Corollary 2(vii), Theorem 2.5 of [28] is acquired. (iii) By choosing,andin Corollary 2(v), an inequality for-convex functions via Riemann–Liouville integrals is acquired. (iv) By choosingandin in Corollary 2(vi), an inequality for extended generalized fractional integrals is acquired. (v) By choosingandin in Corollary 2(vi), Theorem 4 of [26] is acquired. (vi) By choosingin in Corollary 3.2 (vii), Theorem 3.1 of [27] is acquired.

In the subsequent theorem, we offer another type of Hadamard’s inequality.

Theorem 5. Letis nonnegative, nonzero and integrable function and,, be the functions such thatandbe positive andbe differentiable and strictly increasing. Ifis-convex, in that case for generalized-fractional operators (12) and (13), we acquirewhere for all .

Proof. Setting and in (3.2), we haveMultiplying both parts of (48) by and then integrating over , we haveBy taking and in (49), in that case by usage -fractional operators (2.12) and (2.13), the first side of (47) is acquired.
To demonstrate the second side of (47), once again -convexity of over , for , we getMultiplying both sides of (50) by , then integrating over , we acquireChoosing and in (51), in that case by usage -fractional operators (12) and (13), the second side of (47) is acquired.

Corollary 3. By using (47), anymore-fractional inequalities are offered as noted below:(i)By choosingand, we have(ii)By choosingand, we have(iii)By choosingand, we acquire(iv)By choosingand, we have(v)By choosingand, we have(vi)By choosing,and, we have

Remark 5. The above-fractional inequalities are farther in line with foreknown conclusions as noted below: (i) By choosingin Corollary 3(v), an inequality for extended generalized fractional integrals is acquired. (ii) By choosingandin Corollary 3(v), Theorem 2.2 of [28] is acquired.

The second type of the Fejér–Hadamard’s inequality for generalized -fractional integrals is dedicated as noted below:

Theorem 6. Letis nonnegative, nonzero and integrable function and,, be the functions such thatandbe positive andbe differentiable and strictly increasing,is a nonnegative and integrable function. Ifis-convex and, in that case the below inequalities for generalized-fractional operators (12) and (13) occur:where for all .

Proof. We demonstrate the claim as follows:
Multiplying (48) by and then integrating over , we haveBy setting and , that is, , in (59), in that case by usage and -fractional integral operators (12) and (13), the first side of (58) is acquired.
To demonstrate the second side of (58), multiplying both parts of (50) byand then integrating over , we haveSetting and in (59), then by using and -fractional integral operators (12) and (13), the second inequality of (58) is obtained.

Corollary 4. By using (58), some more-fractional inequalities are offered as noted below:(i)By choosingand, we obtain(ii)By choosingand, we obtain(iii)By choosingand, we obtain(iv)By choosingand, we obtain(v)By choosingand, we get(vi)By choosingand, we get

Remark 6. Those as mentioned above-fractional inequalities are farther in line with foreknown conclusions as by choosingin Corollary 4(v), an inequality for extended generalized fractional integrals is obtained.

Data Availability

There are no data required for this paper

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All the authors made equal contributions.

Acknowledgments

Science & Technology Bureau of ChengDu 2020-YF09-00005-SN supported by Sichuan Science and Technology program 2021YFH0107 Erasmus+ SHYFTE Project 598649-EPP-1-2018-1-FR-EPPKA2-CBHE-JP.

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Copyright © 2022 Tao Yan 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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