Hadamard-Type <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 Inequalities for Exponentially <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="M2" 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

Jung, Chahn Yong; Farid, Ghulam; Bibi, Sidra; Nisar, Kottakkaran Sooppy; Kang, Shin Min

doi:https://doi.org/10.1155/2020/6815056

Journal of Mathematics

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Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 6815056 | https://doi.org/10.1155/2020/6815056

Hadamard-Type -Fractional Integral Inequalities for Exponentially -Convex Functions

Chahn Yong Jung,¹Ghulam Farid,²Sidra Bibi,³Kottakkaran Sooppy Nisar,⁴and Shin Min Kang^5,6

Academic Editor: Nan-Jing Huang

Received29 Jun 2020

Accepted29 Aug 2020

Published24 Oct 2020

Abstract

The class of exponentially -convex functions has been discovered to unify different kinds of convexities. This paper finds new Riemann–Liouville fractional Hadamard-type inequalities for this generalized class of convex functions. These results further have their consequences which are presented in the form of corollaries. Moreover, some known results are identified in the form of remarks.

1. Introduction

Fractional calculus has opened a new era in the theoretic and application point of view. Many subjects of science and engineering get wonderful advancements in modeling and solving complex systems with the help of fractional integral and derivative operators. Besides making advancements in other fields, fractional integral and derivative operators have been proved very useful in generalizing and extending mathematical inequalities. In this paper, we will extend the Hadamard-type inequalities for Riemann–Liouville (RL) fractional integral operators.

Definition 1. Let . The left and right RL fractional integrals and of order , , of are defined byrespectively, where is the Euler’s gamma function and .
The -analogue RL fractional integral operators are studied by Mubeen and Habibullah in [1].

Definition 2. Let . The left and right RL -fractional integrals and of order , of are defined bywhere is the -gamma function and .
Fractional integral inequalities are generalizations of classical inequalities which play a key role in the solutions and their uniqueness in fractional boundary value problems. The Hadamard inequality is one of the classical inequalities for convex functions, and it has been studied by a lot of researchers for different kinds of fractional integral and derivative operators, see [2–11] and references therein. It is stated in the undermentioned theorem.

Theorem 1 (see [11]). If is a convex function on the interval of real numbers and with then

Definition 3. A function , where is an interval in , is said to be convex function ifholds for all and .
The Hadamard inequality is the geometric interpretation of convex functions which has been analyzed by many researchers for fractional integral and differentiation operators. The objective of this paper is to obtain -fractional integral inequalities for a generalized class of convex functions, namely, exponentially -convex functions. Convex functions proved very useful for the establishment of very known and vital inequalities. An important and significant generalization of convex functions is exponentially -convex functions. It is defined by He et al. in [12] as follows.

Definition 4. Let be an interval containing and let be a nonnegative function. A function is called an exponentially -convex function, if is nonnegative, and for all , , , and , one has

Remark 1. By selecting a suitable function and a particular value of parameters and , the above definition produces the functions comprised in the following remarks:(i)By setting , -convex function [4] can be obtained(ii)By setting and , -convex function [13] can be obtained(iii)By setting and , -convex function [14] can be obtained(iv)By setting , and , -convex function [15] can be obtained(v)By setting , , and , -convex function [16] can be obtained(vi)By setting , , , and , convex function [17] can be obtained(vii)By setting , , , and , -function [18] can be obtained(viii)By setting and , exponentially -convex function [19] can be obtained(ix)By setting , , and , exponentially -convex function [20] can be obtained(x)By setting and , exponentially -convex function [21] can be obtained(xi)By setting , , and , exponentially convex function [22] can be obtained(xii)By setting , , and , -convex function [23] can be obtained(xiii)By setting , , , and , -convex function [20] can be obtained(xiv)By setting , , , and , Godunova–Levin function [24] can be obtained(xv)By setting , , , and , -Godunova–Levin function of the second kind [25] can be obtainedThe article is organized in the following manner: in Section 2, we prove the -fractional integral inequality of Hadamard type for exponentially -convex functions and deduce some related results. In section 3, we prove a version of the -fractional integral inequality of Hadamard type for differentiable function so that is exponentially -convex. In section 4, we prove the -fractional integral inequality of Hadamard type for the product of two exponentially -convex functions.

2. Main Results

First, we give a -fractional integral inequality for exponentially -convex functions as follows.

Theorem 2. Let be an exponentially -convex function with , also let , . Then, we will havewhere , and .

Proof. Since is exponentially -convex on , for , we havefrom which multiplying both sides with and integrating over , we will haveBy changing variables, we will haveThis completes the proof of first inequality in (6). The second inequality in (6) follows by using Holder’s inequality:Thus from (9), we will get (6).

Some special cases of the above theorem are discussed in the following corollaries.

Corollary 1. The following inequality holds for an exponentially -convex function via an RL fractional integral by setting in inequality (6):

Corollary 2. The following inequality holds for the -convex function via an RL -fractional integral by setting in inequality (6):

Corollary 3. The following inequality holds for the -convex functions via RL -fractional integrals by setting and in inequality (6):

Corollary 4. The following inequality holds for the -convex function via RL fractional integrals by setting , , , and in inequality (6):The next two special cases are already proved in [26].

Corollary 5. The following inequality holds for the exponentially -convex function via RL -fractional integrals:

Proof. By setting and in inequality (6) of Theorem 2, we get the above inequality (15) which is given in [26] (Theorem 1).

Corollary 6. The following inequality holds for the -convex function via RL -fractional integrals:

Proof. By setting , , and in inequality (6) of Theorem 2, we get the above inequality (16) which is given in [26] (Theorem 4). □
The next result holds for exponentially -convex functions, whereas the function is superadditive.

Theorem 3. Let be exponentially -convex functions with , also let be superadditive and , . Then, for RL -fractional integrals, we have

Proof. Since is an exponentially -convex function on , for , we getsince is superadditive; therefore,By multiplying both sides of the above inequality with and integrating over , we will haveBy substituting in the left side of the above inequality, it leads to (17).

Some special cases are given in the form of following corollaries.

Corollary 7. The following inequality holds for the exponentially -convex function via RL fractional integrals by setting in inequality (17):

Corollary 8. The following inequality holds for the -convex function via RL -fractional integrals by setting in inequality (17):

Corollary 9. The following inequality holds for -convex functions via RL -fractional integrals:

Proof. By setting and in inequality (17) of Theorem 3, we get the above inequality (22) which is given in [4] (Theorem 2.6).

3. Fractional Hadamard-Type Inequalities for Functions Whose Derivatives in Absolute Values are Exponentially -Convex

In the following, -fractional integral inequalities of Hadamard type for exponentially -convex functions in terms of the first derivatives have been obtained. For the next result, we use the following lemma.

Lemma 1 (see [4]). Let be a differentiable mapping on the interval with . If , then for -fractional integrals, we will have

Theorem 4. Let be a function such that and . If is exponentially -convex functions with and , . Then, for RL -fractional integrals, we havewhere .

Proof. By using the property of modulus from Lemma 1, we will getBy exponentially -convexity of , we haveNow, by using Holder inequality in the right-hand side of (27), we will getBy some manipulation, one can get inequality (25).

Corollary 10. The following inequality holds for the exponentially -convex function via RL fractional integrals by setting in inequality (25):

Corollary 11. The following inequality holds for the -convex function via RL -fractional integrals by setting in inequality (25):

The following special cases are proved in [4, 26].

Corollary 12. The following inequality holds for the -convex function via RL -fractional integrals:

Proof. By setting and in inequality (25) of Theorem 4, we get the above inequality (31) which is given in [4] (Theorem 3.6).

Corollary 13. The following inequality holds for the exponentially -convex function via RL -fractional integrals:

Proof. By setting and in inequality (25) of Theorem 4, we get the above inequality (32) which is given in [26] (Theorem 2).

Corollary 14. The following inequality holds for the exponentially -convex function via RL -fractional integrals:

Proof. By setting , , and in inequality (25) of Theorem 4, we get the above inequality (33) which is given in [26] (Theorem 5).

4. Fractional Hadamard-Type Inequalities for Product of Two Exponentially -Convex Functions

Now, we obtain some Hadamard-type inequalities for the product of two exponentially -convex functions via RL -fractional integrals.

Theorem 5. Let be functions such that , . If function is exponentially -convex and function is exponentially -convex on with , then the following inequalities hold for RL -fractional integrals:where and .

Proof. Since function is exponentially -convex and function is exponentially -convex, for , we haveby multiplying both sides of the above inequality with and integrating over , we haveBy substituting in the left-hand side of the above inequality, we getBy using Holder inequality, we will haveThus, we will getSimilarly, by changing the roles of and , after a little computation, one can getAdding (39) and (40), we get the required result.

Corollary 15. The following inequality holds for the exponentially -convex function via RL fractional integrals:

Proof. By setting in inequality (34) of Theorem 5, we get the above inequality (41).

Corollary 16. The following inequality holds for the -convex function via RL -fractional integrals:

Proof. By setting in inequality (34) of Theorem 5, we get the above inequality (42).

Corollary 17. The following inequality holds for the -convex function via RL -fractional integrals:

Proof. By setting and in inequality (34) of Theorem 5, we get the above inequality (43) which is given in [4] (Theorem 4.3).

Corollary 18. The following inequality holds for the exponentially -convex function via RL -fractional integrals:

Proof. By setting and in inequality (34) of Theorem 5, we get the above inequality (44) which is given in [26] (Theorem 3).

Corollary 19. The following inequality holds for the -convex function via RL -fractional integrals:

Proof. By setting , , and in inequality (34) of Theorem 5, we get the above inequality (45) which is given in [26] (Theorem 6).

Theorem 6. Let be functions such that , . If function is exponentially -convex and function is exponentially -convex on with , then the following inequalities hold for RL -fractional integrals:

Proof. Since function is exponentially -convex and function is exponentially -convex, then for , we haveBy multiplying both sides of the above inequality with and integrating over , we haveBy substituting in the left-hand side of the above inequality, we getSimilarly, by changing the rules of and , after a little computation, one can getAdding (49) and (50), we get the required result.

Corollary 20. The following inequality holds for the exponentially -convex function via RL fractional integrals:

Proof. By setting in inequality (49) of Theorem 6, we get the above inequality (51).

Corollary 21. The following inequality holds for the -convex function via RL -fractional integrals:

Proof. By setting in inequality (49) of Theorem 6, we get the above inequality (52) which is given in [4] (Theorem 4.6).

Corollary 22. The following inequality holds for the -convex function via RL fractional integrals:and

Proof. From (49) by setting , , and , we get (53). Similarly, using , , and in (50), we get (54) which is given in [27] (Theorem 12).

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The research work of Ghulam Farid was supported by the Higher Education Commission of Pakistan with Project no. 5421.

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Copyright

Copyright © 2020 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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