Generalized Hadamard Fractional Integral Inequalities 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 38.9035 15.2545" width="38.9035pt"><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="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z"/></g><g transform="matrix(.017,0,0,-0.017,12.372,0)"><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,-0.017,19.161,0)"><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,-0.017,32.702,0)"><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

Miao, Chao; Farid, Ghulam; Yasmeen, Hafsa; Bian, Yanhua

doi:https://doi.org/10.1155/2021/6642289

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Volume 2021 | Article ID 6642289 | https://doi.org/10.1155/2021/6642289

Generalized Hadamard Fractional Integral Inequalities for Strongly -Convex Functions

Chao Miao,¹Ghulam Farid,²Hafsa Yasmeen,²and Yanhua Bian¹

Academic Editor: Xiaolong Qin

Received20 Dec 2020

Revised09 Jan 2021

Accepted29 Jan 2021

Published25 Mar 2021

Abstract

This article deals with Hadamard inequalities for strongly -convex functions using generalized Riemann–Liouville fractional integrals. Several generalized fractional versions of the Hadamard inequality are presented; we also provide refinements of many known results which have been published in recent years.

1. Introduction

Fractional calculus is related to the integrals and derivatives of any arbitrary real or complex order. Its history starts from the end of the seventeenth century, but now it has many applications in almost every field of mathematics, science, and engineering such as electromagnetic, viscoelasticity, fluid mechanics, and signal processing. Fractional integral and derivative operators are of great importance in fractional calculus. The Riemann–Liouville fractional integrals are playing key role in its development. Sarikaya et al. [1, 2] studied Hadamard inequality through Riemann–Liouville fractional integrals of convex functions. This study has encouraged a number of researchers to work further in the field of mathematical inequalities by using fractional integral operators. As a consequence, Hadamard’s inequality is generalized and extended by fractional integral operators in many ways (see [3–9] and the references therein). The following inequality is the well-known Hadamard inequality for convex functions which is stated in [10].

Let be a convex function defined on an interval and where . Then, the following inequality holds:

For the history of this inequality, we refer the readers to [11, 12]. Use of convex functions in the fields of statistics [13], economics [14], and optimization [15] is of prime importance because they play an important role in development of new concepts and notions. Various scholars extended the research on integral inequalities to fractal sets [16]. In this paper, the Hadamard inequality is studied for generalized Riemann–Liouville fractional integrals of strongly -convex functions; also, by using two integral identities, some error bounds of already established fractional inequalities are studied. Bracamonte et al. [17] defined the strongly -convex function as follows.

Definition 1. A function is said to be strongly -convex function with modulus in second sense, where , ifholds for all and .

The well-known definition of Riemann–Liouville fractional integral is given as follows.

Definition 2. (see [18]) (see also [19]). Let . Then, left-sided and right-sided Riemann–Liouville fractional integrals of a function of order where are given bywhere is real part of and . The following theorems are the fractional versions of Hadamard inequality by Riemann–Liouville fractional integrals.

Theorem 1. (see [1]). Let be a positive function with and . If is a convex function on , then the following fractional integral inequality holds:with .

Theorem 2. (see [2]). Under the assumptions of Theorem 1, the following fractional integral inequality holds:with .

By establishing an integral identity, the following error estimation of inequality (6) is proved.

Theorem 3. (see [1]). Let be a differentiable mapping on with . If is convex on , then the following fractional integral inequality holds:

A -analogue of Riemann–Liouville integral is defined as follows.

Definition 3. (see [20]). Let . Then, -fractional Riemann–Liouville integrals of order where , , are defined aswhere is defined by [21]

If , (8) and (9) coincide with (3) and (4).

Two -fractional versions of Hadamard inequality for -fractional Riemann–Liouville integrals are given in the next two theorems.

Theorem 4. (see [22]). Let be a positive function with . If is a convex function on , then the following inequality for -fractional integrals holds:

Theorem 5. (see [23]). Under the assumption of Theorem 4, the following inequality for -fractional integrals holds:

By establishing an integral identity, in the following theorem, the error estimation of Theorem 4 is proved.

Theorem 6. (see [22]). Let be a differentiable mapping on with . If is convex on , then the following inequality for -fractional integrals holds:

In the following, we recall the definition of generalized Riemann–Liouville fractional integrals by a monotonically increasing function.

Definition 4. (see [24]). 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 given by

If is identity function, then (14) and (15) coincide with (3) and (4).

The -analogue of generalized Riemann–Liouville fractional integral is defined as follows.

Definition 5. (see [25]). 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 by

For further study of fractional integrals, see [26, 27]. We will utilize the following well-known hypergeometric, beta, and incomplete beta functions in our results [28].

The rest of the paper is organized as follows. In Section 2, we obtain Hadamard inequalities for generalized Riemann-Liouville fractional integrals of strongly -convex functions. Many specific cases are given as outcomes of these inequalities; they are related to the results which have been published in different papers. In Section 3, by using two integral identities for generalized fractional integrals, the error bounds of fractional Hadamard inequalities are established for differentiable strongly -convex functions. This paper reproduces the results which are explicitly given in [1, 2, 22, 23, 29–37].

2. Main Results

This section is dedicated to the Hadamard inequality for strongly -convex functions via generalized Riemann–Liouville fractional integrals. We will give two versions of this inequality. First one is stated and proved in the following theorem.

Theorem 7. Let , be a positive function and . Also, let be strongly -convex function on with modulus , such that . Then, for and , the following -fractional integral inequalities hold for operators given in (16) and (17):with .

Proof. The following inequality holds for strongly -convex functions.By setting , , , in (20), multiplying resulting inequality with , and then integrating with respect to , we getBy setting and in (21) and by applying Definition 5, we get the following inequality:The above inequality leads to the first inequality of (19). On the other hand, is strongly -convex function with modulus ; for , we have the following inequality:By integrating (23) over after multiplying with , the following inequality holds:Again using substitutions as considered in (21), we getThis leads to the second inequality of (19).

Remark 1. Under the assumption of Theorem 7, the following outcomes are noted.(i)If , , then the inequality stated in [[32], Theorem 9] is obtained.(ii)If , , , and is the identity function in (19), then Theorem 4 is obtained.(iii)If , , , , and is the identity function in (19), then Theorem 1 is obtained.(iv)If , , , and is the identity function in (19), then refinement of Theorem 1 is obtained.(v)If , , , , , and is the identity function in (19), then Hadamard inequality is obtained.(vi)If , , and in (19), then the inequality [[34], Theorem 1] is obtained.(vii)If , , , and in (19), then the inequality stated in [[33], Theorem 2.1] is obtained.(viii)If , and is the identity function in (19), then the inequality stated in [[31], Theorem 6] is obtained.(ix)If , , , , and is the identity function in (19), then the inequality stated in [[35], Theorem 6] is obtained.(x)If , , , and is the identity function in (19), then the inequality stated in [[29], Theorem 2.1] is obtained.

Corollary 1. Under the assumption of Theorem 7 with in (19), the following inequality holds:

Theorem 8. Under the assumption of Theorem 7, the following -fractional integral inequality holds:with .

Proof. By setting , , in (20), multiplying resulting inequality with , and then integrating with respect to , we getBy setting and and by applying Definition 5, we get the following inequality:The above inequality leads to the first inequality of (27). On the other hand, is strongly -convex function with modulus ; for , we have the following inequality:By integrating (30) over after multiplying with , the following inequality holds:Again using substitutions as considered in (28), we getThis leads to the second inequality of (27).

Remark 2. Under the assumption of Theorem 8, the following outcomes are noted.(i)If and in (27), then the inequality stated in [[32], Theorem 10] is obtained.(ii)If , , , , and is the identity function in (27), then Theorem 2 is obtained.(iii)If , , , and is the identity function in (27), then refinement of Theorem 2 is obtained.(iv)If , , , , and is the identity function in (27), then Hadamard inequality is obtained.(v)If , , , and is the identity function in inequality (27), then Theorem 5 is obtained.(vi)If , , and in (27), then the inequality stated in [[32], Corollary 5] is obtained.(vii)If , , and is the identity function in inequality (27), then the inequality stated in [[31], Theorem 7] is obtained.

Corollary 3. Under the assumption of Theorem 8 with in (27), the following inequality holds:

3. Error Estimations of Hadamard Inequalities via Strongly -Convex Functions

In this section, we will study error estimations of Hadamard inequalities for generalized Riemann–Liouville fractional integrals of strongly -convex functions. The estimations obtained here provide refinements of many well-known results. The Mathematica program is used for integration. We recall the well-known Hölder’s integral inequality.

Theorem 9. (see [38]). Let and . If and are real functions defined on and if and are integrable functions on , thenwith equality holding iff almost everywhere, where and are constants.

In order to prove the next result, the following lemma is useful.

Lemma 1. (see [34]). Let and be a differentiable mapping on . Also, suppose that , is an increasing and positive monotone function on , having a continuous derivative on , and . Then, for , the following identity holds:

Theorem 10. Let , be a differentiable mapping on such that . Also, suppose that is strongly -convex on with modulus . Then, for and , the following -fractional integral inequality holds for operators given in (16) and (17):with and being the hypergeometric function.

Proof. By Lemma 1, it follows thatSince is strongly -convex function on , for , we haveNow using (38) in (37), we haveWe now evaluate integrals that appear on the right side of the above inequality:Using (40)–(42) in (39), we get (36).

Remark 3. Under the assumption of Theorem 10, the following outcomes are noted.(i)If and in (45), then the inequality stated in [[32], Theorem 11] is obtained.(ii)If , , and in (45), then the inequality stated in [[32], Corollary 10] is obtained.(iii)If , , , and is the identity function in (45), then a refined error estimation of the fractional Hadamard inequality is obtained.(iv)If and in (45), then the inequality stated in [[34], Theorem 2] is obtained.(v)If , , , and is the identity function in (45), then Theorem 6 is obtained.(vi)If , , , , and is the identity function in (45), then Theorem 3 is obtained.(vii)If , , and is the identity function in (45), then the inequality stated in [[31], Theorem 8] is obtained.

Corollary 5. Under the assumption of Theorem 10 with in (3.10), the following inequality holds:

Corollary 6. Under the assumption of Theorem 10 with , , , , and as the identity function in (3.10), the following inequality holds:

Lemma 2. Let be a differentiable mapping on such that . Then, for and , the following identity holds for operators given in (16) and (17):

Proof. LetFirst, we evaluate :Now integrating by parts, we haveSubstituting , so that in (48), we get the following inequality:Now, we evaluate :Integrating by parts, we getSubstituting , so that in (51), we get the following inequality:Adding (49) and (52), (45) is obtained.

Remark 4. Under the assumption of Lemma 2, the following outcomes are noted.(i)If and is the identity function in (45), then the identity stated in [[30], Lemma 2.3] is obtained.(ii)If in (45), then the identity stated in [[32], Lemma 2] is obtained.(iii)If , , and is the identity function in (45), then the identity stated in [[2], Lemma 3] is obtained.(iv)If , , , and is the identity function in (45), then the identity stated in [[2], Corollary 1] is obtained.(v)If and is the identity function in (45), then the identity stated in [[23], Lemma 3.1] is obtained.

Theorem 11. Let , be a differentiable mapping on such that . Also, suppose that is strongly -convex function on for . Then, for and , the following -fractional integral inequality holds for operators given in (16) and (17):with and .

Proof. We divide the proof in two cases.

Case 1. Fix . Applying Lemma 2 and strongly -convexity of , we have

Case 2. For . From Lemma 2 and using power mean inequality, we getHence, we get (53).

Remark 5. Under the assumption of Theorem 11, the following outcomes are noted.(i)If and in (53), then the inequality stated in [[32], Theorem 12] is obtained.(ii)If , , and is the identity function in (53), then the inequality stated in [[31], Theorem 9] is obtained.(iii)If , , , and is the identity function in (53), then the inequality stated in [[30], Theorem 2.4] is obtained.(iv)If , , and is the identity function in (53), then the inequality stated in [[23], Theorem 3.1] is obtained.(v)If , , , , and is the identity function in (53), then the inequality stated in [[2], Theorem 5] is obtained.(vi)If , , , , , and is the identity function in (53), then the inequality stated in [[36], Theorem 2.2] is obtained.

Corollary 7. Under the assumption of Theorem 11 with in (53), the following inequality holds:

Corollary 8. Under the assumption of Theorem 11 with , , , , and as the identity function in (53), the following inequality holds:

Lemma 3. (see [39]). Let be a real number. For and , the following inequality holds:

Theorem 12. Let be a differentiable mapping on with . Also, suppose that is strongly -convex function on for . Then, for and , the following -fractional integral inequality holds for operators given in (16) and (17):with and .

Proof. By applying Lemma 2 and using the property of modulus, we getNow applying Hölder’s inequality for integrals, we getUsing strongly -convexity of , we getHere we have used Lemma 3. This completes the proof.

Remark 6. Under the assumption of Theorem 12, the following outcomes are noted.(i)If and in (59), then the inequality stated in [[32], Theorem 13] is obtained.(ii)If and is the identity function in (53), then the inequality stated in [[31], Theorem 10] is obtained.(iii)If , , and is the identity function in (53), then the inequality stated in [[30], Theorem 2.7] is obtained.(iv)If , , , and is the identity function in (59), then the inequality stated in [[23], Theorem 3.2] is obtained.(v)If , , , , and is the identity function in inequality (59), then the inequality stated in [[2], Theorem 6] is obtained.(vi)If , , , , , and is the identity function in inequality (59), then the inequality stated in [[37], Theorem 2.4] is obtained.

Corollary 9. Under the assumption of Theorem 12 with in (61), the following inequality holds:

Corollary 10. Under the assumption of Theorem 12 with and in (59), the following inequality holds:

4. Conclusion

In this article, we studied the Hadamard inequalities and their estimations for generalized Riemann–Liouville fractional integrals of strongly -convex functions. These inequalities represent the generalizations and refinements of a number of well-known inequalities stated in [1, 2, 22, 23, 29–37]. The error estimations of Hadamard inequalities for differentiable strongly -convex functions are better as compared to those which are obtained for convex functions, strongly convex functions, and strongly -convex functions.

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 © 2021 Chao Miao 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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