Refinements of Some Integral 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 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

Farid, Ghulam; Chu, Yu-Ming; Andrić, Maja; Jung, Chahn Yong; Pečarić, Josip; Kang, Shin Min

doi:https://doi.org/10.1155/2020/8878342

Mathematical Problems in Engineering

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Fractional Mathematical Modelling and Optimal Control Problems of Differential Equations

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Volume 2020 | Article ID 8878342 | https://doi.org/10.1155/2020/8878342

Refinements of Some Integral Inequalities for -Convex Functions

Ghulam Farid,¹Yu-Ming Chu,^2,3Maja Andrić,⁴Chahn Yong Jung,⁵Josip Pečarić,⁶and Shin Min Kang⁷

Academic Editor: Xinguang Zhang

Received29 Aug 2020

Revised24 Sept 2020

Accepted27 Oct 2020

Published21 Nov 2020

Abstract

In this paper, the refinements of integral inequalities for all those types of convex functions are given which can be obtained from -convex functions. These inequalities not only provide refinements of bounds for unified integral operators but also for various associated fractional integral operators containing Mittag–Leffler function. At the same time, presented results give generalizations of many known fractional integral inequalities.

1. Introduction

The following fractional integral operator is the well-known Riemann–Liouville fractional integral operator.

Definition 1. (see [1]). Let . Then, Riemann–Liouville fractional integrals of order where are defined as follows:where is the gamma function.
Next, generalizations of Riemann–Liouville fractional integral operators are given.

Definition 2. (see [2]). Let be an integrable function. Also, let be an increasing and positive function on , having a continuous derivative on . The left-sided and the 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 given as follows.

Definition 3. (see [3]). 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 .
The following integral operator is given in [4].

Definition 4. Let be the functions such that be positive and and be differentiable and strictly increasing. Also, let be an increasing function on . Then, for , the left and right integral operators are defined bywhere .
A fractional integral operator containing an extended generalized Mittag–Leffler function in its kernel is defined as follows.

Definition 5. (see [5]). Let , , and with , , and . Let and . Then, the generalized fractional integral operators and are defined bywhereis the extended generalized Mittag–Leffler function. For further study of the Mittag–Leffler function, see [6, 7]. is the Pochhammer symbol defined by , and is the extended beta function given byThe following identities for the constant function are obtained in [8] (see also [9]):Recently, a unified integral operator is defined as follows.

Definition 6. (see [10]). Let , , be the functions such that be positive and and be differentiable and strictly increasing. Also, let be an increasing function on and , , and . Then, for , the left and the right integral operators are defined bywhere the involved kernel is defined byThe known fractional integrals studied in [2, 11–22] can be reproduced from the above definition, see [23], Remarks 6 and 7.
The aim of this study is to obtain the bounds of all known fractional integral operators defined in [2, 11–22] in a unified form for strongly -convex functions. In the result, we get refinements of many known integral and fractional integral inequalities. Next, we recall definitions of convex, strongly convex, -convex, -convex, -convex, and strongly -convex functions.

Definition 7. (see [24]). A function is said to be a convex function if the inequalityholds for all and .
The concept of a strongly convex function is defined as follows.

Definition 8. (see [25]). Let be a nonempty convex subset of a normed space. A real-valued function is said to be strongly convex with modulus on if for each and , we haveA generalization of the convex function defined on the right half of the real line is called the -convex function, and it is given as follows.

Definition 9. (see [26]). Let . A function is said to be an -convex function in the second sense ifholds for all and .
The notion of the -convex function and strongly -convex function is defined as follows.

Definition 10. (see [27]). A function is said to be an -convex function, where and , if for every and , we have

Definition 11. (see [28]). A function is said to be a strongly -convex function with modulus ifwith and .
A further generalized convexity is given as follows.

Definition 12. (see [29]). A function is said to be an -convex function, where and , if for every and , we haveThe notion of the strongly -convex function is defined as follows.

Definition 13. (see [30]). A function is said to be a strongly -convex function, with modulus , for , ifholds for all and .
Using strongly -convexity and utilizing fractional operators (6) and (7), some fractional integral inequalities are obtained as in [31]. The following result provides the bound of sum of left and right fractional integrals (6) and (7) for strongly -convex functions at an arbitrary point.

Theorem 1. (see [31]). Let be a real-valued function. If is positive and strongly -convex, then for , the following fractional integral inequality holds:The following Hadamard-type inequality holds for generalized fractional integral operators for strongly -convex functions.

Theorem 2. (see [31]). Let , , be a real-valued function. If is positive, strongly -convex and , then for , the following fractional integral inequality holds:In the following, using the strongly -convexity of , a modulus inequality is obtained.

Theorem 3. (see [31]). Let be a real-valued function. If is differentiable and is strongly -convex, then for , the following fractional integral inequality holds:In [32], we studied the properties of the kernel given in (13). Here, we are interested in the following property.
P: let and be increasing functions. Then, for , , the kernel satisfies the following inequality:This can be obtained from the following two straightforward inequalities:The reverse of inequality (13) holds when and are decreasing.
The upcoming section contains the results for unified integral operators dealing with the bounds of several fractional integral operators in a compact form by utilizing strongly -convex functions. A compact version of the Hadamard inequality is presented, and also a modulus inequality is given for the differentiable function such that is a strongly -convex function. In the whole paper, we will use

2. Main Results

The following result provides the upper bound of unified integral operators.

Theorem 4. Let , , be a positive integrable and strongly -convex function, . Then, for unified integral operators (11) and (12), the following inequality holds:

Proof. By , the following inequalities hold:For a strongly -convex function, the following inequalities hold for and , respectively:From (28) and (30), one can havei.e.,On the other hand, from (29) and (31), one can havei.e.,By adding (33) and (35), (27) can be obtained.

Corollary 1. Setting in (27), we can obtain the following inequality involving fractional integral operators defined in [4]:

Remark 1. (i)If we consider in (27), then Theorem 3.1 in [32] can be obtained, and for , we get its refinement(ii)If we consider and in (27), then Theorem 1 can be obtained(iii)If we consider in the result of (ii), then Corollary 1 in [31] can be obtained(iv)If we consider in the result of (ii), then Corollary 3 in [31] can be obtained(v)If we consider in the result of (ii), then Corollary 5 in [31] can be obtained(vi)If we consider in the result of (v), then Corollary 7 in [31] can be obtained(vii)If we consider in the result of (ii), then Corollary 5 in [31] can be obtained(viii)If we consider = (1, 1) in (27), then Theorem 2 in [33] is obtained(ix)If we consider , , and = (1, 1) in (27), then Theorem 8 in [23] is obtained(x)If we consider and in (27), then Theorem 1 in [34] is obtained(xi)If we consider , , , and = (1, 1) in (27), then Theorem 1 in [35] is obtained(xii)If we consider in the result of (xi), then Corollary 1 in [35] is obtained(xiii)If we consider , , , and in (27), then Theorem 2.1 in [36] is obtained(xvi)If we consider in the result of (xiii), then Corollary 2.1 in [36] is obtained(xv)If we consider , , = (1, 1), , and in (27), then Theorem 1 in [37] can be obtained(xvi)If we consider in the result of (xv), then Corollary 1 in [37] can be obtained(xvii)If we consider , , , , and = (1, 1) in (27), then Theorem 1 in [38] is obtained(xviii)If we consider in the result of (xvii), then Corollary 1 in [38] can be obtained(xviii)If we consider and in the result of (xvii), then Corollary 2 in [38] can be obtained(xix)If we consider and in the result of (xvii), then Corollary 3 in [38] can be obtained The following lemma is very helpful in the proof of the upcoming theorem, see [31].

Lemma 1. Let be a strongly -convex function, . If is , , then the following inequality holds:

In the literature, many mathematicians have established many types of Hadamard inequalities, and for their generalizations, see [39–42]. This also motivates us to introduce the more generalized forms of Hadamard-type inequalities. So, by the help of the abovementioned lemma, the following result provides generalized Hadamard inequality for strongly -convex functions.

Theorem 5. Under the assumptions of Theorem 4, in addition to , the following inequality holds:

Proof. By , the following inequalities hold:A strongly -convex function satisfying the following inequalities hold for :From (39) and (41), one can haveFurther, the aforementioned inequality takes the form which involves Riemann–Liouville fractional integrals in the right-hand side, and thus we have upper bound of the unified left-sided integral operator (2) as follows:On the other hand, from (39) and (41), the following inequality holds which involves Riemann–Liouville fractional integrals on the right-hand side and gives the estimate of the integral operator (3):By adding (43) and (44), the following inequality can be obtained:Multiplying both sides of (37) by and integrating over , we haveFrom Definition 6, the following inequality is obtained:Similarly, multiplying both sides of (37) by and integrating over , we haveBy adding (47) and (48), the following inequality is obtained:Using (45) and (49), inequality (38) can be obtained, which completes the proof.

Corollary 2. Setting in (38), we can obtain the following inequality involving fractional integral operators defined in [4]:

Remark 2. (i)If we consider and in (38), then Theorem 7 in [31] can be obtained(ii)If we consider in the result of (i), then Theorem 8 in [31] can be obtained(iii)If we consider = (1, 1) in (38), then Theorem 3 in [33] is obtained(iv)If we consider and = (1, 1) in (38), then Theorem 22 in [23] is obtained(v)If we consider , , , and = (1, 1) in (38), then Theorem 3 in [35] is obtained(vi)If we consider in the result of (v), then Corollary 3 in [35] is obtained(vii)If we consider , , , and in (38), then Theorem 2.4 in [36] is obtained(viii)If we consider in the result of (vii), then Corollary 2.6 in [36] is obtained(ix)If we consider , , = (1, 1), , and in (38), then Theorem 3 in [37] can be obtained(x)If we consider in the result of (ix), then Corollary 6 in [37] can be obtained(xi)If we consider , , , , and in (38), then Theorem 3 in [38] can be obtained(xii)If we consider in the result of (xi), then Corollary 6 in [38] can be obtained

Theorem 6. Let , , be a differential function such that is a strongly -convex function, . Then, for unified integral operators (11) and (12), the following inequality holds:where

Proof. For a strongly -convex function , the following inequalities hold for and , respectively:From (28) and (54), the following inequality is obtained:Similarly, from (29) and (55), the following inequality is obtained:By adding (56) and (57), inequality (52) can be achieved.

Corollary 3. Setting in (52), we can obtain the following inequality involving fractional integral operators defined in [4]:

Remark 3. (i)If we consider in (52), then Theorem 3.4 in [32] can be obtained(ii)If we consider and in (52), then Theorem 6 in [31] can be obtained(iii)If we consider in the result of (ii), then Corollary 13 in [31] can be obtained(iv)If we consider in the result of (ii), then Corollary 11 in [31] can be obtained(v)If we consider = (1, 1) in (52), then Theorem 3 in [33] is obtained(vi)If we consider and = (1, 1) in (52), then Theorem 25 in [23] is obtained(vii)If we consider and in (52), then Theorem 2 in [34] is obtained(viii)If we consider , , , and = (1, 1) in (52), then Theorem 2 in [35] is obtained(ix)If we consider in the result of (viii), then Corollary 2 in [35] is obtained(x)If we consider , , , and in (52), then Theorem 2.3 in [36] is obtained(xi)If we consider in the result of (x), then Corollary 2.5 in [36] is obtained(xii)If we consider , , = (1, 1), , and in (52), then Theorem 2 in [37] can be obtained(xiii)If we consider in the result of (xii), then Corollary 4 in [37] can be obtained(xiv)If we consider and in the result of (xii), then Corollary 5 in [37] can be obtained(xv)If we consider , , , , and = (1, 1) in (52), then Theorem 2 in [38] is obtained(xvi)If we consider in the result of (xv), then Corollary 5 in [38] can be obtained

3. Concluding Remarks

In this paper, bounds of a unified integral operator for strongly -convex functions are studied. The compact form of these bounds lead to further interesting consequences with respect to fractional integrals of various kinds for convex, -convex, -convex, -convex, and convex functions. These findings are generalized in nature and give the refinements of many inequalities for unified and fractional integral operators via different types of convex functions.

Data Availability

No data are required for this work.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research work of Yu-Ming Chu was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485). The work of Josip Pečarić was supported by the Ministry of Education and Science of the Russian Federation (the agreement no. 02.a03.21.0008).

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Copyright © 2020 Ghulam Farid 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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