Bounds of a Unified Integral Operator via 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="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>Convexity and Their Consequences

Hu, Yi; Farid, Ghulam; Zijiang, undefined; Mahreen, Kahkashan

doi:https://doi.org/10.1155/2020/3530591

Journal of Function Spaces

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Research Article | Open Access

Volume 2020 | Article ID 3530591 | https://doi.org/10.1155/2020/3530591

Bounds of a Unified Integral Operator via Exponentially -Convexity and Their Consequences

Yi Hu,^1,2Ghulam Farid ,³ Zijiang,^1,2and Kahkashan Mahreen³

Academic Editor: Mitsuru Sugimoto

Received20 Nov 2019

Accepted23 Apr 2020

Published22 May 2020

Abstract

Various known fractional and conformable integral operators can be obtained from a unified integral operator. The aim of this paper is to find bounds of this unified integral operator via exponentially -convex functions. The resulting bounds provide compact formulas for the bounds of associated fractional and conformable integral operators. Several Hadamard-type inequalities have been produced from a compact version for unified integral operators for exponentially -convex functions.

1. Introduction

Fractional integral inequalities are useful in the field of fractional calculus. Many mathematicians have introduced fractional differential, fractional integral, and fractional conformable integral operators in this field (see [1–14]). Recently, several mathematical inequalities have been introduced via -convexity (see [15, 16]). The goal of this paper is to obtain the bounds of all integral operators explained in Remarks 5 and 6 in a unified form for exponentially -convex functions.

Definition 1 (see [6]). 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 where is the gamma function.

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 where, , are defined by where is the gamma function given as follows [18]:

A fractional integral operator containing an extended generalized Mittag-Leffler function in its kernel is defined as follows:

Definition 3 (see [19]). Let , , with, , and. Letand Then the generalized fractional integral operators and are defined by where is the extended generalized Mittag-Leffler function.

Recently, a unified integral operator is defined as follows:

Definition 4 (see [20]). 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 right integral operators are defined by where .

For suitable settings of functions , , and certain values of parameters included in Mittag-Leffler function (8), some interesting consequences can be obtained which are comprised in the upcoming remarks.

Remark 5. (i)Let , , , and , in unified integral operators (9) and (10). Then, generalized Riemann-Liouville fractional integral operators (3) and (4) are obtained.(ii)For , (3) and (4) fractional integrals coincide with (1) and (2) fractional integrals, which further produce the following fractional and conformable integrals: (1)By taking as identity function, (3) and (4) fractional integrals coincide with -fractional Riemann-Liouville integrals defined by Mubeen et al. in [7](2)For , along with as the identity function, (3) and (4) fractional integrals coincide with Riemann-Liouville fractional integrals [6](3)For and , , (3) and (4) produce fractional integrals defined by Chen and Katugampola in [1](4)For and , (3) and (4) produce generalized conformable fractional integrals defined by Khan and Khan in [5](5)If we take , , in (3) and , , in (4), then conformable -fractional integrals are achieved as defined by Habib et al. in [3](6)If we take , then conformable fractional integrals are achieved as defined by Sarikaya et al. in [11](7)If we take , , in (3) and , , in (4) with , then conformable fractional integrals are achieved as defined by Jarad et al. in [4](8)If we take and in (9) and (10), then generalized -fractional integral operators are achieved as defined by Tunc et al. in [13](9)If we take and , , , in (3) and (4), then generalized fractional integral operators with an exponential kernel are obtained [2]

Remark 6. Let and, , in unified integral operators (9) and (10). Then, fractional integral operators (6) and (7) are obtained, which along with different settings of in generalized Mittag-Leffler function give the following integral operators: (1)By setting , fractional integral operators (6) and (7) reduce to the fractional integral operators defined by Salim and Faraj in [10](2)By setting , fractional integral operators (6) and (7) reduce to the fractional integral operators defined by Rahman et al. in [9](3)By setting and , fractional integral operators (6) and (7) reduce to the fractional integral operators defined by Srivastava and Tomovski in [12](4)By setting and , fractional integral operators (6) and (7) reduce to the fractional integral operators defined by Prabhakar in [8](5)By setting , fractional integral operators (6) and (7) reduce to the left-sided and right-sided Riemann-Liouville fractional integrals(6)By setting in fractional integral operators (9) and (10) we get and where and are defined in [21]Convex functions play a very vital role in the theory of inequalities. Motivated by its analytical interpretation, many other extended and generalized notions have been defined in literature. These notions are used to extend and generalize a lot of classical results (see [15, 16, 22, 23] and references therein). A generalized notion called exponentially -convexity is defined in [24].

Definition 7. Let and be an interval. A function is said to be exponentially -convex in the second sense, if holds for all and

One can note the deducible definitions in the following remark:

Remark 8. (i)For , (11) produces the definition of exponentially -convex function(ii)For , (11) produces the definition of -convex function(iii)For and , (11) produces the definition of -convex function in the second sense(iv)For and , (11) produces the definition of the convex function(v)For and , (11) produces the definition of the -convex function

In the upcoming section, bounds of unified integral operators for exponentially -convex functions are given in different forms. Bounds of associated fractional and conformable integral operators which are known in literature are also deduced. The Hadamard inequality is derived for exponentially -convex functions. Its diverse conformable and fractional versions are presented. A modulus inequality is established by using exponentially -convexity of . In Section 3, boundedness and continuity of these operators are given.

2. Main Results

Bounds of unified integral operators (9) and (10), by using exponentially -convexity, are established in the following theorem:

Theorem 9. Let , , be a positive exponentially -convex function with and be a differentiable and strictly increasing function. Also let be an increasing function on . If , , , and , then for , we have

Proof. Under the suppositions of and , we can obtain Multiplying with , we can obtain By using , the following inequality is obtained: Using exponentially -convexity of , we have Multiplying (15) and (16) and integrating over , we can obtain By using Definition 4 and integrating by parts, the following inequality is obtained: Now on the other hand for and , the following inequality holds true: Using exponentially -convexity of , we have Adopting the same pattern as we did for (15) and (16), we obtained the following inequality from (19) and (20): By adding (18) and (21), (12) can be obtained.

Remark 10. (1)If we put in (12), then result for exponentially -convex functions can be obtained for the integral operators defined in [23].(2)If we put in (12), then Theorem 1 in [23] can be obtained(3)If we put and in (12), Theorem 8 in [20] can be obtained(4)If we put and in (12), then the result for exponentially convex functions can be obtained for the integral operators defined in [23].(5)If we put , , and in (12), then the result for convex functions can be obtained for the integral operators defined in [23].(6)If we put , , , and in (12), then Proposition 10 in [20] can be obtained(7)If we put , , , , and in (12), then Corollary 1 in [26] can be obtained

Proposition 11. Let , and . Then, (12) gives the following bound for fractional integral operators defined in [6] for :

Proposition 12. Let and . Then, (12) gives the following bound for integral operators defined in [27]:

Corollary 13. If we take and , then (12) gives the following bound for fractional integral operators defined in [17] for :

Corollary 14. If we take , , and , then (12) gives the following bound for Riemann-Liouvelle fractional integrals defined in [6] for :

Corollary 15. If we take , , and , then (12) gives the following bound for fractional integral operators defined in [7] for :

Corollary 16. If we take , , and , then (12) gives the following bound for fractional integral operators defined in [1]:

The following lemma is essential to prove the next result.

Lemma 17. Let be an exponentially -convex function with . If and , then the following inequality holds:

Proof. Since is an exponentially -convex, therefore, the following inequality is valid: By using in the above inequality, we get (28).
The following result provides the Hadamard inequality for unified integral operators defined in (9) and (10).

Theorem 18. Under the assumptions of Theorem 9, in addition if , then we have

Proof. Under the assumptions of and , we can obtain: Multiplying with , we can obtain: By using , the following inequality is obtained: Using exponentially -convexity of , we have Multiplying (33) and (34) and integrating the resulting inequality over , we can obtain By using Definition 4 and integrating by parts, the following inequality is obtained: On the other hand, for , the following inequality holds true: Adopting the same pattern of simplification as we did for (33) and (34), the following inequality can be observed for (34) and (37): By adding (36) and (38), following inequality can be obtained: Multiplying both sides of (28) by , and integrating over we have From Definition 4, the following inequality is obtained: Similarly multiplying both sides of (28) by and integrating over , we have By adding (41) and (42) following inequality is obtained: Using (39) and (43), inequality (30) can be achieved.

Remark 19. (1)If we put in (30), then result for exponentially -convex functions can be obtained for the integral operators defined in [23](2)If we put and in (30), then Theorem 22 in [20] can be obtained(3)If we put and in (30), then the result for exponentially convex functions can be obtained for the integral operators defined in [23](4)If we put in (30), then Theorem 3 in [23] can be obtained(5)If we put and in (30), then the result for convex functions can be obtained for the integral operators defined in [23]

Corollary 20. If we put and , then the inequality (30) produces the following Hadamard inequality for fractional integral operators defined in [17]:

Corollary 21. If we put and , then the inequality (30) produces the following Hadamard inequality for fractional integral operators defined in [6]:

Corollary 22. If we put , and take as identity function, then the inequality (30) produces the following Hadamard inequality for fractional integral operators defined in [7]:

Corollary 23. If we put , and take as the identity function, then the inequality (30) produces the following Hadamard inequality for fractional integral operators defined in [6]:

Theorem 24. Let be a differentiable function. If is exponentially -convex with and be a differentiable and strictly increasing function. Also let be an increasing function on then for, . If , , and , then for , we have where

Proof. Using exponentially -convexity of , we have The inequality (50) can be written as follows: Let us consider the second inequality of (51) Multiplying (15) and (52) and integrating over , we can obtain: By using (9) of Definition 4 and integrating by parts, the following inequality is obtained: Now we consider the left-hand side from the inequality (51), and adopting the same pattern as we did for the right-hand side inequality, we have From (54) and (55), the following inequality is observed: Now using exponentially -convexity of , we have On the same pattern as we did for (15) and (50), one can obtain the following inequality from (19) and (57): By adding (56) and (58), inequality (48) can be achieved.

Remark 25. (1)If we put in (48), then Theorem 2 in [23] can be obtained(2)If we put in (48), then the result for exponentially convex functions can be obtained for the integral operators defined in [23](3)If we put in (48), then the result for exponentially -convex functions can be obtained for the integral operators defined in [23](4)If we put and in (48), then Theorem 25 in [20] can be obtained

3. Boundedness and Continuity

Theorem 26. Under the assumptions of Theorem 9, the following inequality holds for -convex functions:

Proof. If we put and in (17), we have By using Definition 4 and integrating by part, the following inequality is obtained: which can be written as Similarly from (21), the following inequality holds: which further simplifies as follows: From (62) and (64), (59) can be obtained.

Corollary 27. If we take in Theorem 26, then the following inequality holds for convex functions:

Theorem 28. With assumptions of Theorem 26, if , then unified integral operators for -convex functions are bounded and continuous.

Proof. From (62), we have which further gives where .
Similarly, from (64), the following inequality holds: The boundedness is established; also, they are linear therefore the continuity is obtained.

Corollary 29. If we take in Theorem 28, then unified integral operators for convex functions are bounded and continuous and following inequalities hold: where .

4. Concluding Remarks

The class of exponentially -convex functions contains -convex, exponentially convex, -convex, -convex, and convex functions. The unified integral operators defined in (9) and (10) produce almost all fractional and conformable integral operators which have been defined independently in recent decades. We have established the bounds of sum of unified integral operators of exponentially -convex functions. Also a Hadamard inequality for these operators is established. In conclusion, the presented results reproduce plenty of results for operators composed in Remarks 5 and 6 for functions deduced in Remark 8.

Data Availability

There is no additional data required for the finding of results of this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2020 Yi Hu 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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