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Zhang, Zhongyi; Farid, Ghulam; Mahreen, Kahkashan

doi:https://doi.org/10.1155/2021/6675826

Journal of Function Spaces

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Research Article | Open Access

Volume 2021 | Article ID 6675826 | https://doi.org/10.1155/2021/6675826

Inequalities for Unified Integral Operators via Strongly -Convexity

Zhongyi Zhang,¹Ghulam Farid,²and Kahkashan Mahreen²

Academic Editor: Fu-Dong Ge

Received21 Dec 2020

Revised14 Apr 2021

Accepted08 May 2021

Published04 Jun 2021

Abstract

In this paper, we give a generalized definition namely strongly -convex function that unifies many known definitions. By applying this new definition, we present inequalities for unified integral operators which have connection with many of the well-known results for different kinds of convex functions. Moreover, this paper at once provides refinements and generalizations of a lot of fractional integral inequalities which are identified in remarks.

1. Introduction

There are many applications of convexity in diverse fields of mathematics including operation research, mathematical statistics, optimization theory, and graph theory. In mathematical inequalities’ point of view, convex functions are very important. They are extended and generalized in different ways to obtain corresponding generalizations and extensions of well-known inequalities. For the detail study of different kinds of convex functions, we refer the readers to [1–7].

In recent years, the researchers are working on fractional versions of mathematical inequalities by utilizing classical and new kinds fractional integral/derivative operators, see [8–11]. Also, several kinds of convex functions are applied to obtain these fractional versions, for example, see [1, 12–17] and references therein. The inequalities for fractional integrals and derivatives are very useful in the study of fractional differential equations. Using fractional differential and integral inequalities, qualitative properties of fractional differential equations involving the Riemann-Liouville and the Caputo derivatives can be found frequently in literature.

Our objective is to investigate integral inequalities for newly defined function called strongly -convex function. Integral operators (5) and (6) are used to establish these inequalities, and they have interesting consequences for distinctive fractional inequalities for various types of functions. Next, we give some definitions of known generalized fractional integral operators which can be directly obtained from operators (5) and (6).

Definition 1 (see [18]). Let be an integrable function and be an increasing positive function defined on has a continuous derivative on . The fractional integrals of a function with respect to another function on of order are defined by here, represents the gamma function.

One can see a -analogue of Definition 1 in [16]. The following generalized integral operator is given in [19].

Definition 2. Let be the functions such that be positive and , and be differentiable and increasing. Also let be an increasing function on . Then, for , the left and right integral operators are defined by

The Mittag-Leffler function was introduced in 1903, which is generalization of the exponential function just by a single parameter with a convergence condition. The further generalization by another parameter was given by Wiman; further, it was extended by Prabhakar and then by other authors; to see the importance of these extensions, we suggest the reader to [20, 21]. By utilizing an extended generalized Mittag-Leffler function, we have defined a fractional integral operator.

Definition 3 (see [12]). Let , , and , with . Let and The generalized fractional integral operators and are defined by: where is the extended generalized Mittag-Leffler function.

Unified integral operator is based on a kernel which also involves a real valued strictly increasing function along with two variables. This integral operator also unifies the above definitions.

Definition 4 (see [22]). Let , be the functions such that be positive and , and be differentiable and strictly increasing. Also, let be differentiable and strictly increasing. Also, let and , , , and . Then, for , the left and right integral operators are defined by where we have

Several recently defined fractional integrals studied in [12, 14, 18, 21, 23–30] can be reproduced from the above definition, see ([31], Remarks 6 and 7). The following results are obtained for strongly convex functions in [32].

Theorem 5. Let be a positive, integrable, and strongly convex function with modulus . Let be a strictly increasing and differentiable function, also let be an increasing function on . If , , and , then for , the following inequality holds: where is the identity function and .

Theorem 6. Under the assumptions of Theorem 5, in addition, if , then, we have

Theorem 7. Let be a differentiable function. If is strongly convex with modulus and be strictly increasing and differentiable, also let be an increasing function on . If , , and , then for , the following inequality holds:

Next, we give some definitions of convex functions. The definition of -convex function is given as follows:

Definition 8 (see [5]). Let be an interval containing and let be a nonnegative function. A nonnegative function is called -convex function if holds for all , , and .

Remark 9. (i)For , (11) gives the definition of -convex function(ii)For , (11) gives the definition of -convex function(iii)For and , (11) gives the definition of convex function(iv)For and , (11) gives the definition of -function(v)For and , (11) gives the definition of -convex function(vi)For and , (11) gives the definition of Godunova-Levin function(vii)For and , (11) gives the definition of -Godunova-Levin function of second

Definition 10 (see [4]). A function is said to be -convex if holds for all and .

Remark 11. (i)For =, (12) provides -convex function(ii)For =, (12) provides convex function(iii)For =, (12) provides star-shaped function

Definition 12 see ([33]). A function is said to be -convex, where if holds for all and

The following definition unifies -convex, -convex, and -convex functions in a single inequality.

Definition 13. Let be an interval containing and let be a nonnegative function. A nonnegative function is said to be -convex function if holds for all , .

Next, we give definitions of strongly convex, strongly -convex, and strongly -convex functions.

Definition 14 (see [34]). Let be a nonempty convex subset of normed space . A real valued function is said to be strongly convex, with modulus , on if for each and ,

Definition 15 (see [35]). A function is said to be strongly -convex function, with modulus , for , if holds for all and .

Definition 16 (see [36]). A function is said to be strongly -convex, where if holds for all and

Next, we give a property of the kernel given in (7), which will be useful for finding the results of this paper.

: Let and be increasing functions. Then, for , , the kernel satisfies the following inequality:

It is easy to prove by using the following inequalities:

If and are of opposite monotonicities, then (18) holds in reverse direction. For further properties, see [37].

In Section 2, we will define a new notion of strongly -convex function which unifies several kinds of convex functions in a single inequality. By applying this new definition, we give generalizations of results for strongly convex functions. The results established here will produce generalizations and refinements of fractional integral inequalities for different kinds of convex and strongly convex functions which have been published in various papers.

2. Main Results

We give the definition of a function will be called strongly -convex function.

Definition 17. Let be an interval containing and let be a nonnegative function. A nonnegative function is said to be strongly -convex function with modulus if holds for all , .

The definition of strongly -convexity can be achieved by taking in (20).

Definition 18. Let be an interval containing and let be a nonnegative function. A nonnegative function is said to be strongly -convex function with modulus if holds for all , .

One can obtain from (20) definitions of strongly convex, strongly -convex, strongly -convex, strongly -convex, strongly -convex, and strongly -convex functions.

Theorem 19. Let be a positive strongly -convex function with modulus , . Let be strictly increasing and differentiable function, also let be an increasing function on and . If , , and , then for , the following inequality holds: while , .

Proof. Using , we can write the following inequalities Using strongly -convexity of , we have From (23) and (25), the following inequality is obtained: By setting on the right side and using (5) on left side of above inequality, we get The inequality (28) can take the following form: On the other hand, multiplying (24) and (26), and adopting the same pattern as we did for (23) and (25), the following inequality holds true: The inequality (30) can take the following form: By adding (29) and (31), (22) can be achieved.☐

Corollary 20. For , (22) gives the following inequality obtained for fractional integral operator defined in [19]:

Remark 21. (i)For , (22) gives ([38], Theorem 1)(ii)For and , (22) gives ([39], Theorem 1)(iii)For , , , and , (22) gives ([31], Theorem 8)(iv)For , , , , and , (22) gives ([40], Theorem 1)(v)For , , the result of (iv) gives ([40], Corollary 1)(vi)For , , and or , the result of (v) gives ([40], Corollary 2)(vii)For , , and , the result of (v) gives ([40], Corollary 3)(viii)For , , , and , (22) gives ([41], Theorem 1)(ix)For , , , and , (22) gives ([41], Corollary 1)(x)For , , , , and , (22) gives ([42], Theorem 2.1)(xi)For , , the result of (x) gives ([42], Corollary 2.2)(xii)For , , , , , and and using ([12], Remark 11), (22) gives ([42], Corollary 2.3)(xiii)For , , , , , and , (22) gives inequality (26) of ([42], Corollary 2.4) similarly, under the same assumptions along with (22) gives inequality (27) of ([42], Corollary 2.4)(xiv)For , , , , and , (22) gives ([43], Theorem 1)(xv)For , , the result of (xiv) gives ([43], Corollary 1)(xvi)For , , , , and , (22) gives ([44], Theorem 1)(xvii)For , , , , , and , (22) gives ([44], Corollary 1)(xviii)For , , , and , , (22) gives ([45], Theorem 2.1)(xix)For , , , , , and , (22) gives ([45], Corollary 2.1)(xx)For , , , and , (22) gives ([46], Theorem 1)(xxi)For , , , , and , (22) gives ([47], Theorem 1)(xxii)For , , the result of (xxi) gives ([47], Corollary 1)(xxiii)For , , , and , (22) gives ([15], Theorem 1)(xxiv)For , , the result of (xxiii) gives ([15], Corollary 1)(xxv)For , , and , (22) gives ([36], Theorem 4)(xxvi)For , the result of (xxiv) gives ([36], Corollary 1)(xxvii)For , the result of (xxiv) gives ([36], Corollary 2)(xxviii)For , the result of (xxvii) gives ([36], Corollary 3)(xxix)For , , and , (22) gives ([48], Theorem 1)(xxx)For , the result of (xxix) gives ([48], Corollary 1)(xxxi)For , the result of (xxix) gives ([48], Corollary 2)(xxxii)For , the result of (xxxi), gives ([48], Corollary 3)

For the proof of next theorem, we need the following lemma.

Lemma 22. Let , be a strongly -convex function with modulus , . If , then the following inequality holds:

Proof. As is strongly -convex function, we have Let . Then, we have , and using , the inequality (33) is obtained.☐

The upcoming theorem provides the Hadamard inequality for strongly -convex function.

Theorem 23. Under the assumptions of Theorem 19, in addition, if , then, we have

Proof. Using , we can write the following inequalities: Using strongly -convexity of , we have Multiplying (36) and (38) and integrating the resulting inequality over , we obtain By setting on the right side and using (5) on left side of above inequality, we get The inequality (40) can take the following form: Adopting the same pattern of simplification as we did for (36) and (38), the following inequality can be observed for (38) and (37): By adding (41) and (42), following inequality can be achieved: Multiplying both sides of (33) by and integrating over , one can get By using the Definition 4, one can obtain the following inequality: Now, multiplying by on both sides of (33), then integrating over , we get From (43), (45), and (46), inequality (35) can be achieved.☐

Corollary 24. For , (35) gives the following inequality obtained for the fractional integral operator that defined in [19]:

Remark 25. (i)For , , , and , (35) gives ([31], Theorem 22)(ii)For , , , , and , (35) gives ([40], Theorem 3)(iii)For , , the result of (ii) gives ([40], Corollary 6)(iv)For , , , , and , (35) gives ([43], Theorem 3)(v)For , , and in the result of (v) gives ([43], Corollary 6)(vi)For , , , , and , (35) gives ([44], Theorem 3)(vii)For , , the result of (vii) gives ([44], Corollary 3)(viii)For , , , and , , (35) gives ([45], Theorem 2.4)(ix)For , , the result of (ix) gives ([45], Corollary 2.6)(x)For , , and , (35) gives ([36], Theorem 6)(xi)For , the result of (xi) gives ([36], Corollary 5)(xii)For , , and , (35) gives ([48], Theorem 4)(xiii)For , the result of (xiii) gives ([48], Corollary 5)

Theorem 26. Let be a differentiable function such that is strongly -convex with modulus , . Let be strictly increasing and differentiable function, also let be a function which is increasing on the interval and . If , , , and , then for the following inequality holds: where

Proof. Since is strongly -convex function, one can have The inequality (50) can take the following form: From inequality (51), we have Multiplying (23) and (52) and integrating over , we obtain which gives Using the other inequality of (51) and doing on the same way as adopted for the right hand inequality, one can get From (54) and (55), the following inequality is observed: By applying strongly -convexity of , one can get By following the same steps as we did for (23) and (50), from (24) and (57), one can get the following inequality: By adding (56) and (58), inequality (48) can be achieved.☐

Corollary 27. For , (48) gives the following inequality obtained for the fractional integral operator that defined in [19]:

Remark 28. (i)For , (48) gives ([38], Theorem 4)(ii)For , , and , ((48) gives ([39], Theorem 3)(iii)For , , , and , (48) gives ([31], Theorem 25)(iv)For , , , , and , (48) gives ([40], Theorem 2)(v)For , , the result of (iv) gives ([40], Corollary 4)(vi)For , , , and , (48) gives ([41], Theorem 2)(vii)For , , and , the result of (vi) gives ([41], Corollary 2)(viii)For , , , , and , (48) gives ([42], Theorem 2.6)(ix)For , , the result of (viii) gives ([42], Corollary 2.7)(x)For , , , , and , (48) gives ([43], Theorem 2)(xi)For , , and , the result of (x) gives ([43], Corollary 4)(xii)For , , , , and , (48) gives ([44], Theorem 2)(xiii)For , , the result of (xii) gives ([44], Corollary 2).(xiv)For , and , , , (48) gives ([45], Theorem 2.3)(xv)For , , the result of (xiv) gives ([45], Corollary 2.5)(xvi)For , , and , (48) gives ([46], Theorem 2)(xvii)For , , , , and , (48) gives ([47], Theorem 3)(xviii)For , , the result of (xvii) gives ([47], Corollary 5)(xix)For , , , and , (48) gives ([15], Theorem 2)(xx)For , , the result of (xix) gives ([15], Corollary 2)(xxi)For , , and , (48) gives ([36], Theorem 5)(xxii)For in the result of (xxi) gives ([36], Corollary 4)(xxiii)For , , and , (48) gives ([48], Theorem 3)(xxiv)For , the result of (xxiii) gives ([48], Corollary 4)

3. Concluding Remarks

A new definition is given and utilized to obtain some integral inequalities via a unified integral operator. The established results provide generalizations of many well-known inequalities. They also give refinements of recently published results for convex, -convex, -convex, -convex, -convex, -convex, -convex and -convex functions. The reader also can obtain more fractional integral inequalities by setting appropriate functions and parameters involved in the kernel of unified integral operators.

Data Availability

There is no data required for preparation of this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research has been financed by the 2018 Nanjing Vocational College of Information Technology Doctoral special fund project “Study on the Climate Ensemble Forecast Model Based on the East Asian Monsoon Region” (YB20180903).

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Copyright © 2021 Zhongyi Zhang 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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