Refinements of Some Integral Inequalities for -Convex Functions

Abstract

In this paper, we are interested to deal with unified integral operators for strongly -convex function. We will present refinements of bounds of these unified integral operators and use them to get associated results for fractional integral operators. Several known results are connected with particular assumptions.

1. Introduction and Preliminaries

Convex functions play an important role in the formation of new definitions of related functions which help to give the generalization of classical results. Therefore, in recent years, many generalizations of convex functions are defined and utilized to study the Hadamard and other well-known inequalities (see [1–9]). In this paper, we deal with the strongly -convex functions to study the bounds of unified integral operators. The obtained results are compared with already known results.

First, we give some definitions of functions which are necessary for the findings of this paper.

Definition 1 (see [7]). A function is said to be convex on ifholds for all and , where is an interval. Reverse of inequality (1) defines as concave function.

Definition 2 (see [10]). A function is said to be strongly convex with modulus ifholds for all and .

Definition 3 (see [3]). A function is said to be -convex on ifholds for all and , where is a bifunction.

Definition 4 (see [2]). A function is said to be strongly -convex on ifholds for all and , where is a bifunction.
It is to be noted that for , strongly -convex function reduces to strongly convex function. Farid in [11] defined the unified integral operators (5) and (6) and has proved the continuity and the boundedness of these integral operators. The aim of this paper is the study of integral inequalities for strongly -convex functions via unified integral operators. Next, we give definition of the unified integral operators.

Definition 5. Let where be the function such that f is positive and integrable over and is differentiable and strictly increasing. Also, let be an increasing function on and , , and . Then, for , the left and right integral operators are defined as follows:whereBy choosing specific functions and fixing parameters involved in the Mittag-Leffler function , various known fractional integrals can be reproduced (see [5], Remarks 6 and 7). In [4], by using unified integral operators, we have obtained integral inequalities for -convex functions. In the following, we give these inequalities in the form of Theorems 1–3.

Theorem 1. Let be a positive -convex function and be differentiable and strictly increasing function. Also, let be an increasing function on , , , , and . Then, for , we have

Theorem 2. Along with the assumptions of Theorem 1, if and , then the following result holds:Also, the following result holds for the convolution of functions and .

Theorem 3. Let be two differentiable functions such that is -convex and be strictly increasing for . Also, be an increasing function on and , and and . Then, for , we haveAlthough we follow the same method which was adopted to prove the results of [4], here we will get refinements of these results by using strongly -convex functions. In Section 2, we give the refinements of bounds of unified integral operators given in Definition 5. In Section 3, we will present refinements of bounds of fractional integral operators.

2. Main Results

Throughout this section, we have adopted the following notations:

Theorem 4. If is positive strongly -convex function with modulus , along with other assumptions of Theorem 1, then we havewhere is the identity function.

Proof. For the kernel defined in (7) and the strongly -convexity of the function on , the following inequalities hold, respectively:The aforementioned inequalities are used to obtain the following integral inequality:In view of Definition 5 and applying integration by parts, from inequality (15), we get the following upper bound of the right-sided unified integral operator:Again for the kernel defined in (7) and the strongly -convexity of the function on , the following inequalities hold, respectively:The aforementioned inequalities (17) and (18) are used to obtain the following integral inequality:In view of Definition 5 and applying integration by parts, from inequality (19), we get the following upper bound of the left-sided unified integral operator:Inequality (12) will be obtained by combining (16) and (20).

Corollary 1. By setting in (12), we get

Remark 1.  For in (12), we get inequality (8) of Theorem 1; if and , then we will get the refinement of (8). For in (21), we get the result for strongly convex function. For and in (21), we get the result of Theorem 8 in [5].We will use the following lemma for our next result.

Lemma 1. Let be strongly -convex function with modulus . If , thenholds for .

Proof. Strongly -convexity of implies Using the condition in the above inequality, we get (22).

Remark 2.  For , Lemma 1 reduces to Lemma 1 of [4]. For , we get its refinement. For and , Lemma 1 reduces to Lemma 21 of [5].

Theorem 5. Let and in addition with the assumptions of Theorem 4. Then, the following inequality holds:

Proof. For the kernel defined in equation (7) and the strongly -convexity of the function on , the following inequalities hold, respectively:The aforementioned inequalities are used to obtain the following integral inequality:In view of Definition 5, applying integration by parts, and using , from inequality (27), we get the following upper bound of the left-sided unified integral operator:Also, the following inequality holds:The aforementioned inequalities (26) and (29) are used to obtain the following integral inequality:In view of Definition 5 and applying integration by parts, from inequality (30), we get the following upper bound of the right-sided unified integral operator:Now, using Lemma 1, we can writeIn view of Definition 5 and , from (32), we get the following upper bound of the left-sided unified integral operator:Also, from Lemma 1, we can writeIn view of Definition 5 and , from (34), we get the following upper bound of the right-sided unified integral operator:Inequality (24) will be obtained by using (28), (31), (33), and (35).

Remark 3.  For in (24), we get (9) of Theorem 2; if , then we will get refinement of (9). For in (24), we get the result for strongly convex function. For and in (24), we get the result of Theorem 22 in [5].

Theorem 6. If is strongly -convex with modulus along with other assumptions of Theorem 3, then the inequalityholds for , whereand is the identity function.

Proof. Using strongly -convexity of over givesUsing absolute value property, we can writeThe aforementioned inequality (13) and second inequality of (40) are used to obtain the following integral inequality:In view of (37) and applying integration by parts, from inequality (41), we get the following upper bound:Also, inequality (13) and the first inequality of (40) are used to obtain the following integral inequality:Now, using -convexity of over , we haveInequalities (17), (38), and (44) are used to obtain the following upper bounds:Inequality (36) will be obtained by using (42)–(46).

Corollary 2. By setting in (36), we get the following inequality:

Remark 4.  For in (36), we get inequality (10) of Theorem 3; if and , then we will get the refinement of (10). For in (47), we get the result for strongly convex function. For and in (47), we get the result of Theorem 25 in [5].

3. Results for Fractional Integral Operators

In this section, we give the bounds of some of the fractional integral operators which will be deduced from the results of Section 2. Throughout this section, we assume that .

Proposition 1. Under the assumptions of Theorem 4, the following result holds:

Proof. For , Theorem 4 gives (48).

Proposition 2. Under the assumptions of Theorem 4, the following inequality holds:

Proof. For as identity function, Theorem 4 gives (49).

Corollary 3. For , (5) and (6) reduce to the fractional integral operators given in [5]. Further, the following bound for is also satisfied:

Corollary 4. For , where , and as identity function, (5) and (6) give fractional integrals defined in [12]. Further, the following bound is also satisfied:

Corollary 5. Using and as identity functions, (5) and (6) reduce to the fractional integral operators given in [13]. Further, the following bound is also satisfied:

Corollary 6. For , and , (5) and (6) reduce to the fractional integral operators given in [14]. Further, the following bound is also satisfied:

Corollary 7. For , and , (5) and (6) give the following fractional integral operators:Further, the following bound is also satisfied:

Corollary 8. For and , (5) and (6) reduce to the fractional integral operators given in [15]. Further, the following bound is also satisfied:

Corollary 9. For , , (5) and (6) reduce to the fractional integral operators given in [16]. Further, the following bound is also satisfied:

Corollary 10. Using and in (5) and in (6), where , fractional integral operators given in [17] are obtained. Further, the following bound is also satisfied:

Corollary 11. For , and in (5) and in (6), where , fractional integral operators given in [18] are obtained. Further, the following bound is also satisfied:

Remark 5.   For , all the results of Section 3 reduce to the results of Section 3 in [4]; if , then all the results of Section 3 give the refinements of the results of Section 3 in [4]. For and , all the results of Section 3 reduce to the propositions and corollaries of [5].Further, various bounds can be obtained by applying Theorems 5 and 6 which we leave for the reader.

4. Concluding Remarks

The paper presents bounds of unified integral operators (5) and (6) for strongly -convex functions. These bounds are refinements of bounds obtained for unified integral operators for -convex functions in [4]. The results for fractional integral operators have been deduced which provide bounds for Riemann–Liouville and other well-known fractional integral operators.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).