Abstract

This article is aimed at studying novel generalizations of Hermite-Mercer-type inequalities within the Riemann-Liouville -fractional integral operators by employing -convex functions. Two new auxiliary results are derived to govern the novel fractional variants of Hadamard-Mercer-type inequalities for differentiable mapping whose derivatives in the absolute values are convex. Moreover, the results also indicate new lemmas for , , and and new bounds for the Hadamard-Mercer-type inequalities via the well-known Hölder’s inequality. As an application viewpoint, certain estimates in respect of special functions and special means of real numbers are also illustrated to demonstrate the applicability and effectiveness of the suggested scheme.

1. Introduction

Recently, two fundamental notions have been introduced in pure and applied analysis having potential utilities in every field and are known as convexity and concavity. Interestingly, the convexity theory is attributed to Jensen. Several monographs and articles have played a prominent role in the developments, speculations, and modifications of convexity in different directions such as -convexity, harmonic convexity, -convexity, and strong convexity. Moreover, a strong connection has been developed between diverse kinds of convex functions and inequality theory. Their fertile applications in optimization theory, functional analysis, physics, and statistical theory have made it a much fascinating subject, and hence, it is assumed as an incorporative subject between combinatorics, orthogonal polynomials, hypergeometric functions, quantum theory, and linear programming. This is the major motivation behind the keen investigation and progress of the integral inequalities in the literature [1, 2].

Let and let be nonnegative weights such that . The famous Jensen’s inequality (see [1]) in the literature states that if is a convex function on the interval , then for all , , and . It is one of the key inequalities that helps to extract bounds for useful distances in information theory (see [3, 4]).

In 2003, a new variant of Jensen’s inequality was introduced by Mercer [5].

If is a convex function on , then for all , , and .

Matkovic and Pećarić proposed several generalizations on Jensen-Mercer operator inequalities [6]. Later on, Niezgoda [7] has provided several extensions to higher dimensions for Mercer-type inequalities. Recently, the Jensen-Mercer-type inequality has made a significant contribution to inequality theory due to its prominent characterizations.

In the present study, we consider , the class of Breckner -convex functions (which for were called in [8, 9]), -convex in the second sense. In [10], Dragomir and Fitzpatrick introduce the concept of a real-valued Breckner -convex function on a convex subset of a linear space as whenever with and . For , it reduces to the usual notion of convexity. As a result, he generalizes Jensen’s inequality (1) as whenever , , and .

In [9], the class of -convex functions in the first and second senses is introduced along with their significant properties.

Definition 1. Let , a real-valued function on an interval is -convex in the second sense provided that (3) holds for all and with .

They denote this class of function by . Moreover, they proved that the class is stronger than convexity in the first and original sense for . Several properties of -convex functions in both senses are presented in a comprehensive manner with supporting examples. It is interesting to see that if and , then is nonnegative. This result may not hold in general when the function is convex (i.e., ). Also, the situation is more interesting when .

Viewing this literature, we intend to extend the Jensen-Mercer inequality for Breckner -convex functions. For this, we use the ideology of Mercer’s concept [5] and give the following important lemma.

Lemma 2. If is a real-valued Breckner -convex function on the interval and such that , , and , then for any finite positive increasing sequence , we have for all .

Proof. Let . Then, , so the pairs and possess the same midpoint. Since that is the case, there exist , with such that and . Therefore, employing (3) and the assumed condition, we get which completes the proof.

Now, we give the result for the Jensen-Mercer inequality in the Breckner -sense.

Theorem 3. Let be positive real numbers such that and . If is a real-valued Breckner -convex function on , then for any finite positive increasing sequence , we have

Proof. One can prove it by following a similar idea as in [5]; however, we need to employ Lemma 2 and generalized Jensen’s inequality (4) for Breckner -convex functions.

For further properties and applications of -convex functions, see [9, 11] and references therein. The following lemma is of great interest for applications.

Lemma 4 (see [11]). Let be a convex function. Then, the following results hold: (i)If is nonnegative, then it is -convex for (ii)If is nonpositive, then it is -convex for One of the famous integral inequalities for convex functions is the Hermite-Hadamard inequality (see [2]): provided that if a mapping is a convex function on and .

Fractional-order calculus deals with more general behavior than integer-order calculus, and it not only provides new mathematical methods for practical systems but also has been applied into various fields due to its accurate description in many active fields, such as fraction-order memristive chaotic circuit, fractional-orderrelaxation-oscillation model, mathematical biology, and economics (see [12]).

The theory of Riemann-Liouville -fractional integrals is a pertinent extension of Riemann-Liouville fractional integrals. It is important to note that if , the properties of Riemann-Liouville -fractional integrals are quite dissimilar from those of general fractional integrals. For this, the Riemann-Liouville -fractional integrals have agitated the interest of many researchers. Now, we demonstrate some essential concepts of -fractional calculus for the investigation of our results.

Definition 5 (see [13]). Diaz and Pariguan have defined the -gamma function , a generalization of the classical gamma function, which is given by the following formula: where is the Pochhammer -symbol given by It is shown that the Mellin transform of the exponential function is the -gamma function clearly given by for with , where stands for the -gamma function.

Many researchers have generalized the classical and fractional operators by introducing a parameter about a decade ago. Mubeen et al. [14] used special -function theory in fractional calculus for the first time in the literature in the form of Riemann-Liouville -fractional integrals.

Definition 6 (see [15]). Let . The Riemann-Liouville -fractional integrals of with and of order with are defined by respectively.

Remark 7. If , then Riemann-Liouville -fractional integrals reduce to classical Riemann-Liouville fractional integrals. And if and , the fractional integral reduces to the classical integral.

Recently, many researchers are presenting new fractional differential and integral operators and they generalized by using the iteration procedure and by introducing a new parameter . They also found relationships of these generalized fractional operators with existing fractional and classical operators under the special values of the parameter . Many -fractional operators, their properties, related identities, and inequalities are proved during the past years. For instance, see [16, 17] and references therein.

2. Main Results

This section contains several new generalizations of Hermite-Hadamard-Mercer-type inequalities for -convex functions in the second sense (Breckner sense) via -fractional calculus theory.

Throughout the paper, we assumed the following assumptions:

: let with , , and for some fixed , and is the -gamma function.

: for , , whenever we use the definition of the Jensen-Mercer inequality for the -convex function.

Theorem 8. Let be the -convex function such that along with assumptions and . Then, the following Riemann-Liouville -fractional integral inequalities hold: where and is the beta function.

Proof. By employing the definition of the -convex function , we get By change of variables and , , we get Now, multiplying the above inequality by and then integrating w.r.t. over yield By change of variable, we have and so the first inequality of (13) is proved.
Now, for the proof of the second inequality of (13), we first note that if is an -convex function, then for , it gives By adding the inequalities of (19) and (20), we have Now, multiplying the above inequality by and then integrating w.r.t. over , we get Consequently, we get Combining (18) and (23), one can get (13). In order to prove (14), we employ the Jensen-Mercer inequality for the -convex function in the second sense; then, for , it yields Now, by change of variables and , and in (25), we have Multiplying by and then integrating w.r.t. over give It follows that and so the first inequality of (14) is proved.
Now, for the proof of the second inequality of (14), we first note that if is an -convex function, then for , it gives Multiplying by and then integrating w.r.t. over give Therefore, we have Adding to both sides in (31), we get the second inequality of (14).