Abstract

The main objective of this article is to introduce the notion of –polynomial harmonically –convex function and study its algebraic properties. First, we use this notion to present new variants of the Hermite–Hadamard type inequality and related integral inequalities, as well as their fractional analogues. Further, we prove two interesting integral and fractional identities for differentiable mappings, and, using them as auxiliary results, some refinements of Hermite–Hadamard type integral inequalities for both classical and fractional versions are presented. Finally, in order to show the efficiency of our results, some applications for special means and error estimations are obtained as well.

1. Introduction and Preliminaries

The theory of convexity has become an astonishing and profound motivation for researchers in various fields of science. This theory has not just intriguing results in branches of mathematics and engineering but also gives mathematical computations and innovative tools for mathematicians to address and handle numerous issues because of its simplicity and wide scope of significance.

The concept of convexity theory is not a new one as it happens in some other structures in determining the orbital length. These days, the utilization of convexity theory can be straightforwardly or in a roundabout way seen in different subjects like geometry, probability theory, mathematical finance, optimization, functional analysis, etc. Convexity theory has shown up as an incredible procedure to solve several real-life issues in different branches of science. Numerous articles have been composed by various mathematicians on different kinds of convex functions for some related inequalities like inequality, Simpson inequality, Ostrowski inequality, Bullen type inequality, Opial inequality, Olsen type inequality, etc.

Inequalities have a lot of applications in the fields of differential equations and applied mathematics (see [1, 2]). This theory also has a wide range of applications in other areas of science, such as convexity theory, mathematical analysis, discrete fractional calculus, mathematical physics, and many more (see [37]). Let us start our main discussion with the notion of convexity, which is defined as follows.

Definition 1. A function is called convex, ifholds for every and .
Toplu et al. [8] very recently determined the inequality for –polynomial convex function.
Let be a convex function with and . Then,In the present paper, we have given more emphasis to introduce the concept of a new type of –polynomial harmonically convexity with some inspiring algebraic properties. To know detailed knowledge about the new concept, we encourage interested readers to get familiarized with the theory of harmonic convexity [9], –convexity [10], –polynomial convexity [8], and –polynomial harmonically convexity [11]. Park et al. in [12] generalized the notion of convexity as –polynomial –convexity. To follow some results associated with generalized –polynomial convex function established via fractional operators, see [1316], and for coordinates, see [17] and the references therein.

Definition 2. (see [10]). Let be any set and be a function. Then, the function is said to be –convex, ifTunç et al. [10] provided the following results related to –convexity involving classical and fractional operators.Chebyshev’s inequality: if and are two nonincreasing sequences such thatthenWe denote in the sequel of the paper the space of Lebesgue integrable functions on .

Definition 3. (see [18]). Let . The Riemann–Liouville fractional integrals of order for a real valued continuous function are defined byrespectively, where is the well-known gamma function.

Definition 4. (see [19]). Let and . Then, the –Riemann–Liouville fractional integrals of order for a real valued continuous function are defined byrespectively, where is the well-known k-gamma function defined by .
Convexity has recently gained popularity due to its numerous applications in various fields of science, and fractional calculus has played an important role in this development (see [20]). Sarikaya et al. [18] proposed the Riemann–Liouville fractional integral operator to present the Hermite–Hadamard inequality. The Hermite–Hadamard inequality for the -Riemann–Liouville fractional operator was generalized by Liu et al. [21], which was motivated by Sarikaya’s work. Similarly, generalized proportional fractional integrals were employed by various scholars, including Mumcu et al. [22]. Gürbüz et al. [23] worked on the Caputo–Fabrizio operator, Mohammed et al. [24] used tempered fractional integrals, Sahoo et al. [25] used Riemann–Liouville fractional integrals, and Khan et al. [26] established a new version of Hermite–Hadamard inequality employing generalized conformable fractional integrals. Butt and his collaborators, for example, worked on Jensen–Mercer type inequalities using novel fractional operators (see [2729]), while Set et al. [30] and Fernandez et al. [31] presented the Hermite–Hadamard inequality using the Atangana–Baleanu fractional operator. For some recent generalizations of the Hermite–Hadamard inequality, we suggest interested readers to see [3236] and the references therein. Recently, Naila et al. [37] presented some results for –convex function via fractional calculus. In [38], the authors worked on exponentially –convex function, and Tunç and Şanal in [39] obtained some perturbed trapezoid type inequalities for –convex function.
We recall the following hypergeometric function [40], which is defined by Euler in integral form:where is the Euler beta function.

Definition 5. (see [8]). Let be an interval and . A nonnegative real valued function is called –polynomial convex, ifholds for all .
Utilizing the concept of –polynomial convexity and harmonic convexity, then the concept of –polynomial harmonically convex function is established.

Definition 6. (see [11]). Let be an interval and . A nonnegative real valued function is said to be –polynomial harmonic convex, ifholds for every .
This paper is organized as follows. After reviewing some preliminary studies about convexity and inequalities, we introduce the notion of –polynomial harmonically –convex function in Section 2. In Section 3, we study some related algebraic properties. Section 4 deals with proving new variants of the type inequality for –polynomial harmonically –convex functions. In Section 5, we present an integral identity and some related inequalities of type for classical integrals. In Section 6, the fractional version of type inequality and its related results are discussed using Riemann–Liouville fractional operators. Some applications for special means and error estimations are derived in Section 7. Section 8 presents a brief conclusion and future scope of this study.

2. A New Definition

Definition 7. Let be an interval and . A nonnegative real valued function is said to be –polynomial –convex, ifholds for all .

Definition 8. Let be an interval and . A nonnegative real valued function is called –polynomial harmonically –convex, ifholds for every .

Remark 1. (i)For , –polynomial harmonically –convex function reduces to the classical –harmonic convex function.(ii)For , we haveSince as for , we get

Remark 2. Every –polynomial harmonically –convex function is a –convex function with .

3. Algebraic Properties of the New Class of Functions

In this section, we begin with discussing some algebraic properties in support of the newly developed class of functions.

Lemma 1. The inequality holds for all .

Proof. The proof is evident from Remark 1.
In what follows, we discuss some algebraic properties of –polynomial harmonically –convex function.

Proposition 1. Let be an interval and . Every harmonic –convex function on is an –polynomial harmonically –convex function.

Proof. Using the definition of –polynomial harmonically –convex function and from Lemma 1, we have

Proposition 2. Every decreasing convex functions defined by , and are harmonically –convex. Then, by Proposition 1, they are –polynomial harmonically –convex functions.

Proposition 3. The sum of two –polynomial harmonically –convex functions is an –polynomial harmonically –convex function.

Proof. The proof is straightforward and hence left for the readers.

Proposition 4. Let be an –polynomial harmonically –convex function and , where ; then, is an –polynomial harmonically –convex function.

Proof. The proof is straightforward and hence left for the readers.

Proposition 5. Let be a family of –polynomial harmonically –convex functions with for every , where and . Then, is an –polynomial harmonically –convex function on .

Proof. Let and ; then, we have

4. Hermite–Hadamard Type Inequalities

In this section, we will establish two Hermite–Hadamard type integral inequalities for –polynomial harmonically –convex functions.

Theorem 1. Let and be an –polynomial harmonically –convex function. If , then the following inequality holds:

Proof. Since is an –polynomial harmonically –convex function, we haveTaking and using the change of variable technique as follows:we getNow, integrating the above inequality with respect to on , we obtainThis completes the proof of the left inequality of (18).
For the right inequality of (18), taking and using the property of the –polynomial harmonically –convex function , we haveFrom inequalities (22) and (23), we getwhich completes the proof.

Remark 3. If we put in Theorem 1, then the following inequality for harmonically –convex function is obtained:

Theorem 2. Let be two –polynomial harmonically –convex functions on with . If , then the following inequality holds:where and .

Proof. Let and be two real valued, –polynomial harmonically –convex functions on . Then, we haveMultiplying above inequalities, we getIntegrating both sides of above inequality with respect to over , due to the fact that is a nonincreasing function, and using Chebyshev’s sum inequality, we obtainTaking , then inequality (29) reduces to desired inequality (26), which completes the proof.

5. Refinements of Hermite–Hadamard Type Inequalities

Here, we will establish a new equality, and taking this equality into account, some novel refinements of type inequalities are discussed.

Lemma 2. Suppose and consider a function , which is differentiable on with . If , then the following identity holds:

Proof. Let us denoteIntegrating by parts, we haveUsing change of variable technique, where , we obtainThis led us to the desired result.

Theorem 3. Suppose and consider a function , which is differentiable on with . If and is an –polynomial harmonically –convex function, then for and , the following inequality holds:where

Proof. Applying Lemma 2 and using Hölder’s inequality, –polynomial harmonically –convexity of , we haveThis led us to the desired result.

Theorem 4. Suppose and consider a function , which is differentiable on with . If and is an –polynomial harmonically –convex function, then for , the following inequality holds:where

Proof. Applying Lemma 2 and using the well-known power mean inequality, –polynomial harmonically –convexity of , we haveThis led us to the desired result.

Theorem 5. Suppose and consider a function , which is differentiable on with . If and is an –polynomial harmonically –convex function, then for , the following inequality holds:whereand the incomplete beta function is defined as

Proof. Applying Lemma 2 and using the well-known power mean inequality, –polynomial harmonically –convexity of , we haveThe proof is completed.

Theorem 6. Suppose and consider a function , which is differentiable on with . If and is an –polynomial harmonically –convex function, then for , the following inequality holds:where

Proof. Applying Lemma 2 and using the well-known power mean inequality, –polynomial harmonically –convexity of , we haveThe proof is completed.

6. Fractional Hermite–Hadamard Type Inequalities

Now applying –Riemann–Liouville fractional integrals, we obtain the following fractional version of Theorem 1.

Theorem 7. Let and be an –polynomial harmonically –convex function. If , then for , the following fractional inequalities hold:where .

Proof. Using the definition of –polynomial harmonically –convex function, we haveTaking and then changing the variables, we getNow, multiplying both sides of above inequality by and integrating with respect to , we obtain . Hence, .
Multiplying both sides by , we haveThis completes the proof of the left inequality.
For the right inequality, using the property of the –polynomial –convex function , we haveAdding above inequalities, we getMultiplying both sides of above inequality by and integrating with respect to , we obtainThen, we haveMultiplying both sides of above inequality by , we derived the desired result. The proof is completed.

Corollary 1. If we take in Theorem 7, we have

6.1. Refinements via Fractional Integral Operator

The following new lemma is very crucial for our next results.

Lemma 3. Suppose and consider a function , which is differentiable on with . If </