Abstract

The understanding of inequalities in convexity is crucial for studying local fractional calculus efficiency in many applied sciences. In the present work, we propose a new class of harmonically convex functions, namely, generalized harmonically --convex functions based on fractal set technique for establishing inequalities of Hermite-Hadamard type and certain related variants with respect to the Raina’s function. With the aid of an auxiliary identity correlated with Raina’s function, by generalized Hölder inequality and generalized power mean, generalized midpoint type, Ostrowski type, and trapezoid type inequalities via local fractional integral for generalized harmonically --convex functions are apprehended. The proposed technique provides the results by giving some special values for the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

1. Introduction

Fractional calculus is a developing arena in mathematics with profound presentations in all connected areas such as wave liquids, thermodynamics, image processing, virology, biological population models, chaos, and signals processing, see [1–5]. It has been employed to model physical phenomena that are originated to be paramountly described by fractional differential equations. Nowadays, the local fractional calculus is trying to report the nondifferentiable problems, for example, heat conduction problem involving local derivatives of fractional order [6, 7], local fractional Tricomi equation [8], and fractal vehicular traffic flow [9].

Also, several noted variants via local fractional integral have been investigated by many researchers, for example, Mo et al. [10] propounded Hermite-Hadamard inequalities for generalized -convex functions. Abdeljawad et al. [11] introduced some local fractional variants for generalized -convex functions in the fractal style with applications. Also, Sun and Liu [12] reported new generalizations of Hermite-Hadamard inequalities for generalized harmonically convex functions. For further investigations concerning to local fractional integral inequalities, we recommend the readers to [13–15] and the references therein.

It is well known that convexity theory has potential applications in many intriguing and captivating fields of research and furthermore played a remarkable role in numerous areas, such as coding theory, optimization, physics, information theory, engineering, and inequality theory. Several new classes of classical convexity have been proposed in the literature (see [16–26]). Many researchers endeavored, attempted, and maintain their work on the concept of convex functions, generalize its variant forms in different ways using innovative ideas and fruitful techniques [27–29]. In this regard, integral inequalities have played an important role in describing real world problems such as in elementary research conducted by Niculescu and Persson [30]. In this context, Hermite-Hadamard [31, 32] like type inequalities are very influential in convexity theory, which describes that if is convex function, then the double inequality holds for all with

Numerous approaches have been done by several researchers to inquire novel generalizations, speculations, and modifications for the Hermite-Hadamard inequality and its variant forms; we prescribe the related phenomena [33–36] to interested readers.

Now, we evoke the concept of harmonically convex function [37] as follows.

Definition 1 ([37]). Let be an interval, and a real-valued function is said to be harmonically convex if the inequality holds for all and

In [37], can derived the Hermite-Hadamard inequality for harmonically convex function. holds for all with and (the set of Lebesgue integrable functions).

To make this collection more comprehensive, we introduce the notion of generalized harmonically --convex functions in the sense of Raina’s function on fractal set. By considering generalized harmonically --convex functions, certain refinements of Hermite-Hadamard type inequalities are derived that can be used to characterize some useful consequences of harmonically -convex functions with the appropriate assumption of , which are connected to Raina’s function. Remarkable special cases are captured, and new Ostrowski type results are proved for the aforesaid class of functions. Additionally, our findings for the new schemes are also constituted some better approaches in various directions for proposing variation in accordingly Remark 12. Application of the -type special mean and Mittag-Leffler functions are presented by mingling the suggested methodology and the theory of the special functions. We hope that the idea and results obtained herein will be a catalyst for further investigation.

2. Prelude

In what follows, we demonstrate a different notion of differentiation that merges the idea of fractional differentiation and fractal derivative; recently, one has established the accompanying sets [6].

For the sake of brevity, we mention (the set of natural numbers), (set of positive integers), (set of rational number), and (set of real numbers), respectively.

Also, whenever the -type set of real line numbers are included, the is supposed to be implicitly One has also characterized by two binary operations the addition “+” and the multiplication “.” (which is conventionally omitted) on the -type set of real line numbers as follows.

For two binary operations the addition “+” and the multiplication “.” are defined as and for

Then, we have the following asserations to show (1) is an abelian group: for (i)(ii)(iii)(iv) is the additive identity for (v) for any is the inverse element of for (2) is an abelian group: for (vi)(vii)(viii)(ix) is the multiplicative identity for (x) for any is the inverse element of for (3)Distributive law holds:

Proposition 2. The following asserations holds true: (i) is a field(ii)The additive identity and the multiplication identity are unique(iii)The additive inverse element and the multiplicative inverse element are unique(iv)For each , its inverse element may be written as but not as ; for each , its inverse element may be written as (v)If the order is defined on as follows: if and only if then is an ordered field like

In [6], Yang introduced the following notions for local fractional continuity and differentiability as follows:

Definition 3 ([6]). A nondifferentiable mapping is said to be local fractional continuous at if for any there exists , satisfying that holds for If is local continuous on then we symbolize it by .

Definition 4 ([6]). The local fractional derivative of of order at is defined by the expression where Let If there exists for any then it is denoted by where .

Definition 5. Let and let be a partition of which satisfies Then, the local fractional integral of on of order is defined as follows where and .
Here, it follows that if and if For any if there exists then it is denoted by

Lemma 6 ([6]). (1)Suppose that then (2)Suppose that and then

Lemma 7 ([6]).

An analog in the fractal set of the classical Hölder’s inequality has been established by [15], which is asserted by the following lemma.

Lemma 8 ([15]. Generalized Hölder’s inequality). For with and let then

In the work of Sun and Liu [13], we present the following concept related with the notion of generalized harmonically convexity.

Definition 9 ([13]). Let be an interval and let then is said to be generalized harmonically convex function if the inequality holds:

Recently, several noted consequences and motivating facts related to fractal theory have been appeared in the literature [5, 14, 15]. In [13], Sun explored the following analog of the Hermite-Hadamard inequality (3) for generalized harmonically convex functions.

Firstly, let be a bounded sequence of real numbers, and denotes the Raina’s function.

In [38], Raina explored a new class of functions stated as: where and is bounded sequence of positive real numbers. Notice that if we choose in (13), then where , and are parameters which can choose aribtrary real and complex values (provided that ,), and we have the notion by then the classical hypergeometric function stated as follows

Also, if with and restricting its domain to in (13), then we have the classical Mittag-Leffler function:

Next, we evoke a novel concept of harmonic set and mappings including Raina’s functions.

Definition 10. Let and be a bounded sequence of real numbers. A nonempty set is said to be generalized harmonically -convex set, if for all

Next, we introduce the notion of generalized harmonically --convex function as follows:

Definition 11. Let and be a bounded sequence of real numbers. If a mapping satisfies the following inequality: holds for all and

Remark 12. In view of Definition 11, we have that (i)Choosing then we acquire harmonically -convex function as follows: (ii)If we take and then we get Definition 3.1 in [13](iii)If we take along with then we get Definition in [37]

Recall the two special functions on Yang’s fractal sets. (1)Generalized beta function (2)Generalized hypergeometric function

3. Hermite-Hadamard Type Inequality

In this section, we derive a novel version of Hermite-Hadamard inequalities for generalized harmonically -convex functions, and this is the main motivation of this paper.

Theorem 13. For a bounded sequence of real numbers and be generalized harmonically --convex function where such that and Then, the following inequalities hold: