Novel Refinements via n–Polynomial Harmonically s–Type Convex Functions and Application in Special Functions

In this work, we introduce the idea of n–polynomial harmonically s–type convex function. We elaborate the new introduced idea by examples and some interesting algebraic properties. As a result, new Hermite–Hadamard, some refinements of Hermite– Hadamard and Ostrowski type integral inequalities are established, which are the generalized variants of the previously known results for harmonically convex functions. Finally, we illustrate the applicability of this new investigation in special functions (hypergeometric function and special mean of real numbers).


Introduction
The theory of convexity presents an amazing, fascinating, and captivating field of research and also played significant and important roles in many areas, such as statistics, economics, optimization, management science, finance, engineering, game theory, and mathematical inequalities. In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen's inequality, can be used to deduce inequalities such as the arithmetic-geometric mean inequality and Hölder's inequality. Due to widespread views, robustness, and a lot of applications, the theory of convexity has become a rich source of motivation and absorbing field for researchers. Using the theory of convexity, mathematicians provided amazing tools and numerical techniques to tackle and to solve a wide class of problems, which arise in pure and applied sciences. This theory has a rich and paramount history and has been the focus and fixed point of intense study for over a century in mathematics. Many generalizations, variants, and extensions for the convexity have attracted the attention of many researchers, see [1][2][3]. This theory also played a meaningful role in the development of the theory of inequalities. In diverse and opponent research, inequalities have a lot of applications in statistical problems, probability, and numerical quadrature formulas. Integral inequalities have an interesting mathematical model due to its robustness and pivotal applications in fractional integral calculus and mathematical analysis. In approximation theory, integral inequalities explain and brief the growth rate of competing mathematical analysis. For the applications, interested readers refer to [4][5][6]. Recently, Toplu et al. [7] introduced a generalization form of convexity namely n -polynomial convex functions.
It is well known that the harmonic mean is the special case of power mean. It is often used for situations when the average rates are desired and have a lot of applications in different fields of sciences which are statistics, computer science, trigonometry, geometry, probability, finance, and electric circuit theory. Harmonic mean is the most appropriate measure for rates and ratios because it equalizes the weights of each data point. Harmonic mean is used to define the harmonic convex set. In 2003, the first harmonic convex set was introduced by Shi and Zhang [8]. Harmonic convex function was first introduced and discussed by Anderson et al. [9]. Awan et al. [10] keeping his work on generalizations introduced a new generalized class of convex function called n-polynomial harmonic convex function. For the literature and attention for the readers about harmonically convex functions, see [11][12][13][14]. Motivated and inspired by the ongoing activities and research in the convex analysis field, we find out that there exists a special class of functions known as stype convex function. Recently, Rashid et al. [15] introduced n-polynomial s-type convex function. Eventually, many mathematicians put an effort, hardworking, and has collaborated with different ideas and concepts in the field of convex analysis. The amazing techniques and remarkable ideas of this article may inspire and motivate for further research in this pivotal, captivating, and valuable field. Before we start, we need the following necessary known definitions and literature.
Motivated by the above results, literature and ongoing activities, and research in this amazing and captivating field, we will give in Section 3 the idea and its algebraic properties of n-polynomial harmonically s-type convex function. In Section 4, we will derive the new version of Hermite-Hadamard inequality by using the newly introduced definition. As a result in Sections 5 and 6, we will give some refinements of the Hermite-Hadamard and Ostrowski type inequalities, and in Section 7, we will give some applications for our proposed new definition. Finally, a brief conclusion will be provided as well.

New n-Polynomial Harmonically s-Type
Convex Function and Its Properties We are going to introduce a new definition called an n-polynomial harmonically s-type convex function and will study some of their algebraic properties.

Remark 7.
(i) Taking n = 1 in Definition 6, we obtain the following new definition about harmonically s-type convex 2 Journal of Function Spaces (ii) Taking s = 1 in Definition 6, then, we get a definition, namely, n-polynomial harmonically convex function, which is defined by Awan et al. [10] (iii) Taking n = 1 and s = 1 in Definition 6, then, we get a definition, namely, harmonically convex function, which is introduced by _ Iscan [16] (iv) Taking n = 2 and s = 1 in Definition 6, we obtain the following new definition about 2-polynomial harmonically convex function That is the beauty of this newly introduce definition if we choosing different values of n and s, as a result, we obtain new amazing integral inequalities and also found some results which connect with previous results. Proof. The proof is evident. Proposition 9. Let I ⊂ ð0,+∞Þ be a harmonically convex set. Every harmonically convex function on a harmonically convex set is an n-polynomial harmonically s-type convex function.
Proof. Using the definition of harmonically convex function and from the Lemma 8, Proposition 10. Every n-polynomial harmonically convex function is an n-polynomial harmonically s-type convex function.
Proposition 11. Every n-polynomial harmonically s -type convex function is harmonically h-convex function with hðκ Now, we make some examples via newly introduce definition n-polynomial harmonically s-type convex function.
Example 12. ψðxÞ = e x is an increasing convex function, so it is harmonically convex function (see [17]). By using Proposition 9, it is an n-polynomial harmonically s-type convex function.
These are the clear advantages of the proposed new definition with respect to other known functions on the topic mentioned above. Now, we will study some of their algebraic properties. (1) Let ψ and φ be an n-polynomial harmonically s-type convex, then (2) Let ψ be an n-polynomial harmonically s-type convex function, then which completes the proof.

Remark 17.
Interested readers can find many other nice properties of this new class of functions. We omit here the details.

Hermite-Hadamard Type Inequality via n-Polynomial Harmonically s-Type Convex Functions
The purpose of this section is to derive a new version of Hermite-Hadamard type using n-polynomial harmonically stype convexity.
Proof. Since ψ is an n-polynomial harmonically s-type convex function, we have which lead to Using the change of variables, we get Integrate the above inequality with respect to κ over [0,1], then which completes the left side inequality. For the right side inequality, changing the variable of integration as x = ½ϑ 1 ϑ 2 / κϑ 2 + ð1 − κÞϑ 1 and using Definition 6 for the function ψ, we have which completes the proof.

Refinements of (H-H) Type Inequality via n-Polynomial Harmonically s-Type Convex Functions
In order to obtain some refinements of (H-H) type inequality using n-polynomial harmonically s-type convex functions, we need the following lemma.
Proof. From Lemma 22, Hölder's inequality, n-polynomial harmonically s-type convexity of jψ ′ j q , and properties of modulus, we have 6 Journal of Function Spaces which completes the proof.

Ostrowski Type Inequalities via n-Polynomial Harmonically s-Type Convex Functions
In this section, we are going to add some new Ostrowski type inequalities via this newly introduced definition namely npolynomial harmonically s-type convex function. In order to obtain the results, we need the following lemma.
If the function jψ′j q is an n -polynomial harmonically s -type convex, for q ≥ 1 and s ∈ ½0, 1, then for all x ∈ ½ϑ 1 , ϑ 2 , one has where

Journal of Function Spaces
Proof. Using Lemma 37, power mean inequality, n-polynomial harmonically s-type convexity of jψ ′ j q and properties of modulus, we have which completes the proof.