Hermite–Hadamard Type Inequalities via Generalized Harmonic Exponential Convexity and Applications

In this work, we introduce the idea and concept of m–polynomial p–harmonic exponential type convex functions. In addition, we elaborate the newly introduced idea by examples and some interesting algebraic properties. As a result, several new integral inequalities are established. Finally, we investigate some applications for means. The amazing techniques and wonderful ideas of this work may excite and motivate for further activities and research in the different areas of science.


Introduction
Theory of convexity present an active, fascinating, and attractive field of research and also played prominence and amazing act in different fields of science, namely, mathematical analysis, optimization, economics, finance, engineering, management science, and game theory. Many researchers endeavor, attempt, and maintain his work on the concept of convex functions and extend and generalize its variant forms in different ways using innovative ideas and fruitful techniques. Convexity theory provides us with a unified framework to develop highly efficient, interesting, and powerful numerical techniques to tackle and to solve a wide class of problems which arise in pure and applied sciences. In recent years, the concept of convexity has been improved, generalized, and extended in many directions. The concept of convex functions also played prominence and meaningful act in the advancement of the theory of inequalities. A number of studies have shown that the theory of convex functions has a close relationship with the theory of inequalities.
The integral inequalities are useful in optimization theory, functional analysis, physics, and statistical theory. In diverse and opponent research, inequalities have a lot of applications in statistical problems, probability, and numerical quadrature formulas [1][2][3]. So eventually due to many generalizations, variants, extensions, widespread views, and applications, convex analysis and inequalities have become an attractive, interesting, and absorbing field for the researchers and for attention; the reader can refer to [4][5][6]. Recently Kadakal and Iscan [7] introduced a generalized 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 the situations when the average rates is desired and have a lot of applications in different field 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, first time harmonic convex set was introduced by Shi [8]. Harmonic and p-harmonic convex function was for the first time introduced and discussed by Anderson et al. [9] and Noor et al. [10], respectively. Awan et al. [11] keeping his work on generalizations, introduced a new class called n-polynomial harmonically convex function. Motivated and inspired by the ongoing activities and research in the convex analysis field, we found out that there exists a special class of function known as exponential convex function, and nowadays, a lot of people working are in this field [12,13]. Dragomir [14] introduced the class of exponential type convexity. After Dragomir, Awan et al. [15] studied and investigated a new class of exponentially convex functions. Kadakal and İşcan introduced a new definition of exponential type convexity in [16]. Recently, Geo et al. [17] introduced n-polynomial harmonic exponential type convex functions. The fruitful benefits and applications of exponential type convexity is used to manipulate for statistical learning, information sciences, data mining, stochastic optimization and sequential prediction [7,18,19] and the references therein. Before we start, we need the following necessary known definitions and literature.

Preliminaries
In this section, we recall some known concepts.
For the harmonic convex function, İşcan [20] provided the Hermite-Hadamard type inequality.
The function ψ is called Jensen p-harmonic convex function.
If we put p = −1 and p = 1, then p-harmonic convex sets and p-harmonic convex functions collapse to classical convex sets, harmonic convex sets, and harmonic convex functions, respectively.
Motivated by the above results, literature, and ongoing activities and research, we organise the paper in the following way. Firstly, we will give the idea and its algebraic properties of m-polynomial p-harmonic exponential type convex functions. Secondly, we will derive the new sort of (H-H) and refinements of (H-H) type inequalities by using the newly introduced idea. Finally, we will give some applications for means and conclusion.

Generalized Exponential Type Convex Functions and Its Properties
We are going to introduce a new definition called m-polynomial p-harmonic exponential type convex function and will study some of their algebraic properties. Throughout the paper, one thing gets in mind m represents finite ℤ + , m-poly p-harmonic exp convex function represents m-polynomial p -harmonic exponential type convex function and (H-H) represents Hermite-Hadamard.

Remark 6.
(i) Taking m = 1 in Definition 5, we obtain the following new definition about p -harmonically exp convex function: 2 Journal of Function Spaces (ii) Taking m = 2 in Definition 5, we obtain the following new definition about 2-poly p-harmonically exp convex function: (iii) Taking p = 1 in Definition 5, then, we get a definition, namely, m-poly harmonically exp convex function which is defined by Geo et al. [17] (iv) Taking p = −1 in Definition 5, we obtain the following new definition about m-poly exp convex function: (v) Taking m = 1 and p = 1 in Definition 5, we obtain the following new definition about harmonically exp type convex function: (vi) Taking m = 1 and p = −1 in Definition 5, then, we get a definition, namely, exponential type convex function which is defined by Kadakal et al. [16] That is the beauty of this newly introduce definition if we put the values of m and p, then, we obtain new inequalities and also found some results which connect with previous results.
Proof. The rest of the proof is clearly seen.

Proposition 8.
Every p-harmonic convex function is m-poly p-harmonic exp convex function.
(i) If p = 1 in Proposition 9, then as a result, we get harmonically convex function, which is introduced by Noor et al. [22] (ii) If p = −1 in Proposition 9, then as a result, we get hconvex function, which is defined by Varošanec [6] Now, we make and investigate some examples by way of newly introduced definition.
Example 12. If ψðνÞ = 1/ν 2p , ∀x ∈ ℝ \ f0g is p-harmonic convex function, then according to Proposition 8, it is an m-poly p-harmonic exp convex function. Now, we will discuss and investigate some of its algebraic properties.

Journal of Function Spaces
(ii) For c ∈ ℝðc ≥ 0Þ, cψ is an m-poly p-harmonic exp convex function Proof.
(i) In case of m = 1, we investigate the following new inequality: (ii) In case of p = 1, the above Theorem 15 collapses to Theorem 3.3 in [17] (iii) In case of m = p = 1, as a result, we obtain the following new inequality: (iv) In case of p = −1, then, the above Theorem 15 collapses to the following new inequality: (v) In case of m = 1 and p = −1, as a result, the above Theorem 15 collapses to the Theorem (2.2) in [16] Theorem 17. Let 0 < ℘ 1 < ℘ 2 ,ψ j : ½℘ 1 , ℘ 2 ⟶ ½0,+∞Þ be a class of m-poly p-harmonic exp convex functions and ψðuÞ = sup j ψ j ðuÞ. Then, ψ is an m-poly p-harmonic exp convex function and U = fu ∈ ½℘ 1 , ℘ 2 : ψðuÞ<+∞g is an interval.

. (H-H) Type Inequality via Generalized Exponential Type Convexity
The main object of this section is to investigate and prove a new version of (H-H) type inequality using m-poly pharmonic exp convexity.
Proof. Since ψ is an m-poly p-harmonic exp convex function, we have which lead to ψ 2x p y p x p + y p Using the change of variables, we get Integrating the above inequality with respect to κ on ½0, 1, we obtain which completes the left side inequality. For the right side inequality, first of all, we change the variable of integration by ν = ½ð℘ which completes the proof.

Corollary 22.
In case of m = 1 in Theorem 21, then, we get the following new (H-H) type inequality for p-harmonic exp convex functions: Remark 24.
(i) In case of p = 1, then as a result, we obtain Theorem 4.1 in [17] (ii) In case of m = 1 and p = −1, then as a result, we obtain Theorem 3.1 in [16] (iii) In case of m = 1 and p = 1, then as a result, we obtain Corollary 1in [17]

Refinements of (H-H) Type Inequality via Generalized Exponential Type Convexity
In this section, in order to prove our main results regarding on some Hermite-Hadamard type inequalities for m-poly p -harmonic exp convex function, we need the following lemmas: where A κ = ½κ℘ Using integration by parts Lemma 26 (see [23]). Let ψ : Journal of Function Spaces Theorem 27. Let ψ : I = ½℘ 1 , ℘ 2 ⊆ ℝ \ f0g ⟶ ℝ be differentiable function on the I ∘ of I. If ψ ′ ∈ L½℘ 1 , ℘ 2 and jψ ′ j q is an m-poly p-harmonic exp convex function on I, q ≥ 1, then where Proof. Using Lemma 25, properties of modulus, power mean inequality, and m-poly p-harmonic exp convexity of the jψ′j q , we have which completes the proof.
Corollary 31. Under the assumptions of Theorem 30 with p = −1, we have the following new result: Corollary 32. Under the assumptions of Theorem 30 with p = 1, we have the following new result: where Theorem 33. Let ψ : I = ½℘ 1 , ℘ 2 ⊆ ℝ \ f0g ⟶ ℝ be differentiable function on the I ∘ of I. If ψ′ ∈ L½℘ 1 , ℘ 2 and jψ ′ j q is an m-poly p-harmonic exp convex function on I, q ≥ , 1 then where Proof. Using Lemma 26, properties of modulus, power mean inequality, and m-poly p-harmonic exp convexity of the jψ′j q , we have which completes the proof.
Corollary 34. Under the assumptions of Theorem 33 with p = −1 and m = 1, we have the following new result: