On inequalities of Hermite-Hadamard type via n-polynomial exponential type s-convex functions

Abstract: In this paper, a new class of Hermite-Hadamard type integral inequalities using a strong type of convexity, known as n-polynomial exponential type s-convex function, is studied. This class is established by utilizing the Hölder’s inequality, which has several applications in optimization theory. Some existing results of the literature are obtained from newly explored consequences. Finally, some novel limits for specific means of positive real numbers are shown as applications.


Introduction
Convex function analysis begins with real-valued functions of a real variable. They serve as a model for deep generalization into the context of numerous variables and a variety of applications; see book [1,2] for more details. Convexity theory gives us a coherent framework for developing extremely efficient, fascinating, and strong numerical tools for tackling and solving a wide range of problems in various domains of mathematics. Zhao et al. [3] utilized the convexity and concavity to modify first kind Bessel functions. Several intriguing generalizations and extensions of classical convexity have been employed in optimization and mathematical inequalities.
Convexity theory was also influential in the development of inequalities theory. Baleanu et al. [4] studied a class of Hermite-Hadamard-Fejer type inequalities using fractional integral. In 2021, Wu et al. [5] explored another class of inequalities via an extended fractional operator. By using this theory, lots of work has been done in the field of inequalities [6][7][8][9]. The theory of convexity covers a wide range of convex functions, including exponentially convex function, m-convex functions, and s-convex function. It has vast applications in many disciplines of theoretical and practical mathematics. A powerful and elegant connection between analysis and geometry characterizes the current perspective on convex functions [10]. Definition 1.1. [1] The function Ω : K → R, where K is known to be convex on a compact subset Ω of real numbers if for any two points θ and ϑ in K and for any 0 ≤ r ≤ 1 holds.
The classical Hermite-Hadamard's inequality is widely studied via convex functions [11][12][13]. It is stated in the following definition. Definition 1.2. Let Ω : K → R be a convex mapping, where K ⊆ R. For any two points θ and ϑ in K with θ < ϑ If Ω is concave, the inequalities are reversed.
The exponentially convex functions are used to modify for statistical learning, sequential prediction, and stochastic optimization; see [14,15]. Antczak et al. and Noor et al. were the first to propose the class of exponentially convex functions [16][17][18]. Furthermore, the concept of an exponentially s-convex function was explored by Mehreen, and Anwar [19].
Definition 1.4. The beta function is defined by the integral for Re(u) > 0 and Re(v) > 0.
The following equality can be built in [21] and is stated as: Lemma 1.5. Let K ⊆ R and Ω : K → R be a differentiable function on K 0 . If θ and ϑ are any two points in K with θ < ϑ, then we have In [22], Kirmaci presented the following lemma.
Lemma 1.6. Let Ω : K → R be a differentiable function on K 0 and K ⊆ R. If θ and ϑ are any two points in K 0 with θ < ϑ, then we have Let Ω : K → R be a differentiable mapping on K 0 , where K ⊆ R and θ < ϑ.
If Ω is a convex mapping, then the following inequalities hold The proof of above lemma can be found in [21].
, then the following equality holds (1.8)

Main results
This section contains significant outcomes on Hermite-Hadamard type inequalities that are evaluated using n-polynomial exponentially s-convex functions by utilizing Hölder's inequality. The definition and properties of the Beta function are used to yield the main results.
, then we have the following inequality holds: (2.10) Proof. By using Lemma 1.6, we have By using Lemma 1.7, we obtain Since |Ω | σ is n-polynomial exponentially s-convex, so there arise two cases. Case (i). For σ = 1. From n-polynomial exponentially s-convexity of |Ω | on 3θ−ϑ 2 , 3ϑ−θ 2 and using the facts e sr ≤ e s and e s(1−r) ≤ e s for any 0 ≤ r ≤ 1, we obtain Similarly, we have By substituting inequalities (2.12) and (2.13) in (2.11), we get Case (ii). For σ > 1. By using the Hölder's inequality for σ > 1 and the facts e sr ≤ e s and e s(1−r) ≤ e s for any 0 ≤ r ≤ 1, we get (2.14) Similarly, we have So the inequalities (2.11), (2.14), and (2.15) gives the proof of required result.
Theorem 2.8. Under the assumptions of Theorem 2.7, we have the following inequality holds: Proof. By using first the Hölder's inequality and then n-polynomial exponentially s-convexity along with the facts e sr ≤ e s and e s(1−r) ≤ e s for any 0 ≤ r ≤ 1, we get (2.26) Keeping in mind (2.22) and (2.26), we obtained (2.25).

Applications to means and trapezoid formulae
In this section, we use the main results of Section 2 to give some applications to special means of positive real numbers. We first need to recall the following basic definitions of different means and techniques of numerical integration.
The midpoint rule for numerical integration is stated as follows: Let D be a partition with points θ = t 0 < t 1 < · · · < t m−1 < t m = ϑ of the interval [θ, ϑ] and consider the quadrature formula for the trapezoidal version and for the midpoint version and E j (Ω, D) represent the approximate error.
Then the following inequality holds: Proof. Using Corollary 1 with the substitution Ω(t) = t m and by simple mathematical calculation, we get (3.1).
Proof. From Corollary 1, we obtain On the other hand, we have , Ω 3t j+1 − t j 2 .

Conclusions
In this paper, we examined key generalizations of convexity known as n-polynomial exponentially s-convex functions. We have utilized the well-known Hölders inequality to explore new identities for Hermite-Hadamard inequalities. We demonstrated how our newly developed results are utilized to establish certain means of two positive real numbers in various ways. In the future, the methodologies used in this study can generate new inequalities for n-polynomial exponentially s-convex functions.