Some Ostrowski type inequalities via n-polynomial exponentially s-convex functions and their applications

1 Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan 2 Department of Mathematics, Institute of Technical Education and Research, Siksha O Anusandhan University, Bhubaneswar 751030, Odisha, India 3 Virtual University Islamabad, Lahore Campus, Pakistan 4 Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia 5 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa 6 China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 7 College of Business Administration-Finance Department, Dar Al Uloom University, Saudi Arabia


Introduction and preliminaries
These days, the investigation on convexity theory is considered as a unique symbol in the study of the theoretical conduct of mathematical inequalities. As of late, a few articles have been published with a special reference to integral inequalities for convex functions. Specifically, much consideration has been given to the theoretical investigations of inequalities on various kinds of convex functions, for example, s-type convex functions, Harmonic convex functions, tgs-convex functions, Exponential type convex functions, GA-convex functions, (α, m)-convex functions, MT -convex functions, Hyperbolic convex functions, Trigonometrically convex functions, Exponential s-type convex functions, and so on. Many researchers have worked on the above mentioned convexities in different directions with some innovative applications. The most interesting aspect of these variants of convex functions is that each definition is generalization of other one for some specific values. For example, if we choose s = 1 in exponentially s-convex function, it simply reduces to exponentially convex function. For the attention of the readers, see the references [1][2][3][4][5][6][7][8].
In 1938, Ostrowski introduced the following useful and interesting integral inequality see [9], page 468 .
Let ϕ : J ⊆ R → R be a differentiable mapping on J o , the interior of the interval J, such that ϕ ∈ L[α 1 , α 2 ], where α 1 , α 2 ∈ J with α 2 > α 1 . If |ϕ (z)| ≤ K, for all z ∈ [α 1 , α 2 ], then the following inequality: holds. The above inequality (1.1), in literature is known as the well known Ostrowski inequality. For some detailed knowledge about the recent researches on this inequality and related generalizations and extensions, see [10][11][12][13][14][15][16] . This inequality yields an upper bound for the approximation of the integral average 1 ϕ (u) du by the value of ϕ (u) at the point u ∈ [α 1 , α 2 ]. Since the time it is established, broad research history on establishing numerous generalizations of Ostrowski type inequalities have been directed. Most of the earlier and current investigations use different properties of the function and additionally use convexity and bounded variation. The posterity of Ostrowski type inequalities has a great role in the numerical integration theory as utilized by numerous mathematicians. They furnish the numerical integration field with a huge class of quadrature and cubature rules as well.
In assorted and rival research, inequalities have a ton of utilizations in measure theory, mathematical finance, statistical problems, probability, and numerical quadrature formulas. Meng et al. in [17,18] applied the convex model for optimization via probabilistic approach. Brown in his article [19], explained the relationship between inequalities and measure theory. Numerous broadly known outcomes about inequalities can be acquired utilizing the properties of convex functions. In 1994, for the first-time Hudzik and Maligranda [20] presented the class of s-convex functions in the second sense. Further towards this path, Dragomir and Fitzpatrick [21] put endeavors and set up new fundamental inequalities by means of s-convex functions. In 2019,İşcan [22] developed some new Hermite-Hadamard type inequalities for s-convex functions with the help of notable inequalities like improved power-mean Integral inequality and Hölder-İscan integral inequality.
In the frame of simple calculus, we explore and attain the novel refinements of Ostrowski type inequalities. To the best of our knowledge, a comprehensive investigation of newly introduced definition, namely n-polynomial exponentially s-convex function in the present paper is new. Recently, it is seen that many scientists are interested in big data analysis, deep learning and information theory utilizing the concept of exponentially convex functions. Motivated by the ongoing research and literature, the present paper is structured in the following way, first in Section 2, we will give some necessary known definitions and literature. Second in Section 3, we will explore the concept of n-polynomial exponentially s-convex functions. In addition, some algebraic properties and examples for the newly introduced definition are elaborated. In Section 4, we attain the new sort of Hermite-Hadamrd type inequality. Further, in Section 5, we investigate some refinements of the Ostrowski type inequality and some special cases. Finally, in the next section we present some applications to special means and a conclusion.

Preliminaries
In this section, we recall some known concepts.
Breckner in his article [26] introduced, s-convex functions. Hudzik in his paper [20] presented a few properties and connections with s-convexity in the first sense. Usually, when we put s = 1 for s-convexity, it reduces to the classical convexity. In [21], Dragomir et al. proved a generalized Hadamard's inequality, which holds for s-convex functions in the second sense.
Recently, many researchers investigated about the importance and development of the theory of exponentially convex functions. In 2020, Kadakal et al. [27], investigated a new class of exponential convexity, which is stated as follows: [27] A non-negative real-valued function ϕ : J ⊂ R → R is known to be an exponential convex function if the following inequality holds: (2.4) Definition 2.4. [28] A non-negative real-valued function ϕ : J → R is called n-polynomial convex, if holds for every α 1 , α 2 ∈ J, χ ∈ [0, 1], s ∈ [0, 1] and n ∈ N.

Generalized exponentially s-convex function and its properties
Next, due to afore-mentioned research activities, we are able and capable to introduce the generalized form of exponential type convexity, which is called an n-polynomial exponentially s-convex function. Further, we will try to discuss and explore its properties.
Definition 3.1. Let n ∈ N and s ∈ [ln 2.4, 1]. Then the non-negative real-valued function ϕ : J ⊂ R → R is known to be an n-polynomial exponentially s-convex function if the following inequality holds: We represent the class of all n-polynomial exponentially type convex functions on the interval J as POLEXPC(J) for each α 1 , α 2 ∈ J and χ ∈ [0, 1].
Proof. The proof is simple.
From the above lemmas, it can be clearly seen that, the new class of n-polynomial exponentially convex function is very larger when compared to the known class of functions like convex, exponentially convex, s-convex and exponentially s-convex. This is an added advantage of the newly proposed Definition 3.1.
Now, we will makes some examples in the support of the newly introduced function.  Example 5. The Great mathematician Dragomir clearly investigated and proved that in published article [21], the function ϕ(x) = x ls , x > 0 is s-convex function, for the all mentioned conditions s ∈ (0, 1) and 1 ≤ l ≤ 1 s . But, using Proposition 2, it is also an n-polynomial exponentially s-convex function for s ∈ [ln 2.4, 1].
(ii) For nonnegative real number c, cψ is an n-polynomial exponentially s-convex function.
Proof. (i) Let ψ and ϕ be n-polynomial exponentially s-convex functions, then (ii) Let ψ be an n-polynomial exponentially s-convex function, then  Proof. For all α 1 , α 2 ∈ P and χ ∈ [0, 1], and for the same fixed numbers s ∈ [ln 2.4, 1], we have This shows that P is an interval.
We have proved that ϕ is bounded above by M.

Hermite-Hadamard type inequality
In this section, we present one Hermite-Hadamard type inequality for the n-polynomial exponentially s-convex function.

Then (4.2) leads to
Now, integrating both sides of the inequality (4.3) with respect to χ from 0 to 1, one has This completes the proof of first part of inequality (4.1). Next, we prove the second part of inequality (4.1). Let χ ∈ [0, 1]. Using the fact that ϕ is an n-polynomial exponentially s-convex function, we obtain and By adding the above inequalities, we obtain Now, integrating both sides of the above inequality with respect to χ from 0 to 1, then making the change of variable, we obtain which leads to the conclusion that The proof is completed.

Refinements of Ostrowski type inequality involving n-polynomial exponentially s-convex functions
In this section, we present some enhancements of the Ostrowski type inequality for differentiable n-polynomial exponentially s-convex function. Here, we need the following lemma as given in [29].
Corollary 1. If we choose n = 1 in Theorem 5.2, we obtain
Corollary 3. If we choose n = 1 in Theorem 5.3, we obtain

Conclusions
In this paper, we have taken into consideration a critical extension of convexity, that is referred as n-polynomial exponentially s-convex functions and acquired new variants of Hermite-Hadamard inequality employing this new definition. We have also obtained refinements of the Ostrowski inequality for functions whose first derivatives in absolute value at certain power are n-polynomial exponential-type s-convex. Moreover, for different values of parameters, i.e., s, n and z, we have deduced some special cases of our main results. We presented some applications of our established results to special means of two positive real numbers. In the future, new inequalities for the other npolynomial convex functions can be obtained by utilising the techniques used in this paper.