Hermite-Hadamard and Ostrowski type inequalities via α -exponential type convex functions with applications

: This paper introduced and investigated a new form of convex mapping known as α - exponential type convexity. We presented several algebraic properties associated with this newly introduced convexity. Additionally, we established novel adaptations of well-known inequalities, including the Hermite-Hadamard and Ostrowski-type inequalities, speciﬁcally for this convex function. We also derived special cases of these newly established results. Furthermore, we provided new estimations for the trapezoidal formula, demonstrating practical applications of this research.

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, strongly quasi convex function, (p, h)-convex functions, tgs-convex functions, Exponential type convex functions, GA-convex functions, MT-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.One intriguing feature of these different forms of convex functions is that each definition can be seen as a generalization of the other under certain specific conditions.For more details, see [9][10][11][12][13][14][15][16][17][18].
Motivated by ongoing developments and studies in this subject, it has been revealed that there is one particular type of convexity known as exponential convexity, and lots of researchers are now trying to enhance it.Dragomir [19] and Antczak [20] presented the concept of exponential type convexity, and Awan [21] investigated another class of exponential convex function.More recently, Mahir Kadakal and Iscan [22] introduced another meaning of exponential-type convexity.
The main purpose of the article is to introduce the notion of an α-exponential type convex function and derive the variants of the classical Hermite-Hadamard and Ostrowski type inequalities by use of the class of α-exponential type convex functions.We also discuss several new special cases for the obtained results, which show that our obtained results are generalizations and extensions of some previously known results.
Researchers have shown a great interest in big data analysis, deep learning and information theory, utilizing the concept of exponential convex functions.As a result, we anticipate that the introduction of the concept of α-exponential convex functions could capture the attention of these scientists, leading to further advancements in the fields of deep learning, data analysis and information theory.Moreover, many mathematicians have done studies in q-calculus analysis; the interested reader can see [23][24][25][26].
Integral inequalities are commonly satisfied by convex functions, including the well-known Hermite-Hadamard inequality.The Hermite-Hadamard inequality for a convex function Φ : This inequality holds for all µ 1 , µ 2 ∈ I with µ 1 < µ 2 .Some refinements and generalizations of the H-H inequality have been obtained by [27] and the references therein.Let a differentiable function Φ : I ⊆ R −→ be defined on the interior of I along with µ 1 , µ 2 ∈ I o , where µ 1 < µ 2 and also 2 .The above inequality is a well-known Ostowski inequality.For more details, see [28][29][30][31].Here, we recall some known concepts.The exponential convex functions are defined as follows.
The present paper is structured in the following way: In section two, we explore the concept of an α-exponential type convex function and give some of its algebraic properties.In section three, we derive the Hermite-Hadamard inequality for an α-exponential type convex function.In section four, we establish an Ostrowski type inequality for an α-exponential type convex function.Additionally, in section five, we provide new estimations for the trapezoidal formula as practical applications.Finally, in the next section, the conclusion is presented.

α-exponential type convex function and its properties:
Now, we introduce an α-exponential type convex function and give some of its algebraic properties for the newly defined class of function.
We study specific relationships between the class of exponential convex functions and other forms of convex functions.
Lemma 2.1.The subsequent inequalities hold Proof.The proof follows directly by expanding the exponential series.
Proposition 1.Every exponential convex function is an α-exponential type convex function.
Proof.Let µ 1 , µ 2 ∈ I with 0 ≤ υ ≤ 1, for α ∈ and we get e αµ 2 and x ∈ [µ 1 , µ 2 ] is any arbitrary point.Also, consider Thus, since e υ ≤ e and e 1−υ ≤ e, for 0 ≤ υ ≤ 1, we have We established that Φ is bounded above by the real number B. Similarly, we can show that Φ is bounded below.

Some new inequalities for α-exponential type convex function
The objective of this section is to investigate various estimates that enhance the H-H inequality for functions in which the first derivative in absolute value at certain power is an α exponential type convex.Dragomir and Agarwal employed the subsequent lemma in their work [32].
Proof.From Lemma 3.1, we have Applying Holder's integral inequality, we find Since |Φ| q is an α-exponential type convex function, we get Theorem 7. Let a differentiable function Φ : I −→ be defined on the interior of I along with µ 1 , µ 2 ∈ I o , where µ 1 < µ 2 and q > 1 and also is a convex function of an α-exponential type then the subsequent inequality satisfied for 0 ≤ υ ≤ 1: Proof.From Lemma 3.1, we have Applying the power mean inequality, we find . (3.17) Since |Φ| q is an α-exponential type convex function, we get

Refinements of Ostrowski type inequality for an α-exponential type convex functions
Here, we introduced several improvements to the Ostrowski type inequality applicable to differentiable α-exponential type convex functions.Cerone and Dragomir employed the subsequent lemma in their work [33].
Proof.Using Lemma 4.1, since |Φ | is an α-exponential type convex function and in Theorem 8 yields the subsequent midpoint inequality: (2) By assuming z = µ 1 in Theorem 8 yields the subsequent inequality: (4.4) (3) By assuming z = µ 2 in Theorem 8 yields the subsequent inequality: Theorem 9. Suppose a mapping Φ : I −→ , which is differentiable on , then the subsequent inequality holds true for 0 ≤ υ ≤ 1: ) Proof.Utilizing both well-known Holder's inequality and Lemma 4.1 given that |Φ | q is an αexponential type convex function and |Φ (z)| q ≤ K, we deduce: e αz + (e − 2) in Theorem 9 yields the subsequent midpoint inequality: (2) By assuming z = µ 1 in Theorem 9 yields the subsequent inequality: (3) By assuming z = µ 2 in Theorem 9 yields the subsequent inequality: Proof.Employing from both Lemma 4.1 and the power mean inequality and considering that |Φ | q is an αexponential type convex function while |Φ(z)| ≤ K, we arrive at the following result: in Theorem 10 yields the subsequent midpoint inequality: (2) If we choose z = µ 1 in Theorem 10 it yields the subsequent inequality: .12) (3) If we choose z = µ 2 in Theorem 10 it yields the subsequent inequality: . (4.13)

Applications
Assuming that d represents a partition of the interval [µ 1 , µ 2 ] such that d : the trapezoidal formula can be expressed as follows: If either the second derivative of Φ does not exist or is unbounded, then (5.1) cannot be used.However, Dragomir and Wang [34][35][36] demonstrated that R(Φ, d) can be calculated using only the first derivative, which can have several practical applications.(5.5) Proposition 3. Let a differentiable function Φ : I ⊆ R 0 → R be defined on the interior of I, along with µ 1 , µ 2 ∈ I where µ 1 < µ 2 .Assuming that |Φ| q is an α-exponentially convex function on the interval [µ 1 , µ 2 ], and given that p, q > 1 such that 1 p + 1 q = 1, then within the context of (5.(5.6)