Exploration of Hermite–Hadamard-Type Integral Inequalities for Twice Differentiable h -Convex Functions

: The signiﬁcance of fractional calculus cannot be underestimated, as it plays a crucial role in the theory of inequalities. In this paper, we study a new class of mean-type inequalities by incorporating Riemann-type fractional integrals. By doing so, we discover a novel set of such inequalities and analyze them using different mathematical identities. This particular class of inequalities is introduced by employing a generalized convexity concept. To validate our work, we create visual graphs and a table of values using speciﬁc functions to represent the inequalities. This approach allows us to demonstrate the validity of our ﬁndings and further solidify our conclusions. Moreover, we ﬁnd that some previously published results emerge as special consequences of our main ﬁndings. This research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by using generalized fractional operators and expanding the concept of convexity.


Introduction and Preliminaries
Fractional calculus deals with arbitrary order integrals and derivatives that are employed in several applications.In recent years, the area related to fractional differential and integral equations has received much attention from numerous mathematicians and specialists [1].The derivatives of fractional order represent physical models of multiple phenomena in many fields, including engineering [2,3], mathematical physics [4], fractional calculus [5] and bio-engineering [6].The idea of convexity has been modernized, extended and expanded in several ways [7,8].Convexity has a significant impact on our daily lives because of its numerous applications in commerce [9], industry [10], medicine [11] and the arts [12].Geometrically convex functions in n-dimensions and s-dimensions have been derived.Some classes of convex functions, such as geometrically convex and s-convex, were investigated by Yang and Hudzik et al. in [13,14], respectively.Functional analysis, optimization and control theory all depend on convexity in significant ways [15].Due to the activities of its definition, convexity has a strong character in the area of inequalities.The idea of convexity has played a significant and astonishing role in integral inequalities that are equally important for both fields of pure and applied mathematics [16,17].
The classical Hermite-Hadamard inequality was first introduced in 1893 [24].Peter Korus [25,26] successfully updated a class of Hermite-Hadamard inequalities by incorporating the class of generalized convex derivatives.Some other authors studied the Fejer-Hadamard inequalities for convex functions in [27][28][29][30][31][32].This inequality basically provides the bounds of the average value of a convex function.Moreover, it provides error boundaries of specific means relations and numerous numerical quadrature rules of integration such as rectangular, trapezoidal and Simpson [33,34].For further studies, we refer the reader to articles [35,36] and book [37].In the proposed work, we first establish some identities involving fractional integrals of the Riemann-type.Such identities will then be used to investigate Hermite-Hadamard-type integral inequalities for twice differentiable h-convex functions.The Hölders inequality will be utilized to create this class.The gamma function is defined in [38] by the following.Definition 1.The Euler gamma function defined for Re(y) > 0, as In [39], Mubeen et al. defined the following definition of the k-gamma function.
Definition 2. The k-gamma functions defined for Re(y) > 0, as , where (y) m,k is the Pochhammer k-symbols for k > 0 and factorial function.The integral form is given by Clearly, Γ(y) = lim k→1 Γ k (y), and the relation between classical gamma and k-gamma is The following expression represents the complete beta function as defined in reference [40].Definition 3. The complete beta function is defined as The incomplete beta function [41] and relation between beta and gamma functions are given in the following.Definition 4. The incomplete beta function is defined as where y ∈ [0, 1] and ı 1 > 0, ı 2 > 0. The notable relation between the beta and gamma function is stated as The definition of the h-convex function presented in [42] is defined as follows.
Definition 5. Let h : Ω → be a positive increasing function.A function , and for all ı 1 , ı 2 ∈ ℵ, we have This definition generalizes the following convexities.
(i) If we set h(ξ) = l in (1), then we obtain the convex function (ii) If we set h(ξ) = l s in (1), then we obtain the s-convex function in the second sense 1), then we obtain the Godunova Levin functions.
, then the Riemann-Liouville fractional integrals of the order ζ are defined by and are known as the left-and right-sided Riemann-Liouville fractional integrals with Γ(.) as the gamma function.

Definition 7 ([39]
).Let ∈ L[ı 1 , ı 2 ], then the fractional integrals of order ζ defined by and are known as the left-and right-sided k-Riemann-Liouville fractional integrals with a k-gamma function.
To establish the main results, we need to recall the following lemmas presented in [44,45], respectively.Lemma 1.For all ξ ∈ [0, 1], we have ; then, the equation for k-fractional integrals is as follows: The motivation behind this work is to explore and analyze a new set of inequalities that involve mean-type concepts by employing more general convexity and fractional integrals.The Höder's inequality is used to establish these results that have applications in diverse areas, including mathematics, statistics, engineering and computer science, where it serves as a valuable tool for analyzing and solving a wide range of problems.Through visual representations and validation of our findings, we seek to contribute to the existing body of knowledge and inspire further research in this area, potentially leading to advancements in mathematical theory and applications.

Main Results
In this section, we first present some important identities using a generalized fractional operator of the Riemann-type.Secondly, we explore Hermite-Hadamard inequalities by first establishing some identities.
; then, we have the identity Integrating by parts, we obtain Substituting ( 5) and ( 6) in (4), we obtain with (7) on both sides, we can write Substituting Lemma 2 into (8), we obtain the required result.Hence, the proof is complete.
By using Lemma 2 and Lemma 1, we obtain the following results.
holds, where h is bounded by M.
Proof.By using Lemma 2, we have By using the h-convexity of | |, we can write The required proof is complete.
Example 1.The inequality presented in Theorem 1 can be verified by sketching the graph of (29).
For this purpose, we substitute (ξ) = e ξ and obtain the following and By utilizing the expressions ( 13) and ( 14) in ( 29), we obtain Corresponding to the choice of the parameters M = 1, . .
The numerical values in Table 1 corresponds to Figure 1.
By using the h-convexity of | | ω 2 , we obtain Hence, the proof is complete.
Example 2. The inequality presented in Theorem 2 can be verified by sketching the graph of (31).
For this purpose, we substitute (ξ) = ξ 2 and obtain the following relations and By utilizing ( 17) and ( 18) in (31), we obtain Corresponding to the choice of the parameters M = 1, The numerical values in Table 2 corresponds to Figure 2. , Remark 4. By substituting h(ξ) = ξ s and k = 1 in Theorem 2, we obtain ([46] Theorem 3.2), i.e., , In the next two results, we used Lemma 3.
Proof.By using Lemma 3, we can write By using the h-convexity of | |, we have .
Hence, the desired result is proved.
Example 3. The inequality presented in Theorem 3 can be verified by sketching the graph of (20).For this purpose, by utilizing the expressions ( 13) and ( 14) in (20), we obtain .
Corresponding to the choice of the parameters M = 1, . .
The numerical values in Table 3 corresponds to Figure 3.
Theorem 4. Consider a function defined on the interval [ı 1 , ı 2 ] such that its second derivative, denoted by , exists on this interval.Assume that the function belongs to the class of integrable functions L[ı 1 , ı 2 ] and is both h-convex and a positive monotonic function.Additionally, suppose that 0 ≤ ı 1 < ı 2 .Then, the inequality holds, where |h(x)| ≤ M and Proof.By using Hölder's inequality and Lemma 3, we have .
Theorem 5. Suppose we have a function defined on the interval [ı 1 , ı 2 ] such that the absolute value of its second derivative, denoted by | |, exists.If | | is both monotonic and positive and it is also h-convex on the interval [ı 1 , ı 2 ], where 0 ≤ ı 1 < ı 2 , then the following inequality holds: where |h(x)| ≤ M.
Proof.By using Lemma 4, we can write By using the h-convexity of | |, we can write However, Additionally, Substituting these values of integrals in (24), we obtain the required result.
By using the h-convexity of | | ω 2 , we can write + However, , and . Additionally , and .
Substituting the values of integrals in (27), we obtain the required result.The proof is complete.

Some Applications to the Main Results in Terms of Means
In mathematics, the means we employ hold profound significance in various domains such as problem-solving, statistical analysis, optimization problems and mathematical proofs.This section contains applications of thhe main results in terms of means.The representation of the means are given as follows: (i) The arithmetic mean: The logarithmic mean: Let ı 1 ,ı 2 ∈ R + , ı 1 < ı 2 ; then, we have the following inequalities.
Proof.Using Theorem (1) and making some simplification, we can write this as By substituting (ξ) = e ξ , k = This proved relation (29).

Concluding Remarks
Convexity, a concept that originated from Archimedes around 250 B.C., is a simple and intuitive notion with far-reaching implications in various aspects of our daily lives, including industry, business, medicine and art.Its application is particularly prominent in the field of inequalities, which holds significant importance in optimization theory.In our recent research, we focused on exploring the Hermite-Hadamard integral inequality by employing h-convex functions and a Riemann-type fractional integral.By leveraging Hölder's inequality, we introduced novel findings that have broad implications for inequality theory.These results were derived based on a newly established identity, allowing us to extend previously published findings and broaden the scope of our investigation.To establish the validity of our obtained results, we represented the double inequalities using graphical representations and tables of values.This comprehensive approach provides concrete evidence supporting our conclusions and further solidifies the significance of our research.Our research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by incorporating generalized fractional operators and expanding the scope of convexities.By embracing these broader perspectives, we anticipate the discovery of more general results that can advance the theory of inequalities and enrich the field of fractional calculus. and

Table 1 .
The comparison results in Example 1 between the double inequality are presented in the following table.Figure 1.The graph of Theorem 1 for the choice of order 1 ≤ ζ ≤ 4 is presented in Figure 1.

Table 2 .
The comparison results in Example 2 between the double inequality are presented in the following table.

Table 3 .
The comparison results in Example 3 between the double inequality are presented in the following table.

Table 4 .
The comparison results in Example 4 between the double inequality are presented in the following table.

Table 5 .
The comparison results in Example 5 between the double inequality are presented in the following table.