Some Further Results Using Green’s Function for s -Convexity

. For s -convex functions, the Hermite–Hadamard inequality is already well-known in convex analysis. In this regard, this work presents new inequalities associated with the left-hand side of the Hermite–Hadamard inequality for s -convexity by utilizing a novel technique based on Green’s function. Also, H¨older, Young, and power-mean inequalities are used to obtain these new inequalities. Finally, some applications to special means of real numbers are provided. In conclusion, we think that the methodology used in this work will encourage more research in this feld.


Introduction
Convexity is a fundamental idea in both applied and pure mathematics, acting as a powerful tool for evaluating functions and sets, proving inequalities, and modeling and resolving practical issues. In many areas of mathematics and beyond, this idea is essential for estimating integrals and setting boundaries. Relevant researchers can access some articles on convex functions from references [1][2][3][4][5][6][7][8][9].
Tus, we recall the elementary notation in convex analysis.

Defnition 1. A set I ⊂ R is said to be convex function if
for each ϖ, ϕ ∈ I and ζ ∈ [0, 1].
Defnition 2. Te mapping ℘: I ⟶ R is said to be a convex function if the following inequality holds: for all ϖ, ϕ ∈ I and ζ ∈ [0, 1]. If (− ℘) is convex, then ℘ is said to be concave. In terms of geometry, this indicates that if B, U, and Z are three separate locations on the graph of ℘, with U between B and Z, then U is on or below the chord BZ. In many diferent felds, convex functions are crucial. For instance, the concavity of ℘ is described in terms of declining returns for a production function φ � ℘(L) in economics. If ℘ is convex, then increasing returns are shown. In addition, a convex function applied to the expected value of a random variable is always limited beyond the convex function's expected value, according to the theory of probability. It is possible to derive other inequalities, including the geometric-arithmetic mean inequality and Hölder's inequality, using this conclusion, known as Jensen's inequality. Te idea of convexity has developed into a rich source of inspiration and a fascinating topic for scholars because of its widespread viewpoints, resilience, and plenty of applications. Mathematicians have developed incredible tools and numerical methods using the notion of convexity to deal with and resolve an enormous number of issues that emerge in the pure and applied sciences. Tis theory has a long and important history, and for more than a century, mathematics has focused on and concentrated on it. On the other hand, there are a lot of new issues in applied mathematics where the idea of convexity is insufcient to adequately characterize them in order to have benefcial consequences. Because of this, the idea of convexity has been expanded upon and developed in various research studies; see [10][11][12][13][14][15][16].
Utilizing various forms of convexity, some important inequalities have been observed. s-convexity is one of the several varieties of convexity. Hudzik and Maligranda in reference [17] took into account, among other things, the class of functions that are s-convex in the second sense. Te following is the defnition of this class: a function ℘: holds for all ϖ, ϕ ∈ [0, ∞), ζ ∈ [0, 1] and for some fxed s ∈ (0, 1]. Te class of s-convexity is frequently denoted by the symbol K 2 s . It is obvious that s � 1 converts s-convexity into the typical convexity of functions defned on [0, ∞].
Te authors of the same paper, namely, [17], demonstrated that all functions from K 2 Example 3. (see [17]). Let s ∈ (0, 1) and l, Z, c ∈ R. We defne function ℘: [0, ∞) ⟶ R as follows: It can be simply confrmed that  [18] by Yildiz et al. Te Hölder inequality, the power-mean integral inequality, and certain extensions connected to these inequalities were utilized to establish these inequalities. In addition, they compared some inequalities. In [19], a new defnition for s-convex functions is given and some properties of this defnition are investigated. In addition, extended versions of the previously well-known conclusions for harmonically convex functions, such as H − H, various H − H refnements, and Ostrowski-type inequalities, are developed. In [20], the expression "extended s -convex functions" was introduced by the authors, who also developed some inequalities of the H − H type for extended s-convex functions. Te authors then used these newly discovered integral inequalities to deduce certain specifc mean inequalities. In reference [21], the authors established an equation for a function whose third derivative is integrable, developed some novel integral inequalities of the H − H type for extended s-convex mappings using the Hölder inequality, and then used these integral inequalities to produce inequalities for various kinds of special means. In reference [22], the authors established some new inequalities of the H − H type for extended s-convex mappings and obtained new inequalities with respect to λ and μ using Lemma 2.1. Finally, utilizing the s-convexity for the Raina function, diferent inequalities are obtained with fractional integral operators in reference [23].
Convex mappings and sets have been improved and expanded in many disciplines of mathematics due to their robustness (as was described above); in particular, the convexity theory has been used to prove a number of inequalities that are prevalent in the literature. In the practical literature on mathematical inequalities, the H − H type integral inequality is, to the best of our knowledge, a well-known, important, and incredibly helpful inequality. Tere are several classical inequalities that are closely associated with the classical H − H type integral inequality, such as Simpson, Opial, Hardy, Hölder, Ostrowski, Minkowski, arithmetic-geometric, Young, and Gagliardo-Nirenberg inequalities. Tese inequalities are of pivotal signifcance. Following is a statement of this double inequality: assume that ℘ is a convex mapping on [ϖ, ϕ] ⊂ R, where ϖ ≠ ϕ. Terefore, Te reader who is interested can refer to references [24][25][26][27] for a number of recent fndings pertaining to H − H inequality.
In [28], the researchers proved a diferent form of H − H inequality which holds for s-convex mappings in the second sense.
Te famous Young inequality is defned as follows: where ϖ and ϕ are nonnegative numbers, p > 1, and [30]. Te reversed version of inequality (7) reads Te defnition of the Hölder inequality is as follows. Let p > 1 and (1/p) + (1/q) � 1. If ℘ and κ are real functions defned on [ϖ, ϕ] such that |℘| p and |κ| q are integrable functions on [ϖ, ϕ], then 2 Journal of Mathematics Te well-known Hölder inequality, one of the most signifcant inequalities in analysis, was demonstrated in this way using inequality (7). It makes a signifcant contribution to many felds of applied and pure mathematics and is essential in helping to solve several issues in the social, cultural, and natural sciences.
Te most popular form of Young's inequality, which is frequently used to demonstrate the well-known inequality for L p functions, is as follows: where ϖ, ϕ > 0 and 0 ≤ ζ ≤ 1.

Preliminaries
Tis paper uses a relatively new approach based on Green's function to illustrate the H − H inequalities for s-convex functions. First, we will start by giving the defnition of Green's function.
Let ϖ < ϕ and the following four new Green's functions defned on [ϖ, ϕ] × [ϖ, ϕ] are defned by Mehmood et al. in [31] as follows: In [31], the authors developed the following Lemma, which we will utilize to demonstrate our main conclusions. Lemma 6. Let ϖ < ϕ and ℘: [ϖ, ϕ] ⟶ R be a twice differentiable function and G κ (κ � 1, 2, 3, 4) be the new Green's functions defned by (12). Ten, Proof. By using the procedures of integration by parts in ϕ ϖ G 1 (ξ, μ)℘ ″ (μ)dμ, the above equation may be easily computed. Te specifcs of proof are thus left to readers who are interested. Likewise, a similar method can be used in other equations. □ Remark 7. Troughout this study, G � G 1 will be used.
Tere are many studies on Green's function in the literature. In these studies, diferent inequalities were obtained by using diferent methods as well as Green's function. For example, in [32], by utilizing Green's function, Jensen's inequality, convexity, and monotone functions, the authors developed the left Riemann-Liouville fractional H − H type inequalities as well as the extended H − H type inequalities. In [33], the H − H inequalities have again been established using Green's function and convexity. In [34], the authors revisited the H − H inequalities for the Riemann-Liouville fractional operators with the help of Green's function, and fnally, in [35], Li et al. established H − H inequalities for the left generalized fractional integral via Green's function.
Te aim of this study is to obtain new integral inequalities with the use of Green's function for functions whose q-th power is s-convex and s-concave. In other words, it is to develop a new method using Green's function. In addition to the defnition of Green's function, Young, Hölder, and power-mean inequalities were used to obtain these new identities. As a consequence, these inequalities are associated with the left-hand side of H − H inequality. Finally, new propositions are given for special means.
Proof. Utilizing the defnition of Green's function, we have identity (17). From the properties of absolute value and triangle inequality for integrals in (17), we obtain inequality (18). By the change of variable in (18), we have Using the Young inequality in (23), we get By using the s-convexity of |℘ ″ | q in (24), we have Tis completes the proof. □ Corollary 11. Let all the assumptions of Teorem 10 be satisfed and since 0 < (1/p(2p + 1)4 p ) < (1/12), for p > 1, we obtain Corollary 12. If we choose s � 1 in inequality (22), we get For s-convex maps, the following theorem provides a new upper bound for the left-hand side of H − H inequality.
Proof. By using the same procedure in Teorem 8, we have inequality (19) as follows: Utilizing the Hölder inequality for p > 1 and (1/p) + (1/q) � 1 in (29), we get 6 Journal of Mathematics Hence, the proof is done.

Applications for Special Means
Now, let us look at the means for the random real numbers σ 1 and σ 2 (σ 1 ≠ σ 2 ).

Results and Discussion
If we use Green's functions G 2 , G 3 , and G 4 for various convexities, we may obtain the same conclusions as in this article. Te H − H inequality for s-convex functions, as well as the H − H inequality for fractional operators, preinvex, co-ordinate convex functions, and so on, may all be studied using Green's function or any other new Green's function. Readers who are interested in an exercise can utilize the other three Green's functions to produce the results that correspond to them.

Conclusion
Producing novel and special integral inequalities is the primary motivation for inequality theory, one of the most crucial areas of mathematical analysis. Researchers sometimes utilize novel function classes, sometimes new integral operators, and sometimes try to obtain modifcations of a few well-known inequalities in diferent spaces for this aim. Especially in the last decade, many methods have been developed with diferent results, and many remarkable refnements, extensions, and generalizations have been obtained. Using these diferent methods, many diferent types of inequalities, lemmas, and diferent identities can be found in articles, and relevant researchers can access these studies in the literature. One of these important methods is the wellknown Green's function.
In this study, a new method was developed using Green's function to prove the new results obtained. In addition, conclusions pertaining to the left-hand side of H − H were obtained for derivatives of the q-th power of s-convex functions. Using the identity (2.11) (in [37]) and diferent types of convexity, researchers can derive both H − H type and diferent type well-known inequalities. In this sense, we hope that this study will inspire researchers to obtain further results.

Data Availability
No data were used to support the fndings of this article.