Hermite–Hadamard-Type Inequalities for the Generalized Geometrically Strongly Modified 
 h
 -Convex Functions

Convexity theory becomes a hot area of research due to its applications in pure and applied mathematics, especially in optimization theory. The aim of this paper is to introduce a broader class of convex functions by unifying geometrically strong convex function with 
 
 h
 
 convex functions. This new class of functions is called as generalized geometrically strongly modified 
 
 h
 
 -convex functions. We established Hermite–Hadamard-type inequalities for the generalized geometrically strongly modified 
 
 h
 
 -convex functions. Our results can be considered as generalization and extension of literature.


Introduction
e modern analysis involves the applications of convexity, and no mathematician, working in applied mathematics, especially, in nonlinear programming and optimization theory, can ignore the significant role of convex sets and convex functions. It can be viewed as one of the most natural and simple notation in mathematics. e convexity of sets and the convexity of functions have been the object of many studies during the last century. Early contributions to convex analysis were made by Hölder [1] and Minkowski [2]. Convexity also plays a fundamental role in mathematical economics and statistics [3][4][5], and the importance of convex functions is well known in statistics, management sciences, and engineering. Since the classical convexity is not enough to attain certain goals in applied mathematics, the classical convexity has been generalized in many ways, see [6,7]. e theory of inequalities produced important contributions in convex analysis, for example, the Hermite-Hadamard, Schur, and Fejér inequalities are very important and produce many applications [8,9]. e Hermite-Hadamard inequalities give us an estimate of the mean value of a convex function and are defined as follows. e Hermite-Hadamard inequality for a convex function ϕ: (1) ere are many generalizations of the Hermite-Hadamard inequality, see [10][11][12]. Different notions of convex functions and different variants of fractional intergrades have been used in literature to establish Hermite-Hadamard inequalities, see, for example, Hermite-Hadamard-type inequalities for harmonically convex functions [6], Hermite-Hadamard inequality for Caputo-Fabrizio fractional integrals [13], Hermite-Hadamard-type inequalities for h-convexity [14], and Hermite-Hadamard inequalities for s-convex and s-concave functions [15]. For more details about inequality theory, we refer the readers to [16][17][18][19][20] and references therein.
In this paper, we introduced a broad class of convex functions and established Hermite-Hadamard-type inequalities in the setting of newly introduced class of functions. e paper is organized as follows. In Section 2, we will give some basic definitions and basic algebraic properties for this class of functions. Sections 3 and 4 are devoted to Hermite-Hadamard and some other inequalities for the generalized geometrically strongly modified h-convex functions.

Definitions and Basic Results
We start with some basic definitions related to our work. roughout this paper, J is an interval in R and η: N × N ⟶ M ⊂ R is a bi-function.
Definition 1 (convex function, see [31]). Let ϕ: J ⟶ R be an extended real-valued function defined on a convex set J ⊂ R n . en, the function ϕ is convex on J if holds, for all r, s ∈ J and α ∈ (0, 1).
Remark 1. If we take η(r, s) � r − s in Definition 1, then the generalized convex function reduces to a convex function.
Definition 8 (nonnegatively homogeneous). A function η is said to be nonnegatively homogeneous if Definition 9 (supermultiplicative function, see [35]). A function ϕ: Definition 10 (logarithmic mean). Let r, s ∈ R such that r, s ≠ 0 and |r| ≠ |s|; then, logarithmic mean for real numbers is defined as Definition 11 (geometrically convex function, see [14]). A function ϕ: J ⊂ R + � (0, ∞) ⟶ R is said to be a geometrically convex function or geometric arithmetic convexity with respect to a bi-function η(., .): Definition 12 (generalized geometrically convex function, see [24]). A function ϕ: J ⊂ R + � (0, ∞) ⟶ R is said to be generalized geometrically convex function with respect to a bi-function η(., .): Definition 13 (generalized geometrically strongly modified h-convex functions). A function ϕ: J ⊂ R + � (0, ∞) ⟶ R is said to be generalized geometrically strongly modified h-convex function with respect to a bi-function η(., .): Remark 2. e abovementioned definitions can be related with each other as follows: (1) For η(r, s) � r − s, the generalized convex function reduces to the convex function (2) For h(α) � α, the generalized modified h− convex function reduces to the generalized convex function proof. Let ϕ and ψ be two generalized geometrically strongly modified h-convex functions; then, by definition, we have Now, by additive property of η and taking 2μ � μ * , we obtain is completes the proof. e next result is about the scalar multiplication of a function. □ Proposition 2. Let ϕ be generalized geometrically strongly modified h-convex function with λ ≥ 0 and η is nonnegatively homogeneous. en, λϕ is also a generalized geometrically strongly modified h-convex function.
Proof. Since ϕ is the generalized geometrically strongly modified h-convex function, so, by definition, we have As η is nonnegatively homogeneous and by taking λμ � μ * , we obtain

Journal of Mathematics 3
is completes the proof. □ Proposition 3. Let ϕ and ψ be two generalized geometrically strongly modified h-convex functions, where η is additive and nonnegatively homogeneous. en, rϕ + sψ is also the generalized geometrically strongly modified h-convex function ∀ r, s ∈ R.
Proposition 4. Let ψ be a supermultiplicative function and ϕ be a generalized geometrically strongly modified h-convex function. If ψ(r) − ψ(s) � r − s, then ϕ°ψ is also a generalized geometrically strongly modified h-convex function.
Proof. Let ψ be a supermultiplicative function and ϕ is a generalized geometrically strongly modified h-convex function. Also, since ψ(r) − ψ(s) � r − s, so ∀, r, s ∈ J and α ∈ [0, 1], and we obtain is shows that ϕ°ψ is a generalized geometrically strongly modified h-convex function. en, the linear combination ψ: R ⟶ R is also a generalized geometrically strongly modified h-convex function.
is completes the proof. □ Proposition 6. If ϕ is a generalized geometrically strongly modified h-convex function, then ϕ is also a generalized geometrically modified h-convex function.
Proof. We have ∀r, s ∈ J ⊂ R. is completes the proof.
□ Corollary 1. If ϕ is the generalized geometrically strongly convex function and α ≤ h(α), then ϕ is also a generalized geometrically strongly modified h-convex function.
(37) Lemma 2. Let ϕ: J ⊂ R ⟶ R be a differentiable function on the interior J°o f J, where r, s ∈ J with r < s and ϕ ″ ∈ L 1 [r, s]. en, we have Theorem 4. Let ϕ: J ⊂ R + ⟶ R + be a twice differentiable function on J°, r, s ∈ J with r < s and ϕ ″ ∈ L 1 [r, s]. If |ϕ ″ | q is a generalized geometrically strongly modified h− convex function, monotonically decreasing on [r, s] for q ≥ 1 and α ∈ [0, 1], then the following inequality holds: Proof. From Lemma 2 and using the well-known powermean inequality, we have Since |ϕ ″ | q is generalized geometrically strongly modified h-convex function and monotonically decreasing on [r, s], we obtain Journal of Mathematics erefore, we have is completes the proof.

Some Other Inequalities
Theorem 5. Let ϕ: J ⊂ R + � (0, ∞) ⟶ R be a differentiable function on the interior J°o f J, where r, s ∈ J with r < s and ϕ ′ ∈ L[r, s]. If |ϕ ′ | is generalized geometrically strongly modified h-convex function for q ≥ 1, then the following inequality holds: Proof. Using Lemma 3 and Holder's inequality, we have Now, consider Similarly, I 2 becomes Also, (48) Combining (45)-(48), we have Journal of Mathematics 11 is completes the proof. If we take η(r, s) � r − s, then (45) reduces to the following result. □ Corollary 4. Let ϕ: J ⊂ R + � (0, ∞) ⟶ R be a differentiable function on the interior J°o f J, where r, s ∈ J with r < s and ϕ ′ ∈ L[r, s]. If |ϕ ′ | is a generalized geometrically strongly modified h-convex function for q ≥ 1, then the following inequality holds: (50) Theorem 6. Let ϕ: J ⊂ R + � (0, ∞) ⟶ R be a differentiable function on the interior J°o f J, where r, s ∈ J with r < s and ϕ ′ ∈ L[r, s]. If |ϕ ′ | is generalized geometrically strongly modified h-convex function for q > 1, then the following inequality holds: Now, consider Similarly, I 2 becomes Also, ln s − ln r 2 L r (q/q− 1) , s (q/q− 1) 1− (1/q) s ϕ ′ (s) is completes the proof.

Conclusions
e theory of the convex function is applicable in almost every field of mathematics specially in approximation theory. e inequality theory got application in diverse areas of pure and applied mathematics. In this paper, we introduced a broader class of convex functions by unifying geometrically strong convex function with η h convex function and developed the Hermite-Hadamardtype and fractional integral inequalities for this class of function.

Data Availability
All data used to support the findings of the study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Xishan Yu analyzed the results and designed the whole paper, Muhammad Shoaib Saleem proved the results, Shumaila Waheed wrote the first version of the paper, and Ilyas Khan wrote the final version of the paper and arranged the funding for this research.