Some Interesting Inequalities for the Class of Generalized Convex Functions of Higher Order

. In this paper, we study a generalized version of strongly reciprocally convex functions of higher order. Firstly, we prove some basic properties for addition, scalar multiplication, and composition of functions. Secondly, we establish Hermite-Hadamard and Fejér type inequalities for the generalized version of strongly reciprocally convex functions of higher order. We also include some fractional integral inequalities concerning with this class of functions. Our results have applications in optimization theory and can be considered extension/generalization of many existing results.


Introduction
Convexity is a very simple and ordinary concept. Due to its massive applications in industry and business, convexity has a great influence on our daily life. In the solution of many real-world problems, the concept of convexity is very decisive. Problems faced in constrained control and estimation are convex. Geometrically, a real-valued function is said to be convex if the line segment joining any two of its points lies on or above the graph of the function in Euclidean space.
Convexity of a function in classical sense is defined as a function f 1 : M ⟶ ℝ,f 1 is convex if we have If the above inequality is reversed, then the function is said to be concave.
Using different techniques, the notion of convexity is being extended day by day [1][2][3]. Many extensions and generalizations are made speedily due to its applications in modern engineering, optimization, economics, and nonlin-ear programming [4][5][6][7]. For recent generalizations, one can see [8,9] and the references therein.
Using the definition of convex functions, several important inequalities can be proved, and the Hermite-Hadamard inequality is one of them. The Hermite-Hadamard inequality is for any convex function f 1 : M ⊆ ℝ ⟶ ℝ with a 1 , b 1 ∈ M and a 1 b 1 , the Hermite-Hadamard double inequality is In [9], using the weight function wðxÞ, Fejér gave a generalization of the Hermite-Hadamard inequality as follows: Let f 1 : ½a 1 , b 1 ⊆ ℝ ⟶ ℝ be a convex function and w : ½a 1 , b 1 ⟶ ℝ is nonnegative, integrable, and symmetric about ða 1 + b 1 Þ/2, then we have In [10], the notations of p-convex set and p-convex functions are introduced. The strongly convex functions of modulus μ are introduced in [11]. In [12,13], the strongly p-convex and harmonic convex functions were introduced, respectively. The p-harmonic convex set and p-harmonic convex functions were studied in [14], and in [15], the strongly reciprocally convex of modulus μ are introduced. The strongly reciprocally p-convex and h-convex functions were introduced in [16,17], respectively. The (p,h)-convex functions are introduced in [18], and the higherorder strongly convex with modulus μ are introduced in [19]. Now, we present the notation of strongly reciprocally (p,h)-convex functions of higher order (SRHO). Definition 1. Let μ ∈ ð0,∞Þ and M is any interval. Then the function for all x, y ∈ M, j ∈ ½0, 1, and l ≥ 1, where ϕðjÞ = jð1 − jÞ.
As we know that ℝ is a Norm space under the usual modulus norm, thus, for any x ∈ ℝ, Using (5), the inequality 1 can be written as ∀x, y ∈ M and j ∈ ½0, 1 with l ≥ 1, where ϕðjÞ is same as in Definition 1. The aim of this paper is to study a generalized version of strongly reciprocally convex functions of higher order and establish the Hermite-Hadamard and Fejér type inequalities for this new class of convex functions. We also presented fractional versions of the above mentioned inequalities for the strongly reciprocally (p,h)-convex of higher order. It is worthy to mention here that the results presented in this paper are more generalized and can be considered extensions of many existing results.

Basic Results
Now, we present some basic properties for strongly reciprocally (p,h)-convex of higher order.

Proposition 5.
Consider a sequence of SRHO, f 1i are defined on an interval M, provide 1 ≤ i ≤ n, then for positive constants

Journal of Function Spaces
Proof. For a p-harmonic convex set M, we have ∀x, y ∈ M and j ∈ ½0, 1, where γ = ∑ n i=1 λ i μ. Hence, the result is proved.
Proof. Let M be a p-harmonic convex set. Then, ∀x, y ∈ M and j ∈ ½0, 1, we have This completes the proof.

Hermite-Hadamard Type Inequality
In this section, we establish Hermite-Hadamard's type inequality for the function belonging to SRðphÞ.

Theorem 7.
Consider an interval M not containing zero and SRHO f 1 : M ⟶ ℝ of nonnegative modulus μ and Proof. Substituting j = 1/2 in Definition 1, gives ð13Þ which is left side of the inequality (12).
Finally, for the right side of the inequality (12), setting x = a 1 and y = b 1 in Definition 1 gives Integrating (15) that is right hand side of (12) and proof is completed.

Fejér Type Inequality
Now, we are going to develop the Fejér type inequality for the function belonging to SRðphÞ.  x p Proof. Substituting j = 1/2 in Definition 1, yields Considering x = ½ða By the properties of w, −μϕ Integrating inequality (21), which is left side of the inequality (17).
Finally, for the right side of the inequality (17), setting x = a 1 in Definition 1 gives By the properties of w, Integrating inequality (25), that is right hand side of (17) and the proof is completed.

Fractional Integral Inequalities
where Proof. Using Lemma 9, we have Using power mean inequality, Since j f 1 ′ðxÞj q is in SRðphÞ, so ! q+q/p " Hence, the desired result is obtained.