Strongly Reciprocally p-Convex Functions and Some Inequalities

)e importance of convex functions and convex sets cannot be ignored, especially in nonlinear programing [1–5] and optimization theory [6], see, for instance, [7–14]. Generalization in the convexity is always appreciable. Also, many generalizations and extensions have been made in the theory of inequalities as well as in convexity. Several inequalities have been studied and established for the convexity of functions, and many generalizations, applications, and refinements take place, see [7, 9, 13, 15–18], for further study. In the theory of inequalities, the famous inequality, Hermite–Hadamard inequality was established by Jaques Hadamard [19]. If σ: L⟶ R is a convex function, then

In the theory of inequalities, the famous inequality, Hermite-Hadamard inequality was established by Jaques Hadamard [19]. If σ: L ⟶ R is a convex function, then (1) holds for all c 1 , c 2 ∈ L with c 1 ≤ c 2 .
Mathematically, Jensen-type inequality is stated as if σ is a convex function defined on L ⊂ R, then holds for all n ∈ R, x 1 , x 2 , . . . , x n ∈ L and μ 1 , μ 2 , . . . , μ n ≥ 0 with μ 1 + μ 2 + · · · + μ n � 1. is inequality has applications in probability and statistics. e article is organized as follows: Section 2 is devoted to preliminaries and basic results, whereas in the last section, we will develop the main results for strongly reciprocally p-convex functions.

Preliminaries
is section concerns preliminaries and basic results for the strongly reciprocally p-convex functions.
Definition 3 (strongly convex function; see [14]). Let μ be a positive number. A function σ: L ⟶ R is called a strongly convex function if for all c 1 , c 2 ∈ L and r ∈ [0, 1].
Definition 7 (strongly reciprocally convex function; see [18]). Let L⊆R and μ ∈ (0, ∞). A function σ: L ⟶ R is said to be strongly reciprocally convex with modulus μ on L if the inequality holds for all c 1 , c 2 ∈ L and r ∈ [0, 1]. Now, we are ready to introduce a new class of convexity named as strongly reciprocally p-convex function.
Definition 8 (strongly reciprocally p-convex function). A function σ: L ⟶ R is called strongly reciprocally p-convex with modulus μ on L if the inequality σ x p y p rx p +(1 − r)y p
where μ * � λμ and μ ≥ 0. e next lemma establishes the connection between the strong and reciprocal p-convexity and harmonic p-convexity. □ Lemma 1. Let σ: L ⟶ R be a function; σ is strongly reciprocally p-convex iff the function φ: Proof. Let σ be strongly reciprocally p-convex; then, we have φ x p y p rx p +(1 − r)y p Journal of Mathematics 3 is shows that φ is a harmonic p-convex function.
Conversely, if φ is harmonically p-convex, then σ x p y p rx p +(1 − r)y p is implies that σ is a strongly reciprocally p-convex function for all x, y ∈ L and r ∈ [0, 1].

Main Results
In this section, Hermite-Hadamard-, Fejér-, and Jensentype inequalities are investigated. e next theorem gives the generalization of the Hermite-Hadamard inequality for strongly reciprocally p-convex functions.
Proof. We start by the definition; set r � (1/2) in inequality (10), and we have σ 2x p y p x p + y p (1/p) which is the left side of the inequality. For the right side of inequality (15), set x � c 1 and y � c 2 in (10); we have Integrating w.r.t r over [0, 1], the above inequality yields σ(x) then we obtain From (18) and (22), we get (15).