Some Properties and Inequalities for the (h, s)-Nonconvex Functions

Convexity, the study of convex functions, has scope in various fields of science and pure mathematics, as well as applied mathematics. From the last few decades, many extensions and generalizations of convexity have been expressed to support different research ideas in mathematics, see, for instance, [1–12]. Suppose f: I � [μ1, μ2]⟶ R be a convex function and μ1 < μ2, where μ1, μ2 ∈ I � [μ1, μ2], then the inequality

In this article, we generalized the concept of (h, s)-convex functions and defined strongly (h, s)-convex functions. Moreover, we present some basic properties and results of strongly (h, s)-nonconvex functions, and Jensen inequality, Schur inequality, Hermite-Hadamard inequality, and weighted version of Hermite-Hadamard-type inequalities are obtained for this class of functions.

Preliminaries and Some Properties
In this section, we recall some definitions from the literature which are helpful for the study of strongly (h, s)-nonconvex functions.
is completes the proof. □ Proposition 2. If f and g are strongly (h, s)-nonconvex functions and λ > 0, then (i) Since f and g are strongly (h, s)-nonconvex functions, then we have where ϑμ * 1 � 2ϑ, and this completes the proof. (ii) Since λ > 0 and f is a strongly (h, s)-nonconvex function, then we have where ϑμ * 1 � λϑ.
then g is a (h, s)-nonconvex function.
then g is a strongly (h, s)-nonconvex function.
Proof. We start the proof with (20)

Main Results
In this section, we intend to make the reformulations of the Jensen-type inequality, Hermite-Hadamard type inequalities, Fejér type inequality, and Schur type inequality for strongly (h, s)-nonconvex functions.
□ Now, the following results are obtained.
Fixing h(ξ s ) � ξ s in inequality (22), we obtain Corollary 1. Similarly, for p � 1, the inequality (22) yields Corollary 2. If we impose the above conditions with s � 1 on inequality (22), we obtain the Jensen inequality for a strongly convex function.
Proof. e proof starts with the assumption that f : I � [μ 1 , μ 2 ] ⟶ R be a strongly (h, s)-nonconvex function with modulus ϑ ≥ 0 and u Journal of Mathematics (30) Using the property of h and multiplying on both sides of the above inequality by h(u which completes the proof.
Integrating the above inequality with respect to "t" over