Generalized convexity and inequalities

Abstract

Let R + = (0, ∞) and let M be the family of all mean values of two numbers in R + (some examples are the arithmetic, geometric, and harmonic means). Given m 1 , m 2 ∈ M, we say that a function f :for all x, y ∈ R + . The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m 1 , m 2 )-convexity on m 1 and m 2 and give sufficient conditions for (m 1 , m 2 )-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.

Some Properties of Generalized Strongly Harmonic Convex Functions

“…al. [1] and Iscan [5] have investigated various properties of harmonic convex functions. Noor and Noor [9] have shown that the minimum of the differentiable harmonic convex functions on the harmonic convex set can be characterized by a class of variational inequalities, which is called harmonic variational inequality.…”

Some Properties of <em>h</em>-MN-Convexity and Jensen&rsquo;s Type Inequalities

“…In 2007, Anderson et al in [2] developed a systematic study to the classical theory of continuous and midconvex functions, by replacing a given mean instead of the arithmetic mean. for all x, y ∈ I and t ∈ [0, 1].…”

Inequalities via strongly p-harmonic log-convex functions

“…For useful details and generalization of Hermite-Hadamard inequalities, see [1,2,3,4,5,6,7,8]. The harmonic convex function, was introduced and studied by Anderson et al [9] and Iscan [1]. Iscan [3] introduced the concept of harmonic s-convex function in second sense.…”