Abstract
This paper deals with a new extension of Jensen's discrete inequality to a partially convex function f , which is defined on a real interval I, convex on a subinterval (a, b) ⊂ I, decreasing for u ≤ c and increasing for u ≥ c ,w herec ∈ (a, b). Several relevant applications are given to show the effectiveness of the proposed partially convex function theorem. MSC: 26D07; 26D10; 41A44
Highlights
1 Introduction Let x = {x, x, . . . , xn} be a sequence of real numbers belonging to a given real interval I, and let p = {p, p, . . . , pn} be a sequence of given positive weights associated to x and satisfying p + p + · · · + pn =
In [ ], we extended the weighted Jensen discrete inequality to a half convex function f, defined on a real interval I and convex for u ≤ s or u ≥ s, where s ∈ I
WHCF-Theorem Let f be a function defined on a real interval I and convex for u ≤ s or u ≥ s, where s ∈ I, and let p, p, . . . , pn be positive real numbers such that p = min{p, p, . . . , pn}, p + p + · · · + pn =
Summary
WHCF-Theorem Let f be a function defined on a real interval I and convex for u ≤ s or u ≥ s, where s ∈ I, and let p , p , . HCF-Theorem Let f be a function defined on a real interval I and convex for u ≤ s or u ≥ s, where s ∈ I. We will use HCF-Theorem and WHCF-Theorem to extend Jensen’s inequality to partially convex functions, which are defined on a real interval I and convex only on a subinterval [a, b] ⊂ I. WPCF-Corollary Let g be a function defined on a positive interval I, decreasing for t ≤ r and increasing for t ≥ r , where r ∈ I, and let p , p , .
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