\(a(x)\)-convex functions and their inequalities
type: Item
instance of: scholarly article
title: \(a(x)\)-convex functions and their inequalities
zbMATH Open document ID: 1248.26031
author: K. Smoljak
author: K. Krulić-Himmelreich
author: J. Pečarić
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2012-07-04T00:00:00Z
full work available at URL: vol35 3 10.html
review text: The authors define \(a(x)\)-convex functions as follows: Let \(f\) and \(a\) be real functions defined on interval \(I\subseteq\mathbb{R}\) such that \(f\) is differentiable and \(a\,f^{\prime}\) integrable. The function \(f\) is called \(a(x)\)-convex on the interval \(I\) if for every \(x,\,y\in I\) \[ (y-x)\,(f^{\prime}(y)-f^{\prime}(x))\geq (y-x)\,\int_{x}^{y}a(t)\,f^{\prime}(t)\,dt \] holds. For \(a(x)=0\), \(a(x)\)-convexity reduces to ordinary convexity. The authors give properties and characterizations of \(a(x)\)-convex functions. They also prove analogues of several inequalities for ordinary convex functions, such as Jensen, Hermite-Hadamard, Lah-Ribarić inequalities, in the case of \(a(x)\)-convex functions.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 6052953
zbMATH Keywords: \(a(x)\)-convex function
zbMATH Keywords: convex functions
zbMATH Keywords: Jensen inequality
zbMATH Keywords: Hermite-Hadamard inequality
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6052953