Remarks on strongly convex functions
type: Item
instance of: scholarly article
title: Remarks on strongly convex functions
zbMATH Open document ID: 1214.26007
author: Kazimierz Nikodem
author: N. Merentes
published in: Aequationes Mathematicae
publication date: 2011-02-14T00:00:00Z
review text: Let \(I\subseteq \mathbb{R}\) be an interval and \(c\) be a positive number. A function \(f:I\rightarrow \mathbb{R}\) is called strongly convex with modulus \( c\) if \[ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-ct(1-t)(x-y)^{2}, \] for all \(x,y\in I\) and \(t\in [0,1]\). Using the property that a function \(f:I\rightarrow \mathbb{R}\) is strongly convex with modulus \(c\) if and only if the function \(g:I\rightarrow \mathbb{R}\) defined by \( g(x)=f(x)-cx^{2}\) is convex, the authors prove the following properties related to strong convex functions: a sandwich theorem, a Hyers-Ulam stability result, an integral Jensen-type inequality, a Hermite-Hadamard-type inequality. A relationship between strong convexity and the generalized convexity in the sense of Beckenbach is also shown.
reviewed by: Gheorghe Toader
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 5851467
zbMATH Keywords: strongly convex functions
zbMATH Keywords: generalized convex functions
zbMATH Keywords: Jensen inequality
zbMATH Keywords: Hermite-Hadamard inequality
MaRDI profile type: Publication
full work available at URL: s00010-010-0043-0
OpenAlex ID: W2019761544
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DOI: 10.1007/S00010-010-0043-0
description: scientific article; zbMATH DE number 5851467