A companion to Jensen-Steffensen's inequality
type: Item
instance of: scholarly article
title: A companion to Jensen-Steffensen's inequality
zbMATH Open document ID: 0606.41015
DOI: 10.1016/0021-9045(85)90099-1
published in: Journal of Approximation Theory
publication date: 1985-01-01T00:00:00Z
review text: Suppose that f is a convex function on (a,b). Let \(a<x_ i\leq...\leq x_ n<b\); \(p_ 1,...,p_ n\in R\); \(P_ n=p_ 1+...+p_ n\) and \(A=\sum^{n}_{i=1}p_ ix_ if_+'(x_ i)/\sum^{n}_{i=1}p_ if_+'(x_ i).\) The following companion to Jensen-Steffensen's inequality is proved: If \(0\leq \sum^{K}_{i=1}p_ i\leq P_ n\) (1\(\leq k\leq n-1)\), \(P_ n>0,\sum^{n}_{i=1}p_ if_+'(x_ i)\neq 0\) and \(A\in (a,b)\), then \[ P_ n^{-1}\sum^{n}_{i=1}p_ if(x_ i)\leq f(\sum^{n}_{i=1}p_ ix_ if_+'(x_ i)/\sum^{n}_{i=1}p_ if_+'(x_ i)). \] For a nondecreasing function f and nonnegative \(P_ i\) this implies \textit{M. L. Slater's} result [ibid. 32, 160-166 (1981; Zbl 0521.41011)].
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 3979676
zbMATH Keywords: convex function
zbMATH Keywords: Jensen-Steffensen's inequality
author: J. Pečarić
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
full work available at URL: 0021-9045(85)90099-1
OpenAlex ID: W2078261139
cites work: A companion inequality to Jensen's inequality
description: scientific article; zbMATH DE number 3979676