A new improvement of Jensen's inequality
type: Item
instance of: scholarly article
title: A new improvement of Jensen's inequality
zbMATH Open document ID: 0862.26006
author: Silvestru Sever Dragomir
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1997-04-21T00:00:00Z
review text: Let \(f:C \to\mathbb{R}\) be a convex mapping on a convex subset \(C\) of a linear space. For \(x_i\in C\), \(p_i \geq 0\), \(1\leq i\leq n\), \(P_n: =\sum^n_{i=1} p_i>0\), \(1\leq k<n\), and \(0 \leq t\leq 1\) define \[ L_n^{[k+1]} (t): =P_n^{-k-1} \sum^n_{i_1, \dots, i_{k+1} = 1} p_{i_1} \dots p_{i_{k+1}} f \bigl(t(x_{i_1} + \cdots + x_{i_k})/k + (1-t) x_{i_{k+1}} \bigr). \] Then the author proves the following main result. On \(0\leq t\leq 1\quad L_n^{[k+1]} (t)\) is convex and satisfies \[ L_n^{[k+1]} \bigl(k/(k+1) \bigr) \leq L_n^{[k+1]} (t)\leq t L_n^{[k]} \bigl((k-1)/k \bigr)+ (1-t) L_n^{[1]} (0)\leq L_n^{[k+1]} (0)= L_n^{[1]} (0). \] Furthermore, \(L_n^{[k+1]} \searrow\) on \([0,k/(k+1)]\) and \(\nearrow\) on \([k/(k+1),1]\). As application numerous new inequalities are derived for concave \(\log f\), for sublinear (superlinear) \(f\) on a convex cone, for normed linear spaces and for various functionals defined for positive definite matrices.
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 52A05
zbMATH DE Number: 871079
zbMATH Keywords: convex functions
zbMATH Keywords: Jensen's inequality
reviewed by: Hans-J. Runckel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 871079