Inequalities for Convex Functions

Abstract

We consider inequalities of the form $\sum _{i=0}^{m}a_{i}\varphi (b_{i})\geq 0,$ and we give necessary and sufficient conditions on the nodes $b_{0},b_{1},\ldots ,b_{m},$ and the weights $a_{i}$ for such an inequality to be true for every real convex function $\varphi.$ In the case the nodes are integers with $b_{0}$ the smallest of them, then $\sum_{i=0}^{m}a_{i}\varphi (b_{i})\geq 0$ if and only if $x^{-b_{0}}\sum_{i=0}^{m}a_{i}x^{b_{i}}/(x-1)^2$ is a polynomial with positive coefficients.

2010 Mathematics Subject Classification. Primary: 26A24.