On the Generalization of Weinberger's Inequality with Alternating Signs

TL;DR

This paper generalizes Weinberger's inequality for sums of convex functions with alternating signs, providing a rigorous proof and characterizing the class of functions for which the inequality holds.

Contribution

It offers a mathematical proof of the generalized Weinberger inequality, clarifies the conditions on the convex function, and introduces a set of functions ensuring the inequality's validity.

Findings

Established a rigorous proof for the generalized inequality.

Identified the set of functions satisfying the inequality conditions.

Extended Szegő's inequality to sums with an odd number of terms.

Abstract

For given set of $m$ positive numbers satisfying the conditions: $$ a_1 \geq a_2 \geq , ... \geq a_m \geq 0, $$ the inequality $$ \sum_{s=1}^{m} (-1)^{s-1}a^r_s \geq \left[ \sum_{s=1}^{m} (-1)^{s-1}a_s\right]^r, \quad r > 1, $$ was proved by H. Weinberger. The generalization of Weinberger's result takes the form $$ \sum_{s=1}^{m} (-1)^{s-1}f(a_s) \geq f\left( \sum_{s=1}^{m} (-1)^{s-1}a_s\right), $$ where $f$ is a convex function satisfying the condition $f(0)\leq 0 $. The condition $f(0)\geq 0 $ in the generalization proposed by Bellman was corrected by Olkin as $f(0) \leq 0 $. Bellman gave only a graphical proof for differentiable convex functions. In this paper, we give a mathematical proof for the generalized inequality including the importance of the condition $f(0)\leq 0$. We introduce a set $\mathcal{W}$ of functions so that functions in the intersection of $\mathcal{W}$ and the set of all convex functions are the ones that are desirable in the generalization. In addition, we give a proof of Szeg\"{o}'s inequality which applies to sums with odd number of terms.