On the Generalization of Weinberger's Inequality with Alternating Signs

Abstract

For given set of $m$ positive numbers satisfying the conditions: $$a_1\ge a_2\ge ,...\ge a_m\ge 0,$$ the inequality $$\sum _{s=1}^m(-1{)}^{s-1}{a}_s^r\ge {\left[\sum _{s=1}^m(-1{)}^{s-1}{a}_s\right]}^r,r>1,$$ was proved by H. Weinberger. The generalization of Weinberger's result takes the form $$\sum _{s=1}^m(-1{)}^{s-1}f(a_s)\ge f\left(\sum _{s=1}^m(-1{)}^{s-1}{a}_s\right),$$ where $f$ is a convex function satisfying the condition $f(0)\le 0$. The condition $f(0)\ge 0$ in the generalization proposed by Bellman was corrected by Olkin as $f(0)\le 0$. 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 $f(0)\le 0$. 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 Szegö's inequality which applies to sums with odd number of terms.

10 pages