  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.
