## Inequalities involving generalized symmetric means.(English)Zbl 0601.26013

Let $$t_ 0,...,t_ k$$ be non-negative real numbers. The generalized nth symmetric mean $$h_ n$$ is defined by $h_ n=\left( \begin{matrix} n+k\\ k\end{matrix} \right)^{-1}\sum_{i_ 0+...+i_ k=n}t_ 0^{i_ 0}...t_ k^{i_ k}$ (n$$=0,1,...;i_ 0,...,i_ k\in {\mathbb{Z}}_+)$$. Further let $$x(t)=\sum^{m}_{i=\ell}a_ it^ i\quad (0\leq \ell \leq m;\quad a_ i\in {\mathbb{R}})$$ be an algebraic polynomial of degree $$\leq m$$, and let $$a=\min \{x(t): c\leq t\leq d\},$$ $$b=\max \{x(t): c\leq t\leq d\}.$$ The main result of this note is contained in the following Theorem. Let f be convex on (a,b). Then $f(\sum^{m}_{i=\ell}a_ ih_ i)\leq \int^{d}_{c}M_ k(t)f(x(t))dt,$ where $$M_ k$$ stands for the kth order B-spline with knots $$t_ 0\leq...\leq t_ k$$, $$(t_ 0<t_ k)$$. With the aid of this result several inequalities involving the means $$h_ i$$ were obtained. Among other things the log-convexity of $$\{h_ n\}_ 0^{\infty}$$ was established. Results of this paper can be applied to the Stirling numbers of the second kind.

### MSC:

 26D15 Inequalities for sums, series and integrals 05A05 Permutations, words, matrices
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### References:

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