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.


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