×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DeTemple, D.W; Robertson, J.M, On generalized symmetric means of two variables, Univ. beograd. publ. eleklrotehn. fak. ser. mat. fiz., Nos. 634-677, 236-238, (1979) · Zbl 0449.26012
[2] Menon, K.V, Inequalities for symmetric functions, Duke math. J., 35, 37-45, (1968) · Zbl 0157.10603
[3] Mitrinović, D.S, Analytic inequalities, (1970), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0199.38101
[4] Neuman, E, On generalized symmetric means and Stirling numbers of the second kind, Zastos. mat., 18, 645-656, (1985) · Zbl 0604.26015
[5] Riordan, J, Combinatorial identities, (1968), Wiley New York/London/Sydney · Zbl 0194.00502
[6] Roberts, A.W; Varberg, D.E, Convex functions, (1973), Academic Press New York · Zbl 0289.26012
[7] Schumaker, L.L, Spline functions: basic theory, (1981), Wiley New York · Zbl 0449.41004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.