English

A Computer Proof of Turan's Inequality

Classical Analysis and ODEs 2007-05-23 v1 Combinatorics

Abstract

We show how Turan's inequality Pn(x)2Pn1(x)Pn+1(x)0P_n(x)^2-P_{n-1}(x)P_{n+1}(x)\geq 0 for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities. For instance, we have found the stronger inequality xPn(x)2Pn1(x)Pn+1(x)0|x|P_n(x)^2-P_{n-1}(x)P_{n+1}(x)\geq 0, 1x1-1\leq x\leq 1, effortlessly with the aid of our method.

Keywords

Cite

@article{arxiv.math/0509468,
  title  = {A Computer Proof of Turan's Inequality},
  author = {S. Gerhold and M. Kauers},
  journal= {arXiv preprint arXiv:math/0509468},
  year   = {2007}
}
R2 v1 2026-07-22T17:24:45.877Z