A Computer Proof of Turan's Inequality

S. Gerhold; M. Kauers

paper

A Computer Proof of Turan's Inequality

arXiv:math/0509468

Abstract

We show how Turan's inequality $P_{n} (x)^{2} - P_{n - 1} (x) P_{n + 1} (x) \geq 0$ 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 $∣ x ∣ P_{n} (x)^{2} - P_{n - 1} (x) P_{n + 1} (x) \geq 0$ , $- 1 \leq x \leq 1$ , effortlessly with the aid of our method.