2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94764We 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.Classical Analysis and ODEsCombinatorics26D07; 33C45; 33F10text