A Computer Proof of Turan's Inequality

We show how Turan’s inequality $P_n(x{)}^2-P_{n-1}(x)P_{n+1}(x)\ge 0$ 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 $\mathrm{\mid }x\mathrm{\mid }{P}_n(x{)}^2-P_{n-1}(x)P_{n+1}(x)\ge 0$, $-1\le x\le 1$, effortlessly with the aid of our method.