  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.
