A Computer Proof of Turan's Inequality
  We show how Turan's inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)\geq 0$ 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. 
S. Gerhold and M. Kauers