Turán's inequalities

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by  and other authors.

If $$P_n$$ is the $$n$$th Legendre polynomial, Turán's inequalities state that
 * $$\,\! P_n(x)^2 > P_{n-1}(x)P_{n+1}(x)\ \text{for}\ -10 ,$$

whilst for Chebyshev polynomials they are
 * $$T_n(x)^2 - T_{n-1}(x)T_{n+1}(x)= 1-x^2>0 \ \text{for}\ -1<x<1 .$$