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 nth Legendre polynomial, Turán's inequalities state that


For H_n, the nth Hermite polynomial, Turán's inequality is

and for Chebyshev polynomials

Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...

it is