Turán's inequalities

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

If $P_n$ is the $n$th Legendre polynomial, Turán's inequalities state that

$P_n(x{)}^2>P_{n-1}(x)P_{n+1}(x)\text{for}-1<x<1.$

For $H_n$, the $n$th Hermite polynomial, Turán's inequalities are

$H_n(x{)}^2-H_{n-1}(x)H_{n+1}(x)=(n-1)!\cdot \sum _{i=0}^{n-1}\frac{2^{n-i}}{i!}{H}_i(x{)}^2>0,$

whilst for Chebyshev polynomials they are

$T_n(x{)}^2-T_{n-1}(x)T_{n+1}(x)=1-x^2>0\text{for}-1<x<1.$

See also

• Askey–Gasper inequality
• Sturm's theorem