# Turán's inequalities

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Template:Harvs (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 Pn is the nth Legendre polynomial, Turán's inequalities state that

${\displaystyle \,\!P_{n}(x)^{2}>P_{n-1}(x)P_{n+1}(x){\text{ for }}-1

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

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

and for Chebyshev polynomials it is

${\displaystyle \!T_{n}(x)^{2}-T_{n-1}(x)T_{n+1}(x)=1-x^{2}>0{\text{ for }}-1

## References

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