Turán's inequalities

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In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Paul Turán (1950) (and first published by Szegö (1948)). There are many generalizations to other polynomials, also often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors.

If P_n is the nth Legendre polynomial, Turán's inequalities state that

P_n(x)² > P_{n − 1}(x)P_{n + 1}(x) for − 1 < x < 1.

[edit] References

• Beckenbach, E. F.; Seidel, W. & Szász, Otto (1951), “Recurrent determinants of Legendre and of ultraspherical polynomials”, Duke Math. J. 18: 1-10, MR0040487 
• Szegö, G. (1948), “On an inequality of P. Turán concerning Legendre polynomials.”, Bull. Amer. Math. Soc. 54,: 401-405, MR0023954, doi:10.1090/S0002-9904-1948-09017-6, <http://www.ams.org/bull/1948-54-04/S0002-9904-1948-09017-6/> 
• Turán, Paul (1950), “On the zeros of the polynomials of Legendre”, Časopis Pěst. Mat. Fys. 75: 113-122, MR0041284