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, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors.

If Pn is the nth 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 Hn, the nth Hermite polynomial, Turán's inequality is

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

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

See also[edit]


  • Beckenbach, E. F.; Seidel, W.; Szász, Otto (1951), "Recurrent determinants of Legendre and of ultraspherical polynomials", Duke Math. J. 18: 1–10, doi:10.1215/S0012-7094-51-01801-7, MR 0040487 
  • Szegö, G. (1948), "On an inequality of P. Turán concerning Legendre polynomials", Bull. Amer. Math. Soc. 54 (4): 401–405, doi:10.1090/S0002-9904-1948-09017-6, MR 0023954 
  • Turán, Paul (1950), "On the zeros of the polynomials of Legendre", Časopis Pěst. Mat. Fys. 75: 113–122, MR 0041284