In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál 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
is the
th Legendre polynomial, Turán's inequalities state that
![{\displaystyle \,\!P_{n}(x)^{2}>P_{n-1}(x)P_{n+1}(x)\ {\text{for}}\ -1<x<1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For
, the
th Hermite polynomial, Turán's inequalities are
![{\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,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
whilst for Chebyshev polynomials they are
![{\displaystyle T_{n}(x)^{2}-T_{n-1}(x)T_{n+1}(x)=1-x^{2}>0\ {\text{for}}\ -1<x<1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
See also
References
- 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 (3): 113–122, doi:10.21136/CPMF.1950.123879, MR 0041284