Turán's inequalities

Turán's inequalities

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by harvs|txt=yes|first=Paul |last=Turán|authorlink=Paul Turán|year=1950 (and first published by harvtxt|Szegö|1948). There are many generalizations to other polynomials, also often called Turán's inequalities, given by harvs|id=MR|0040487
last=Beckenbach|first= E. F.|last2= Seidel|first2= W.|last3= Szász|first3= Otto
title=Recurrent determinants of Legendre and of ultraspherical polynomials
journal=Duke Math. J.|volume= 18|year=1951|pages= 1-10
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) ext{ for }-1

References

*citation|id=MR|0040487
last=Beckenbach|first= E. F.|last2= Seidel|first2= W.|last3= Szász|first3= Otto
title=Recurrent determinants of Legendre and of ultraspherical polynomials
journal=Duke Math. J.|volume= 18|year=1951|pages= 1-10

*citation|id=MR|0023954
last=Szegö|first= G.
title=On an inequality of P. Turán concerning Legendre polynomials.
journal=Bull. Amer. Math. Soc. 54, |year=1948|pages= 401-405
url=http://www.ams.org/bull/1948-54-04/S0002-9904-1948-09017-6/
doi=10.1090/S0002-9904-1948-09017-6

*citation|id=MR|0041284
last=Turán|first= Paul
title=On the zeros of the polynomials of Legendre
journal=Časopis Pěst. Mat. Fys. |volume=75|year=1950|pages= 113-122


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