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In mathematics, the '''Askey–Gasper inequality''' is an inequality for [[Jacobi polynomial]]s proved by {{harvs|txt|first=Richard|last=Askey|authorlink=Richard Askey|first2=George|last2=Gasper|author2-link=George Gasper|year=1976}} and used in the proof of the [[Bieberbach conjecture]].

==Statement==

It states that if ''β'' ≥ 0, ''α'' + ''β'' ≥ −2, and −1 ≤ ''x'' ≤ 1 then

:<math>\sum_{k=0}^n \frac{P_k^{(\alpha,\beta)}(x)}{P_k^{(\beta,\alpha)}(1)} \ge 0</math>

where

:<math>P_k^{(\alpha,\beta)}(x)</math>

is a Jacobi polynomial.

The case when β=0 can also be written as
:<math>\displaystyle {}_3F_2(-n,n+\alpha+2,(\alpha+1)/2;(\alpha+3)/2,\alpha+1;t)>0\mbox{ for }0\leq t<1,\;\alpha>-1.</math>

In this form, with α a non-negative integer, the inequality was used by [[Louis de Branges]] in his proof of the [[de Branges's theorem|Bieberbach conjecture]].

==Proof==

{{harvs|txt|authorlink=Shalosh B. Ekhad|last=Ekhad|year=1993}} gave a short proof of this inequality, by combining the identity
:<math>\displaystyle \frac{(\alpha+2)_n}{n!}{}_3F_2(-n,n+\alpha+2,(\alpha+1)/2;(\alpha+3)/2,\alpha+1;t)</math>
:<math>\displaystyle =\frac{(1/2)_j(\alpha/2+1)_{n-j}(\alpha/2+3/2)_{n-2j}(\alpha+1)_{n-2j}}
{j!((\alpha/2+3/2)_{n-j}(\alpha/2+1/2)_{n-2j}(n-2j)!} </math>
:<math>\displaystyle \times{}_3F_2(-n+2j,n-2j+\alpha+1,(\alpha+1)/2;(\alpha+2)/2,\alpha+1;t)</math>
with the [[Clausen inequality]].

==Generalizations==

{{harvtxt|Gasper|Rahman|2004|loc=8.9}} give some generalizations of the Askey–Gasper inequality to [[basic hypergeometric series]].

==See also==

*[[Turán's inequalities]]

==References==
*{{Citation | author1-link=Richard Askey | last1=Askey | first1=Richard | last2=Gasper | first2=George | title=Positive Jacobi polynomial sums. II | jstor=2373813 | mr=0430358 | year=1976 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=98 | issue=3 | pages=709–737 | doi=10.2307/2373813 | publisher=American Journal of Mathematics, Vol. 98, No. 3}}
*{{Citation | last1=Askey | first1=Richard | last2=Gasper | first2=George | editor1-last=Baernstein | editor1-first=Albert | editor2-last=Drasin | editor2-first=David | editor3-last=Duren | editor3-first=Peter | editor4-last=Marden | editor4-first=Albert | title=The Bieberbach conjecture (West Lafayette, Ind., 1985) | url=http://books.google.com/books?id=HcDl0D4Y6WoC&pg=PA7 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Math. Surveys Monogr. | isbn=978-0-8218-1521-2 | mr=875228 | year=1986 | volume=21 | chapter=Inequalities for polynomials | pages=7–32}}
*{{Citation | last1=Ekhad | first1=Shalosh B. | editor1-last=Delest | editor1-first=M. | editor2-last=Jacob | editor2-first=G. | editor3-last=Leroux | editor3-first=P. | title=A short, elementary, and easy, WZ proof of the Askey-Gasper inequality that was used by de Branges in his proof of the Bieberbach conjecture | series=Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991) | doi=10.1016/0304-3975(93)90313-I | mr=1235178 | year=1993 | journal=[[Theoretical Computer Science (journal)|Theoretical Computer Science]] | issn=0304-3975 | volume=117 | issue=1 | pages=199–202}}
*{{Citation | last1=Gasper | first1=George | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}

{{DEFAULTSORT:Askey-Gasper inequality}}
[[Category:Inequalities]]
[[Category:Special functions]]
[[Category:Orthogonal polynomials]]

Clearing factor - Revision history

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