# Difference between revisions of "Askey–Gasper inequality"

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en>Sodin m (→Statement: clean up using AWB) |
en>Specfunfan m (→Statement) |
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is a Jacobi polynomial. | is a Jacobi polynomial. | ||

The case when β=0 | 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== | ==Proof== | ||

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*{{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}} | *{{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:Inequalities]] | ||

[[Category:Special functions]] | [[Category:Special functions]] | ||

[[Category:Orthogonal polynomials]] | [[Category:Orthogonal polynomials]] | ||

## Revision as of 13:50, 6 January 2014

In mathematics, the **Askey–Gasper inequality** is an inequality for Jacobi polynomials proved by Template:Harvs and used in the proof of the Bieberbach conjecture.

## Statement

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

where

is a Jacobi polynomial.

The case when β=0 can also be written as

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

## Proof

Template:Harvs gave a short proof of this inequality, by combining the identity

with the Clausen inequality.

## Generalizations

Template:Harvtxt give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.

## See also

## References

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