Difference between revisions of "Askey–Gasper inequality"

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en>Sodin
m (→‎Statement: clean up using AWB)
 
en>Specfunfan
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is a Jacobi polynomial.
 
is a Jacobi polynomial.
  
The case when β=0 and α is a non-negative integer was used by [[Louis de Branges]] in his proof of the Bieberbach conjecture.
+
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>
  
The inequality 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 [[de Branges's theorem|Bieberbach conjecture]].
:<math>\displaystyle {}_3F_2(-n,n+\alpha+2,(\alpha+1)/2;(\alpha+3)/2,\alpha+1;t)>0</math> for 0≤''t''<1, α>–1
 
  
 
==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}}
  
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{{DEFAULTSORT:Askey-Gasper inequality}}
 
[[Category:Inequalities]]
 
[[Category:Inequalities]]
 
[[Category:Special functions]]
 
[[Category:Special functions]]
 
[[Category:Orthogonal polynomials]]
 
[[Category:Orthogonal polynomials]]
 
[[bs:Askey–Gasperova nejednakost]]
 
[[km:វិសមភាព អាស្គី-ហ្គាស្ពើ]]
 

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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