Difference between revisions of "Askey–Gasper inequality"

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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]].
+
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==
 
==Statement==
 
+
It states that if {{math|''β'' ≥ 0, ''α'' + ''β'' ≥ −2}}, and {{math|−1 ≤ ''x'' ≤ 1}} then
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>
 
:<math>\sum_{k=0}^n \frac{P_k^{(\alpha,\beta)}(x)}{P_k^{(\beta,\alpha)}(1)} \ge 0</math>
  
where  
+
where
  
 
:<math>P_k^{(\alpha,\beta)}(x)</math>
 
:<math>P_k^{(\alpha,\beta)}(x)</math>
Line 13: Line 12:
 
is a Jacobi polynomial.
 
is a Jacobi polynomial.
  
The case when β=0 can also be written as
+
The case when {{math|''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]].
+
:<math>{}_3F_2 \left (-n,n+\alpha+2,\tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+3),\alpha+1;t \right)>0, \qquad 0\leq t<1, \quad \alpha>-1.</math>
 +
 
 +
In this form, with {{mvar|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==
 +
{{harvs|txt|authorlink=Shalosh B. Ekhad|last=Ekhad|year=1993}} gave a short proof of this inequality, by combining the identity
 +
 +
:<math>\begin{align}
 +
\frac{(\alpha+2)_n}{n!} &\times {}_3F_2 \left (-n,n+\alpha+2,\tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+3),\alpha+1;t \right) = \\
 +
&= \frac{\left(\tfrac{1}{2} \right)_j\left (\tfrac{\alpha}{2}+1 \right )_{n-j} \left (\tfrac{\alpha}{2}+\tfrac{3}{2} \right )_{n-2j}(\alpha+1)_{n-2j}}{j!\left (\tfrac{\alpha}{2}+\tfrac{3}{2} \right )_{n-j}\left (\tfrac{\alpha}{2}+\tfrac{1}{2} \right )_{n-2j}(n-2j)!} \times {}_3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+2),\alpha+1;t \right )
 +
\end{align}</math>
  
{{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]].
 
with the [[Clausen inequality]].
  
 
==Generalizations==
 
==Generalizations==
 
 
{{harvtxt|Gasper|Rahman|2004|loc=8.9}} give some generalizations of the Askey–Gasper inequality to [[basic hypergeometric series]].
 
{{harvtxt|Gasper|Rahman|2004|loc=8.9}} give some generalizations of the Askey–Gasper inequality to [[basic hypergeometric series]].
  
 
==See also==
 
==See also==
 
 
*[[Turán's inequalities]]
 
*[[Turán's inequalities]]
  
 
==References==
 
==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 | 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=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 |displayeditors=4| 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=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}}
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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| doi_brokendate=2015-01-10 }}
  
 
{{DEFAULTSORT:Askey-Gasper inequality}}
 
{{DEFAULTSORT:Askey-Gasper inequality}}

Latest revision as of 20:11, 10 January 2015

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 Template:Mvar 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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