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Askey–Gasper inequality
Encyclopedia
In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture.
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-1.gif)
where
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-2.gif)
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 inequality can also be written as
for 0≤t<1, α>–1
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-4.gif)
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-5.gif)
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-6.gif)
with the Clausen inequality.
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Statement
It states that if β ≥ 0, α + β ≥ −2, and −1 ≤ x ≤ 1 then![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-1.gif)
where
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-2.gif)
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 inequality can also be written as
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-3.gif)
Proof
gave a short proof of this inequality, by combining the identity![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-4.gif)
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-5.gif)
![](http://image.absoluteastronomy.com/images/formulas/3/7/4371008-6.gif)
with the Clausen inequality.
Generalizations
give some generalizations of the Askey–Gasper inequality to basic hypergeometric seriesBasic hypergeometric series
In mathematics, Heine's basic hypergeometric series, or hypergeometric q-series, are q-analog generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series....
.