# Askey–Gasper inequality

Revision as of 23:55, 26 October 2011 by en>Sodin (→Statement: clean up using AWB)

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 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

## 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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