Askey–Gasper inequality

In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976) and used in the proof of the Bieberbach conjecture.

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.

Ekhad (1993) gave a short proof of this inequality, by combining the identity

with the Clausen inequality.

Gasper & Rahman (2004, 8.9) give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.

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