Askey–Gasper inequality

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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.

Statement[edit]

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[edit]

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

with the Clausen inequality.

Generalizations[edit]

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

See also[edit]

References[edit]