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.
Contents
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 α a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbach conjecture.
Proof
Ekhad (1993) gave a short proof of this inequality, by combining the identity
with the Clausen inequality.
Generalizations
Gasper & Rahman (2004, 8.9) give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.
References
Askey–Gasper inequality Wikipedia(Text) CC BY-SA