Askey–Gasper inequality

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

Statement For and ,

if and only if ,

where is a Jacobi polynomial.

The case when can also be written as

In this form, with a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbach conjecture.

Proof Ekhad gave a short proof of this inequality in 1993, by combining the identity

with the Clausen inequality.

Generalizations Gasper and Rahman (2004) give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.

See also Turán's inequalities

References