# Difference between revisions of "Askey–Gasper inequality"

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

$\sum _{k=0}^{n}{\frac {P_{k}^{(\alpha ,\beta )}(x)}{P_{k}^{(\beta ,\alpha )}(1)}}\geq 0$ where

$P_{k}^{(\alpha ,\beta )}(x)$ is a Jacobi polynomial.

The case when β=0 can also be written as

$\displaystyle {}_{3}F_{2}(-n,n+\alpha +2,(\alpha +1)/2;(\alpha +3)/2,\alpha +1;t)>0{\mbox{ for }}0\leq t<1,\;\alpha >-1.$ 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

Template:Harvs gave a short proof of this inequality, by combining the identity

$\displaystyle {\frac {(\alpha +2)_{n}}{n!}}{}_{3}F_{2}(-n,n+\alpha +2,(\alpha +1)/2;(\alpha +3)/2,\alpha +1;t)$ $\displaystyle ={\frac {(1/2)_{j}(\alpha /2+1)_{n-j}(\alpha /2+3/2)_{n-2j}(\alpha +1)_{n-2j}}{j!((\alpha /2+3/2)_{n-j}(\alpha /2+1/2)_{n-2j}(n-2j)!}}$ $\displaystyle \times {}_{3}F_{2}(-n+2j,n-2j+\alpha +1,(\alpha +1)/2;(\alpha +2)/2,\alpha +1;t)$ with the Clausen inequality.

## Generalizations

Template:Harvtxt give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.