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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

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

where

${\displaystyle P_{k}^{(\alpha ,\beta )}(x)}$

is a Jacobi polynomial.

The case when β=0 and α is a non-negative integer was used by Louis de Branges in his proof of the Bieberbach conjecture.

The inequality can also be written as

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

## Proof

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

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

with the Clausen inequality.

## Generalizations

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

## References

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