Difference between revisions of "Askey–Gasper inequality"

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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

where

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

for 0≤t<1, α>–1

Proof

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

with the Clausen inequality.

Generalizations

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

See also

References

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