Askey–Gasper inequality

From formulasearchengine
Revision as of 20:11, 10 January 2015 by en>Citation bot ([579]Alter: doi_brokendate. You can use this bot yourself. Report bugs here.)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

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 can also be written as

In this form, with Template:Mvar 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

with the Clausen inequality.

Generalizations

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

See also

References

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}