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

It states that if $\beta\geq 0$, $\alpha+\beta\geq -2$, and $-1\leq x\leq 1$ then

:$\sum_^n \frac \ge 0$

where

:$P_k^(x)$

is a Jacobi polynomial.

The case when $\beta=0$ can also be written as

:$_3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right)>0, \qquad 0\leq t<1, \quad \alpha>-1.$

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

Proof

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

:$\begin \frac &\times _3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right) = \\ &= \frac \times _3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac(\alpha+1);\tfrac(\alpha+2),\alpha+1;t \right ) \end$

with the Clausen inequality.

Generalizations

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

See also

* Turán's inequalities

References

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{{DEFAULTSORT:Askey-Gasper inequality
Inequalities
Special functions
Orthogonal polynomials