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 $\beta \ge 0$, $\alpha +\beta \ge -2$, and $-1\le x\le 1$ then

$\sum _{k=0}^n\frac{P_k^{(\alpha ,\beta )}(x)}{P_k^{(\beta ,\alpha )}(1)}\ge 0$

where

$P_k^{(\alpha ,\beta )}(x)$

is a Jacobi polynomial.

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

${}_3{F}_2(-n,n+\alpha +2,\frac{1}{2}(\alpha +1);\frac{1}{2}(\alpha +3),\alpha +1;t)>0,0\le t<1,\alpha >-1.$

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

$\begin{array}{cc}& \frac{(\alpha +2{)}_n}{n!}\times {}_3{F}_2(-n,n+\alpha +2,\frac{1}{2}(\alpha +1);\frac{1}{2}(\alpha +3),\alpha +1;t)\\ =& \sum _j\frac{{\left(\frac{1}{2}\right)}_j{(\frac{\alpha }{2}+1)}_{n-j}{(\frac{\alpha }{2}+\frac{3}{2})}_{n-2j}(\alpha +1{)}_{n-2j}}{j!{(\frac{\alpha }{2}+\frac{3}{2})}_{n-j}{(\frac{\alpha }{2}+\frac{1}{2})}_{n-2j}(n-2j)!}\times {}_3{F}_2(-n+2j,n-2j+\alpha +1,\frac{1}{2}(\alpha +1);\frac{1}{2}(\alpha +2),\alpha +1;t)\end{array}$

with the Clausen inequality.

Generalizations

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

See also

• Turán's inequalities

References

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