Askey–Gasper inequality

In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976) and used in the proof of the Bieberbach conjecture.

Statement

For $\beta \ge 0$ and $-1\le x\le 1$,

$\sum _{k=0}^n\frac{P_k^{(\alpha ,\beta )}(x)}{P_k^{(\beta ,\alpha )}(1)}\ge 0$ if and only if $\alpha +\beta \ge -2$,

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

Proof

Ekhad gave a short proof of this inequality in 1993, by combining the identity

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

with the Clausen inequality.

Generalizations

Gasper and Rahman (2004) give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.

See also

• Turán's inequalities

References

• Askey, Richard; Gasper, George (1986), "Inequalities for polynomials", in Baernstein, Albert; Drasin, David; Duren, Peter et al., The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr., 21, Providence, R.I.: American Mathematical Society, pp. 7–32, ISBN 978-0-8218-1521-2, https://books.google.com/books?id=HcDl0D4Y6WoC&pg=PA7