Askey–Gasper inequality

In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976)^[1] 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 _{k=0}^{n}{\frac {P_{k}^{(\alpha ,\beta )}(x)}{P_{k}^{(\beta ,\alpha )}(1)}}\geq 0

where

P_{k}^{(\alpha ,\beta )}(x)

is a Jacobi polynomial.

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

{}_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)>0,\qquad 0\leq t<1,\quad \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.

Ekhad gave a short proof of this inequality in 1993,^[2] by combining the identity

{\begin{aligned}&{\frac {(\alpha +2)_{n}}{n!}}\times {}_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)\\=&\sum _{j}{\frac {\left({\tfrac {1}{2}}\right)_{j}\left({\tfrac {\alpha }{2}}+1\right)_{n-j}\left({\tfrac {\alpha }{2}}+{\tfrac {3}{2}}\right)_{n-2j}(\alpha +1)_{n-2j}}{j!\left({\tfrac {\alpha }{2}}+{\tfrac {3}{2}}\right)_{n-j}\left({\tfrac {\alpha }{2}}+{\tfrac {1}{2}}\right)_{n-2j}(n-2j)!}}\times {}_{3}F_{2}\left(-n+2j,n-2j+\alpha +1,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +2),\alpha +1;t\right)\end{aligned}}

Gasper and Rahman (2004)^[3] give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.

^ Askey, Richard; Gasper, George (1976), "Positive Jacobi polynomial sums. II", American Journal of Mathematics, 98 (3): 709–737, doi:10.2307/2373813, ISSN 0002-9327, JSTOR 2373813, MR 0430358
^ Ekhad, Shalosh B. (1993), Delest, M.; Jacob, G.; Leroux, P. (eds.), "A short, elementary, and easy, WZ proof of the Askey-Gasper inequality that was used by de Branges in his proof of the Bieberbach conjecture", Theoretical Computer Science, Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991), 117 (1): 199–202, doi:10.1016/0304-3975(93)90313-I, ISSN 0304-3975, MR 1235178
^ Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719

Askey, Richard; Gasper, George (1986), "Inequalities for polynomials", in Baernstein, Albert; Drasin, David; Duren, Peter; Marden, Albert (eds.), The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr., vol. 21, Providence, R.I.: American Mathematical Society, pp. 7–32, ISBN 978-0-8218-1521-2, MR 0875228