Askey-Gasper eşitsizliği

Eğer β ≥ 0, α + β ≥ −2 ve −1 ≤ x ≤ 1 ise,

${\displaystyle \sum _{k=0}^{n}{\frac {P_{k}^{(\alpha ,\beta )}(x)}{P_{k}^{(\beta ,\alpha )}(1)}}\geq 0}$ eşitsizliği yazılır. Burada :${\displaystyle P_{k}^{(\alpha ,\beta )}(x)}$ bir Jacobi polinomudur.

β=0 drumunda şu şekilde yazılabilir:

${\displaystyle \displaystyle {}_{3}F_{2}(-n,n+\alpha +2,(\alpha +1)/2;(\alpha +3)/2,\alpha +1;t)>0}$ (0≤t<1, α>–1 için)

α'nın negatif olmayan bir tam sayı olduğu eşitsizliğin bu biçimi, Louis de Branges tarafından Bieberbach varsayımının ispatlanmasında kullanılmıştır.

Shalosh B. Ekhad 1993'te eşitsizliğe kısa bir kanıt sunmuştur.

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

• Askey, Richard; Gasper, George (1976), "Positive Jacobi polynomial sums. II", American Journal of Mathematics, American Journal of Mathematics, Vol. 98, No. 3, 98 (3), ss. 709-737, doi:10.2307/2373813, ISSN 0002-9327, JSTOR 2373813, MR 0430358
• Askey, Richard; Gasper, George (1986), "Inequalities for polynomials", Baernstein, Albert; Drasin, David; Duren, Peter; Marden, Albert (Edl.), The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr., 21, Providence, R.I.: Amerikan Matematik Topluluğu, ss. 7-32, ISBN 978-0-8218-1521-2, MR 0875228
• Ekhad, Shalosh B. (1993), Delest, M.; Jacob, G.; Leroux, P. (Edl.), "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), ss. 199-202, doi:10.1016/0304-3975(93)90313-I, ISSN 0304-3975, MR 1235178