Askey–Gasper inequality

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For ${\displaystyle \beta \geq 0}$ and ${\displaystyle -1\leq x\leq 1}$,

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

where ${\displaystyle P_{k}^{(\alpha ,\beta )}(x)}$ is a Jacobi polynomial.

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

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

In this form, with α a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbach conjecture.^[2]

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Ekhad gave a short proof of this inequality in 1993,^[3] by combining the identity

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

with the Clausen inequality.

Gasper and Rahman (2004)^[4] give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.

• Turán's inequalities