Askey–Gasper inequality

For \beta\geq 0 and -1\leq x\leq 1, :\sum_{k=0}^n \frac{P_k^{(\alpha,\beta)}(x)}{P_k^{(\beta,\alpha)}(1)} \ge 0 if and only if \alpha+\beta\geq -2, where P_k^{(\alpha,\beta)}(x) is a Jacobi polynomial. The case when \beta=0 can also be written as :{}_3F_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. In this form, with 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{align} &\frac{(\alpha+2)_n}{n!}\times {}_3F_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 {}_3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+2),\alpha+1;t \right ) \end{align} with the Clausen inequality. ==Generalizations==

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