Askey–Gasper inequality

In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by 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


 * $$\sum_{k=0}^n \frac{P_k^{(\alpha,\beta)}(x)}{P_k^{(\beta,\alpha)}(1)} \ge 0$$

where


 * $$P_k^{(\alpha,\beta)}(x)$$

is a Jacobi polynomial.

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


 * $${}_3F_2 \left (-n,n+\alpha+2,\tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+3),\alpha+1;t \right)>0, \qquad 0\leq t<1, \quad \alpha>-1.$$

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

Proof
gave a short proof of this inequality, 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) = \\ &= \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
give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.