Askey–Gasper inequality

In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Template:Harvs^[1] and used in the proof of the Bieberbach conjecture.

Statement

For $β \geq 0$ and $- 1 \leq x \leq 1$ ,

\sum_{k = 0}^{n} \frac{P_{k}^{(α, β)} (x)}{P_{k}^{(β, α)} (1)} \geq 0

α + β \geq - 2

,

where $P_{k}^{(α, β)} (x)$ is a Jacobi polynomial.

The case when $β = 0$ can also be written as

_{3} F_{2} (- n, n + α + 2, \frac{1}{2} (α + 1); \frac{1}{2} (α + 3), α + 1; t) > 0, 0 \leq t < 1, α > - 1.

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

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

\begin{matrix} \frac{(α + 2)_{n}}{n!} \times_{3} F_{2} (- n, n + α + 2, \frac{1}{2} (α + 1); \frac{1}{2} (α + 3), α + 1; t) \\ = & \sum_{j} \frac{{(\frac{1}{2})}_{j} {(\frac{α}{2} + 1)}_{n - j} {(\frac{α}{2} + \frac{3}{2})}_{n - 2 j} (α + 1)_{n - 2 j}}{j! {(\frac{α}{2} + \frac{3}{2})}_{n - j} {(\frac{α}{2} + \frac{1}{2})}_{n - 2 j} (n - 2 j)!} \times_{3} F_{2} (- n + 2 j, n - 2 j + α + 1, \frac{1}{2} (α + 1); \frac{1}{2} (α + 2), α + 1; t) \end{matrix}

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