Askey–Gasper inequality
In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976) and used in the proof of the Bieberbach conjecture. == Statement == For β ≥ 0 {\displaystyle \beta \geq 0} and − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} , ∑ k = 0 n P k ( α , β ) ( x ) P k ( β , α ) ( 1 ) ≥ 0 {\displaystyle \sum _{k=0}^{n}{\frac {P_{k}^{(\alpha ,\beta )}(x)}{P_{k}^{(\beta ,\alpha )}(1)}}\geq 0} if and only if α + β ≥ − 2 {\displaystyle \alpha +\beta \geq -2} , where P k ( α , β ) ( x ) {\displaystyle P_{k}^{(\alpha ,\beta )}(x)} is a Jacobi polynomial.