Askey–Gasper inequality explained

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 ^{n} _{k =0} (P ^{(\alpha,\beta)} _{k} (x))/(P ^{(\beta,\alpha)} _{k} ( 1 )) \ge 0

where

P ^{(\alpha,\beta)} _{k} (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 , 0 \leq t< 1 , \alpha> -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

gave a short proof of this inequality, by combining the identity

\begin{align} ((\alpha +2 ) _{n})/(n!) & x {} _{3F} _{2} \left ( - n,n + \alpha +2 ,\tfrac{ 1 }{ 2 }(\alpha +1 );\tfrac{ 1 }{ 2 }(\alpha +3 ),\alpha +1 ;t \right) = \\ & = (\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} } \times _3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac(\alpha+1);\tfrac(\alpha+2),\alpha+1;t \right)\end

with the Clausen inequality.

Generalizations

give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.

See also

• Turán's inequalities