title: Askey–Gasper inequality
text: # 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
  
  It states that if $\beta \geq 0$, $\alpha +\beta \geq -2$, and $-1\leq x\leq 1$ then
  
      $\sum _{k=0}^{n}{\frac {P_{k}^{(\alpha ,\beta )}(x)}{P_{k}^{(\beta ,\alpha )}(1)}}\geq 0$
  
  where
  
      $P_{k}^{(\alpha ,\beta )}(x)$
  
  is a Jacobi polynomial.
  
  The case when $\beta =0$ can also be written as
  
      $_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)>0,\qquad 0\leq t<1,\quad \alpha >-1.$
  
  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
  
  Ekhad (1993) gave a short proof of this inequality, by combining the identity
  
      ${\begin{aligned}{\frac {(\alpha +2)_{n}}{n!}}&\times_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)=\\&={\frac {\left({\tfrac {1}{2}}\right)_{j}\left({\tfrac {\alpha }{2}}+1\right)_{n-j}\left({\tfrac {\alpha }{2}}+{\tfrac {3}{2}}\right)_{n-2j}(\alpha +1)_{n-2j}}{j!\left({\tfrac {\alpha }{2}}+{\tfrac {3}{2}}\right)_{n-j}\left({\tfrac {\alpha }{2}}+{\tfrac {1}{2}}\right)_{n-2j}(n-2j)!}}\times_{3}F_{2}\left(-n+2j,n-2j+\alpha +1,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +2),\alpha +1;t\right)\end{aligned}}$
  
  with the Clausen inequality.
  
  ## Generalizations
  
  Gasper & Rahman (2004, 8.9) give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.
  
  