Zeros of extremal functions in weighted Bergman spaces. (Q1880013)

Let \(D\) be the open unit disk in the complex plane and let \(dA\) be area measure on \(D\) normalized so that \(D\) has area 1. For \(p>0\) and \(\alpha> -1\) let \(A^p_\alpha\) denote the weighted Bergman space of \(D\) consisting of analytic functions \(f\) in \(D\) such that \[ \| f\|^p_{p,\alpha}= (\alpha+1)\int_D \bigl| f(x)\bigr|^p \bigl(1-| z|^2\bigr)^\alpha dA(z)<\infty. \] The main result of the paper is the following: if \(0<\alpha\leq 1\) and \(Z= \{z_k\}\) is an \(A^p_\alpha\)-zero set and none of \(z_k\) is 0, then the solution of the extremal problem \[ \sup\bigl\{\text{Re}\,f(0):\| f\|_{p, \alpha} \leq 1,\;f(z_k)=0,\;k=1,2,\dots \bigr\} \] does not have any zero other than \(\{z_k\}\). An application to hypergeometric functions is obtained as a consequence.