Canonical divisors in weighted Bergman spaces (Q2759018)

Let \(A^p_\alpha\) (\(0<p<\infty\), \(\alpha>-1\)) be the weighted Bergman space of functions analytic in the unit disc \(\mathbb D\) for which \(\|f\|_{p,\alpha}:= ((\alpha+1)\int_{\mathbb D} |f(z)|^p (1-|z|^2)^\alpha dm(z))^{1/p}<\infty\), where \(dm\) is the Lebesgue area measure. NEWLINENEWLINENEWLINEThe author shows that for \(1\leq p<\infty\) and any \(\beta\in\mathbb D\setminus\{0\}\), the solution to the extremal problem NEWLINE\[NEWLINE \sup\{\text{Re } f(0): f\in A^p_\alpha,\;f(\beta)=0,\;\|f\|_{p,\alpha}=1 \} NEWLINE\]NEWLINE is given by NEWLINE\[NEWLINE f(z) = \frac{\overline\beta}{|\beta|} \varphi_\beta(z) \Biggl[ \frac {F(-\alpha-1,\frac p2,\frac p2+1,\overline\beta \varphi_\beta(z))} {(\alpha+1) B(\alpha+1,\frac p2+1) F(-\alpha-1,\frac p2,\frac p2+1, |\beta|^2)} \Biggr]^{1/p} NEWLINE\]NEWLINE whenever \(F(-\alpha-1,\frac p2,\frac p2+1,w)\neq 0\) in the disc \(|w|<|\beta|\); here \(\varphi_\beta(z)=\frac{z-\beta}{1-\overline\beta z}\) and \(F\) stands for the hypergeometric function. This generalizes the corresponding result for \(\alpha=0\) from \textit{P. Duren, D. Khavinson, H. S. Shapiro} and \textit{C. Sundberg} [Pac. J. Math. 157, No. 1, 37-56 (1993; Zbl 0739.30029)], and as therein has applications to construction of contractive zero divisors in the spaces \(A^p_\alpha\).