Hypercontractive inequalities for weighted Bergman spaces

\begin{abstract} We obtain sharp $L^p\to L^q$ hypercontractive inequalities for the weighted Bergman spaces on the unit disk $\mathbb{D}$ with the usual weights \ $\frac{\alpha -1}{\pi }(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{\alpha -2},\alpha >1$ for $q\ge 2,$ thus solving an interesting case of a problem from \cite{JANSON}. We also give some estimates for $0<q<2.$ \end{abstract}