Hypercontractive inequalities for weighted Bergman spaces

We obtain sharp L p → q $L^p\rightarrow L^q$ hypercontractive inequalities for the weighted Bergman spaces on unit disk D $\mathbb {D}$ with usual weights α − 1 π ( | z 2 ) , > $\frac{\alpha -1}{\pi }{(1-|z|^2)}^{\alpha -2},\alpha >1$ ⩾ $q\geqslant 2$ thus solving an interesting case of a problem from Janson [Ark. Math. 21 (1983), no. 1, 97–110]. also give some estimates 0 < $0<q<2$ .