We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results on composition operators between the classical unweighted Bergman and Hardy 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 < $0<q<2$ .