Weissler and Bernoulli type inequalities in Bergman spaces

Abstract

We consider Weissler type inequalities for Bergman spaces with general radial weights and give conditions on the weight $w$ in terms of its moments ensuring that $\Vert f_r{\Vert }_{A^{2n}(w)}\le \Vert f{\Vert }_{A^2(w)}$ whenever $n\in \mathbb{N}$ and $0<r\le 1/\sqrt{n}$. For noninteger exponents a special case of this inequality is proved which can be considered as a certain analog of the Bernoulli inequality. An example of a monotonic weight is constructed for which these inequalities are no longer true.

14 pages