Weissler and Bernoulli type inequalities in Bergman spaces. (English) Zbl 1529.46016
Summary: 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 \(\Vert f_r \Vert_{A^{2n}(w)} \leq \Vert f \Vert_{A^2(w)}\) whenever \(n \in {\mathbb{N}}\) and \(0< r \leq 1/\sqrt{n}\). 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.
MSC:
| 46E20 | Hilbert spaces of continuous, differentiable or analytic functions |
| 30D10 | Representations of entire functions of one complex variable by series and integrals |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
| 47A55 | Perturbation theory of linear operators |
| 47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |
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