Publications de l’Institut Mathématique, Nouvelle Série Vol. 100[114], No. 1/1, pp. 1–16 (2016) 

LOGARITHMIC BLOCH SPACE AND ITS PREDUALMiroslav PavlovićFaculty of Mathematics, University of Belgrade, Belgrade, SerbiaAbstract: We consider the space ${\U0001d505}_{{log}^{\alpha}}^{1}$, of analytic functions on the unit disk $\mathbb{D}$, defined by the requirement ${\int}_{\mathbb{D}}{f}^{\text{'}}\left(z\right)\left\varphi \right(\leftz\right)\phantom{\rule{0.166667em}{0ex}}dA\left(z\right)<\infty $, where $\varphi \left(r\right)={log}^{\alpha}(1/(1r))$ and show that it is a predual of the “${log}^{\alpha}$Bloch” space and the dual of the corresponding little Bloch space. We prove that a function $f\left(z\right)={\sum}_{n=0}^{\infty}{a}_{n}{z}^{n}$ with ${a}_{n}\downarrow 0$ is in ${\U0001d505}_{{log}^{\alpha}}^{1}$ iff ${\sum}_{n=0}^{\infty}{log}^{\alpha}(n+2)/(n+1)<\infty $ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in ${\U0001d505}_{{log}^{\alpha}}^{1}$. Some properties of the Cesàro and the Libera operator are considered as well. Keywords: Libera operator; Cesaro operator; Hardy spaces; logarithmic Bloch type spaces; predual Classification (MSC2000): 30D55 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 8 Nov 2016. This page was last modified: 14 Nov 2016.
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