Logarithmic Bloch space and its predual

TL;DR

This paper introduces a new logarithmic Bloch space and its predual, characterizes its elements via coefficient conditions, and explores properties of related operators like Cesàro and Libera transforms.

Contribution

It defines the space k^1_{\u03b1} as a predual of the loch space, characterizes its elements through coefficient decay, and analyzes operator properties within this framework.

Findings

Characterization of k^1_{\u03b1} via coefficient conditions

Identification of the space as a predual of loch space

Analysis of Cesro and Libera operators' properties

Abstract

We consider the space $\bk^1_{\log^\alpha}$, of analytic functions on the unit disk $\D,$ defined by the requirement $\int_\D|f'(z)|\phi(|z|)\,dA(z)<\infty,$ where $\phi(r)=\log^\alpha(1/(1-r))$ 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 $f(z)=\sum_{n=0}^\infty a_nz^n$ with $a_n\downarrow 0$ is in $\bk^1_{\log^\alpha}$ 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 $\bk^1_{\log^\alpha}.$ Some properties of the Ces\'aro and the Libera operator are considered as well.