Logarithmic Bloch space and its predual

Abstract

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