  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.
