On a class of spaces of analytic functions

The spaces $B(p,q,\lambda)$ ($0<p<q\le\infty$, $0<\lambda\le\infty$) of functions, analytic in the circle $|z|<1$, are introduced, and an unimprovable estimate is obtained for the Taylor coefficients of a function $f\in B(p,q,\lambda)$. It is shown that $B(p,q,\lambda)$ is the space of fractional derivatives $f^{(\alpha}$ of order $\alpha$ ($-\infty<\alpha<1/p-1/q$) of a function $f$ of $B(s,q,\lambda)$, where $s=p/(1-\alpha p)$.