NOTES ON <TEX>${\alpha}$</TEX>-BLOCH SPACE AND <TEX>$D_p({\mu})$</TEX>

Abstract

In this paper, we will show that if <TEX>${\mu}$</TEX> is a Borel measure on the unit disk D such that <TEX>${\int}_{D}\frac{d{\mu}(z)}{(1-\left|z\right|^2)^{p\alpha}}$</TEX> < <TEX>${\infty}$</TEX> where 0 < <TEX>${\alpha},{\rho}$</TEX> < <TEX>${\infty}$</TEX>, then a bounded sequence of functions {<TEX>$f_n$</TEX>} in the <TEX>$\alpha$</TEX>-Bloch space <TEX>$\mathcal{B}{\alpha}$</TEX> has a convergent subsequence in the space <TEX>$D_p({\mu})$</TEX> of analytic functions f on D satisfying <TEX>$f^{\prime}\;{\in}\;L^p(D,{\mu})$</TEX>. Also, we will find some conditions such that <TEX>${\int}_D\frac{d\mu(z)}{(1-\left|z\right|^2)^p$</TEX>.