On spaces of some analytic functions defined on the unit disk (Q1915881)

Let \(f(z)\) be an analytic function on the unit disk \(D\). If \(\rho(t)\), \(0< t< 1\), is a continuous positive non-decreasing function, then \(f(z)\) belongs to the space \(HL^p_\rho\), \(1< p\leq + \infty\), if \(M_p(r, f')= O({\rho(1- r)\over 1-r})\), \(0< r< 1\). In Theorem 1, the authors prove that \(f\in HL^p_\rho\), \(1< p< +\infty\), if and only if the Hadamard product \(f* g\in HL^\infty_\rho\) for each \(g\in H^q\), \({1\over p}+ {1\over q}= 1\). Theorem 2 of the paper gives a similar result for the case \(p= + \infty\) under additional restrictions on \(\rho\). The authors then note as a corollary a characterization for Bloch functions: \(f(z)\) is a Bloch function if and only if \(f* g\) is Bloch function for every \(g\in H^1\).