Cauchy transforms and multipliers (Q1378397)

Let \(U\) denote the open unit disc and \(T\) its boundary. For \(\alpha>0\) the author considers the class \(F_\alpha\) of functions analytic in \(U\) which have a representation \(f(z)= \int_T(1- \overline \xi z)^{-\alpha} d\mu(\xi)\), \(z\in U\), where \(\mu\) is a complex valued Borel measure on \(T\). For \(\alpha=0\) the family \(F_0\) of functions \(f(z)= \int_T\log(1-\overline \xi z)^{-1} d\mu(\xi) +f(0)\), \(z\in U\), \(\mu\) as above, is considered. A function \(g\) analytic in \(U\) is called a multiplier of \(F_\alpha\) if \(fg\in F_\alpha\) for all \(f\in F_\alpha\). A number of nice results on such functions is proved.