The Theoretical Properties Of Some Function Spaces In The Unit Disk

In this paper, the theoretical properties of some function spaces in the unit disk are studied. Our main results are contained in the following two parts which generalize some known results.1. Function spaces and Cesaro means.First, we charaterize α—Bloch type functions by Cesaro means σ_n(f) and obtain certain equivalent conditions for an analytic function to belong to α—Bloch spaces.Second, we study the relationship between Q_K spaces , F(p, q, s) spaces and Cesaro means and show thatsup ||σ_n{f)||Q_K =||f||Q_K, sup ||σ_n(f)||_[F(p,q,s)] =||f||_[F(p,q,s)]. 2.Q_{p, 0} spaces and random power series.We discuss Q_{p, 0} spaces and random power series f_ω(z), and give that ifthen f_ω ∈ Q_{p, 0} almost surely. Here 0 1.