Range Sets And Bmo Norms of Analytic Functions

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Abstract
In this paper we are concerned with the space BMOA of analytic functions of bounded mean oscillation for Riemann surfaces, and it is shown that for any analytic function on a Riemann surface the area of its range set bounds the square of its BMO norm, from which it is seen as an immediate corollary that the space BMOA includes the space AD of analytic functions with finite Dirichlet integrals.Let R be an open Riemann surface which possesses a Green's function, i.e., R ∉ OG, and f b e an analytic function defined on R. The Dirichletintegral DR(f) = D(f) of f on R is defined by 1.1 and we denote by AD(R) the space of all functions f analytic on R for which D(f) < +∞.

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