Modulus of continuity of harmonic functions

Let G be a bounded plane domain, the diameters of whose boundary components have a fixed positive lower bound. Let u be harmonic in G and continuous in the closure G of G. Suppose that the modulus of continuity of u on the boundary of G is majorized by a function of a suitable type. We shall then obtain upper bounds for the modulus of continuity of u in G. Further, we shall show that in some situations these estimates cannot be essentially improved. We shall also consider the same problem for certain bounded domains in space.