We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function which is lower semi-continuous on we give necessary and sufficient conditions in order that there exists a holomorphic function such that
We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For and a positive lower semi-continuous function on with for , we construct a holomorphic function such that for , where .
We study the idea of the boundary subordination of two analytic functions. Some basic properties of the boundary subordination are discussed. Applications to classes of univalent functions referring to a boundary point are demonstrated.
Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk , we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior and on the boundary respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators.