Harmonic Functions with Some Condition for Coefficients

Complex functions which are harmonic in the unit disk can be represented by the sum of a power series and the conjugate of a power series. In geometric theory, classes of functions with some condition imposed on the coefficients of these power series have often been considered. In particular, Silverman introduced and studied harmonic functions with various coefficients or negative coefficients. In the present paper we introduce and study harmonic functions with fixed argument of coefficients. We should notice here that there is a certain symmetry between these problems and the analogous ones of the theory of analytic functions, i.e., problems considered in the geometric theory of analytic functions are adapted to the theory of harmonic functions.