On some boundary properties of bounded analytic functions and the maximum modulus principle in domains of arbitrary connectedness

A characterization from the point of view of the maximum modulus principle is given for harmonic domains, that is, domains for which there is an affirmative answer to the question of the existence of an analytic function on the universal covering surface which is automorphic with respect to the covering group and whose boundary values have prescribed modulus. It is shown that harmonic domains are distinguished from other domains by the “sameness” of the maximum modulus principle for the classes of bounded harmonic and bounded analytic functions.
It is shown that the maximum modulus principle plays an important role in the study of a series of questions from the classical theory of cluster sets. In particular, it is noted that the assertions of some of the well-known theorems of the theory of cluster sets are equivalent to the corresponding maximum modulus principle being satisfied.
Bibliography: 16 titles.