A UNIQUENESS THEOREM FOR SUBHARMONIC FUNCTIONS OF FINITE ORDER

Abstract

Let and be subharmonic functions of finite order on . The main theorem of this paper shows that, if , the relation "" is preserved, in a certain sense, for mass distributions and . This result yields new uniqueness theorems for both subharmonic and entire functions on the complex plane.Corollaries include a broad class of sufficient conditions for the completeness of systems of exponential functions in a complex domain . The conditions for completeness are stated entirely in terms of the distribution of the points of the sequence in the neighborhood of infinity and in terms of the geometric properties (mixed areas) of .