A UNIQUENESS THEOREM FOR SUBHARMONIC FUNCTIONS OF FINITE-ORDER

Let u and v be subharmonic functions of finite order on R(m) . The main theorem of this paper shows that, if u less-than-or-equal-to v , the relation "less-than-or-equal-to" is preserved, in a certain sense, for mass distributions mu(u) and mu(v) . 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 {e(lambdanz)} of exponential functions in a complex domain G. The conditions for completeness are stated entirely in terms of the distribution of the points of the sequence {lambda(n)} in the neighborhood of infinity and in terms of the geometric properties (mixed areas) of G.