Abstract
This theorem of F. Riesz' has the distinction of being true only for analytic functions and not for harmonic functions in general as many similar theorems are. Correspondingly, the proof is very analytic. It is based on the canonical decomposition f(z) = g(z).B(z) where g(z) $ 0 and B(z) has boundary values 1 almost everywhere (Blaschke product). Now, for functions of several complex variables no comparable decomposition exists, and yet we will succeed in generalizing the theorem of Riesz proper. If f(z, ***,z) is analytic for O < I zj I < 1,j = 1, ... k, and if
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