Abstract
In the usual statement of this theorem, for example Ostrowski [9, pp. 195— 196](2), it is supposed that <p(C) is holomorphic on ^(f) =0. If we note, however, that if f(x) ??0 is holomorphic and bounded in |a;| <1, and continuous on |jc| =1 except perhaps at x = l, then the set of points on |ac| =1 where f(x) =0 is of measure zero(3), it is evident that a slight modification of Ostrowski's proof yields Theorem A(4). Since 0(f) is bounded, (1) is equivalent to stating that the left-hand member of (1) is finite. Conversely there follows from well known results:
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