Generalization of a theorem of Men'shov on monogenic functions

It is shown that in Men'shov's theorem on the holomorphicity of continuous functions monogenic at each point of a domain with respect to two intervals intersecting at this point the condition of continuity of $f(z)$ may be replaced by the condition of summability of $(\log^+|f(z)|)^p$ for all positive $p<2$. As a collateral result a theorem of Phragmén–Lindelöf type is proved in which a certain summability condition is imposed in place of a condition on the growth of the function.
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