A Generalization of Men'shov's Theorem on Functions Satisfying Condition $K''$

We consider functions $f(z)$, $z\!\in\! D\!\subset\!\mathbb C$, determining the mappings $w = f(z)$ that, at the points $\zeta$ of the domain $D$, have the same dilatation ratio along the three pairwise noncollinear rays issuing from $\zeta$. Under an additional condition on the disposition of rays, the Trokhimchuk generalization of Men'shov's theorem on the holomorphy of such functions can be extended to functions for which the assumption that they are continuous is replaced by the assumption that $(\log^+|f(z)|)^p$ is integrable with respect to the plane Lebesgue measure for each positive $p< 2$.