Borel's direction of a meromorphic function in a unit circle.

Analogue of Biernacki-Rauch's theorem.Let $f(z)$ be a meromorphic function of finite order $\rho>0$ for $|z|<\infty$ , then Valiron1) proved that there exists a Borel's direction $J$ : $\arg z=\theta_{0}$ , which satisfies the following condition.Let $\omega:|\arg z-\theta_{0}|<\delta$ be any small angular domain, which contains $J$ and $z_{\nu}(a, \omega)$ be zero points of $f(z)-a$ in $\omega$ , multiple zeros being counted only once, then for any $e>0$, $\sum_{\nu}\frac{1}{|z_{v}(a,\omega)|^{\rho-\epsilon}}=\infty$ with two possible exceptions for $a$ .If $f(z)$ is of divergence type, then $\sum_{\nu}\frac{1}{|z_{\nu}(a,\omega)|^{\rho}}=\infty$ with two possible exceptions for $a$ .This Valiron's theorem is generalized by Biernacki and Rauch as follows.Let $g(z)$ be a meromorphic function of order $<\rho$ for $|z|<\infty$ , and $z_{\nu}(f=g, \omega)$ be zero points of $f(z)-g(z)$ in $\omega$ , multiple zeros being counted only once, then for any $e>0$, $\sum_{v}\frac{1}{|z_{\nu}(f=g,\omega)|^{\rho-\epsilon}}=\infty$ 1) G. Valiron: