Borel direction relative to function-values of meromorphic functions with finite logarithmic order

Abstract

It is shown that if $ f(z )$ is meromorphic in the complex plane $ \mathbb C $ with finite positive logarithmic order $ \lambda $ and its characteristic function $ T(r,f) $ satisfies the growth condition
$$ \ls \ T(r,f)/(\log r)^2 = + \infty,$$
then there is a number $ \theta $ with $ 0 \le \theta < 2\pi $ such that for each positive number $ \epsilon $, the expression
$$ \ls \ \dfrac{\log \bigg\{\displaystyle{\sum^3_{i=1}} \ n(r,\theta, \epsilon, f = a_i(z)) \bigg\}}{\log \log r} = \lambda - 1, $$
holds for any three distinct meromorphic functions $ a_i(z) (i = 1,2,3) $ with $ T(r,a_i) = o(U(r,f)/ $ $ (\log r)^2), $ as $ r \to + \infty $, where $ n(r,\varphi ,\epsilon ,f = a(z)) $ denotes the number of roots counting multiplicitie s of the equation $ f(z) = a(z) $ for $ z$ in the angular domain $ \Omega (r,\varphi ,\epsilon ) = \{z: |\arg z - \varphi | < \epsilon $, $ |z| < r \} $ where $ 0 \leq \varphi < 2\pi $, $ \epsilon > 0$, $U(r,f) = (\log r)^{\lambda (r)} $, and $ \displaystyle{\ls \ \lambda (r) = \lambda} $.