ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS

In this paper, we shall prove Theorem 1 Let $f$ be nonconstant meromorphic in $\mathbb{C}$ with finite positive order $\lambda$, $\lambda(r)$ be a proximate order of $f$ and $U(r, f)=r^{\lambda(r)}$, then for each number $\alpha$,$0<\alpha<\pi/2$, there exists a number $\phi_0$ with $0\le \phi_0 < 2\pi$ such that the inequality \[ \limsup_{r\to\infty}\sum_{i=1}^3 n(r, \phi_0, \alpha, f=a_i(z))/U(r, f)>0,\] holds for any three distinct meromorphic function $a_i(z)(i=1, 2, 3)$ with $T(r,a_i)=o(U(r, f))$ as $r\to\infty$.

T-Direction and Borel Direction of Algebroid Functions of Finite and Positive Order

In the present paper, the connection between a T-direction and Borel direction of algebroid functions is investigated. Two algebroid functions (not meromorphic) are obtained to prove that T-directions and Borel directions are two different classes of singular directions for algebroid functions. The existence of an algebroid function (not meromorphic) with a given Borel direction is proved.

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

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} $.