Characteristic function and deficiency of algebroid functions on annuli

In this paper, we develop the value distribution theory for meromorphic functions with maximal deficiency sum for algebroid functions on annuli and we study the relationship between the deficiency of algebroid function on annuli and that of their derivatives. Let $W(z)$ be an $\nu$-valued algebroid function on the annulus $\mathbb{A}\left(\frac{1}{R_{0}},R_{0}\right)$ $(1<R_{0}\leq +\infty)$ with maximal deficiency sum and the order of $W(z)$ is finite. Then
i. $\limsup\limits_{r\rightarrow\infty}\frac{T_{0}(r,W')}{T_{0}(r,W)}= 2-\delta_{0}(\infty,W)-\theta_{0}(\infty,W);$

ii. $\limsup\limits_{r\rightarrow\infty}\frac{N_{0}(r,\frac{1}{W'})}{T_{0}(r,W')}=0;$

iii. $\frac{1-\delta_{0}(\infty,W)}{2-\delta_{0}(\infty,W)}\leq K_{0}(W')\leq \frac{2(1-\delta_{0}(\infty,W))}{2-\delta_{0}(\infty,W)},$

where
$$K_{0}(W')=\limsup\limits_{r\rightarrow\infty}\frac{N_{0}(r,W')+N_{0}(r,\frac{1}{W'})}{T_{0}(r,W')}.$$