Several results and problems in the theory of value distribution (Q1322706)

A survey of some recent results in Nevanlinna Theory. Examples: Theorem 1. (\textit{Yaofei Wang} and \textit{Yang Lo}) Let \(f(z)\) be a transcendental meromorphic function and let \(k\) be a positive integer. Then \[ \sum_{a\in \mathbb{C}} \delta(a,f)+ \sum_{b\in\widehat{\mathbb{C}}} \delta(b,f^{(k)})\leq 3. \] Theorem 2. (\textit{Yaofei Wang}) For transcendental meromorphic functions \(f\) and all positive integers with at most 4 exceptions \[ \sum_{a\in\mathbb{C}} \delta(a,f^{(k)})\leq 1. \] A set \(E\subset\mathbb{C}\) is a Borel removable set for meromorphic functions of order \(\lambda\), if for all \(a\in \mathbb{C}\) with at most 2 exceptions \[ \limsup_{r\to\alpha} {\log n(\{| z|\leq r\}\backslash E, 1/f(z-a))\over \log r}=\lambda. \] Theorem 3. (\textit{Shengjian Wu}) Let \(\{a_ n\}\) be a sequence of complex numbers satisfying \(| a_{n+1}|> | a_ n|^{1+\sigma}\) \((\sigma>0,\;n=1,2,3,\dots)\). Then for any sequence \(\{\varepsilon_ n\}\) tending to 0, \[ E=\bigcup \{z: | z- a_ n|< \varepsilon_ n| a_ n|\} \] is a Borel removable set of all entire functions. Theorem 4. (\textit{Shengjian Wu}) Let \(\{D_ n\}\) be an arbitrary sequence of circular disks whose centers tend to \(\infty\). There is a meromorphic function of order \(\lambda\) \((0< \lambda< 1)\) which is uniformly bounded in \(\mathbb{C}\backslash\bigcup D_ n\).