Nevanlinna characteristics T(r,f(z + z0)) grows as T(r,f(z)) for finite order meromorphic functions f

You need to know: Complex numbers, notation $|z|$ for the absolute value of a complex number $z$ , function in a complex variable, meromorphic function $f$ , poles of $f$ and their multiplicity, integration, notation $\log^+ x = \max(\log x, 0)$ , $\limsup$ notation, big O notation, notation $f(r) \sim g(r)$ if $\lim\limits_{r \to \infty}\frac{f(r)}{g(r)}=1$ .

Background: For a meromorphic function $f$ and real $r \geq 0$ , let $n(r,f)$ be the number of poles $z_i$ of $f$ , counting multiplicity, such that $|z_i|\leq r$ . The Nevanlinna characteristic $T(r,f)$ of $f$ is $T(r,f)=m(r,f)+N(r,f)$ , where $m(r,f) = \frac{1}{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|d\theta$ and $N(r,f)=\int_0^r(n(t,f)-n(0,f))\frac{dt}{t}+n(0,f)\log r$ . The order of a meromorphic function $f$ is $\sigma(f)=\limsup\limits_{r\to\infty}\frac{\log^+ T(r,f)}{\log r}$ .

The Theorem: On 6th May 2005, Yik-Man Chiang and Shao-Ji Feng submitted to The Ramanujan Journal a paper in which they proved that for every meromorphic function $f$ of order $\sigma(f) < \infty$ , any fixed complex number $\eta\neq 0$ , and any $\epsilon>0$ , we have $T(r,f(z + \eta)) = T(r,f) + O(r^{\sigma(f)-1+\epsilon}) + O(\log r)$ .

Short context: Nevanlinna characteristic $T(r,f)$ measures the rate of growth of a meromorphic function $f$ , and can be used to describe the asymptotic distribution of solutions of the equation $f(z)=a$ as $a$ varies. The Theorem implies that $T(r,f(z + \eta)) \sim T(r,f)$ for finite order meromorphic functions. The authors demonstrate various applications of this result to, for example, difference equations.