Counting, chordal proximity and characteristic

Let $f$ be meromorphic on $\mathbb{C}$ and not identically $\mathrm{\infty }$. Fix $a\in \widehat{\mathbb{C}}$ with $f\equiv ̸a$. For $r>0$, let $n(r,a;f)$ be the sum of the local multiplicities of the $a$-points in the closed disc $\mathrm{\mid }z\mathrm{\mid }\le r$; when $a=\mathrm{\infty }$, these are the poles with their pole orders. Define

$N(r,a;f)=n(0,a;f)\log r+{\int }_0^r\frac{n(t,a;f)-n(0,a;f)}{t}dt\mathrm{.}$

For finite $w,a$, put

$\delta (w,a)=\frac{\mathrm{\mid }w-a\mathrm{\mid }}{\sqrt{1+\mathrm{\mid }w{\mathrm{\mid }}^2}\sqrt{1+\mathrm{\mid }a{\mathrm{\mid }}^2}},\delta (w,\mathrm{\infty })=\delta (\mathrm{\infty },w)=\frac{1}{\sqrt{1+\mathrm{\mid }w{\mathrm{\mid }}^2}},\delta (\mathrm{\infty },\mathrm{\infty })=0.$

At a pole or a point where $f=a$, the expressions below are understood as logarithmic singularities in the angular Lebesgue integral. Define the proximity and characteristic by

$m(r,a;f)=\frac{1}{2\pi }{\int }_0^{2\pi }\log \frac{1}{\delta (f(r{e}^{it}),a)}dt,T(r,f)=m(r,\mathrm{\infty };f)+N(r,\mathrm{\infty };f)\mathrm{.}$

For stereographic projection of the unit sphere, $\delta$ is half the Euclidean chord length, so the chordal sphere has diameter one. This definition allows constant finite maps, but excludes their attained target from $n$ and $m$.