The logarithmic derivative of a meromorphic function

A well-known lemma on the logarithmic derivative for a function $f(z)$, $f(0)=1$ ($0<r<\rho<R$), meromorphic in $\{|z|<R\le\infty\}$ is proved in the following form:
$$ m\Bigl(r,\frac{f'}f\Bigr)<ln+\Bigl\{\frac{T(\rho,f)}r\frac\rho{\rho-r}\Bigr\}+5,\!8501. $$

This estimate is more exact than the one previously obtained by Kolokol'nikov and is, in a certain sense, unimprovable.