On the logarithmic derivative of a meromorphic function

We derive the following estimate for the quantity $m\bigl(r,\frac{f'}f\bigr)$ of the Nevanlinna theory of the distribution of values characterizing the growth of the logarithmic derivative of a meromorphic function $f(z)$, $f(0)=1$, $0<r<R<\infty$:
$$ m\bigl(r,\frac{f'}f\bigr)<\ln+\biggl[\frac{T(R,f)}r\Bigl(\frac R{R-r}\Bigr)^2\biggr]+6,0684. $$
This estimate is more accurate than that obtained earlier by Vu Ngoyan and I. V. Ostrovskii.