Growth of entire and meromorphic functions

The influence of the number of 'separated' maximum modulus points of a meromorphic function $f(z)$ on the circle $\{z:|z|=r\}$ on the quantity
$$ b(\infty ,f)=\liminf _{r\to \infty }\log ^+ \max _{|z|=r}\frac {|f(z)|}{rT'_-(r,f)}\,, $$
is investigated, where $T'_-(r,f)$ is the left-hand derivative of the Nevanlinna characteristic. Sharp estimates of the corresponding values are obtained. Sharp estimates of the quantities $b(a,f)$ and $\sum _{a\in \mathbb C}b(a,f)$ in terms of the Valiron deficiency $\Delta (a,f)$ and the Valiron deficiency $\Delta (0,f')$ of zero for the derivative, respectively, are also obtained.