Geodesic
An analogue of the second main theorem for uniform metric
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 5 (1998) no. 3, pp. 212-227 
I. I. Marchenko. An analogue of the second main theorem for uniform metric. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 5 (1998) no. 3, pp. 212-227. http://geodesic.mathdoc.fr/item/JMAG_1998_5_3_a5/
@article{JMAG_1998_5_3_a5,
     author = {I. I. Marchenko},
     title = {An analogue of the second main theorem for uniform metric},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {212--227},
     year = {1998},
     volume = {5},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/JMAG_1998_5_3_a5/}
}
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Let $f$ be a meromorphic function of finite lower order $\lambda$, and order $\rho$, $T(r,f)$ be Nevanlinna's characteristic, $0<\gamma<\infty$, $B(\gamma)$ be Paley's constant. We obtain the estimates for upper and lower logarithmic density of set $$ E(\gamma)=\{r:\sum\limits_{k=1}^{q}\log^{+}\max\limits_{|z|=r}|f(z)-a_k|^{-1}<2B(\gamma)T(r,f)\}. $$ It is shown that $$ \overline{log dens}E(\gamma)\ge 1-\frac{\lambda}{\gamma}, \quad \underline{log dens}E(\gamma) \ge 1-\frac{\rho}{\gamma}\,. $$