An analogue of the second main theorem for uniform metric

I. I. Marchenko

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 5 (1998) no. 3, pp. 212-227

Résumé

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 $$ \overline{log dens}E(\gamma)\ge 1-\frac{\lambda}{\gamma}, \quad \underline{log dens}E(\gamma) \ge 1-\frac{\rho}{\gamma}\,. $$