Estimates of upper and lower logarithmic densities in the theory of meromorphic functions (Q1397565)

In addition to standard notations of the Nevanlinna theory, let \[ \Lambda(\Delta)=\{\lambda: 0\leq\lambda\leq 0.5,\;\sin(\pi \lambda /2)\leq\sqrt{\Delta/2}\}, \] \[ B(\lambda,\Delta)= \begin{cases}\pi\lambda\sqrt{\Delta (2-\Delta)},\quad \lambda\not\in\Lambda(\Delta),\\ \frac{\pi\lambda}{\sin\pi\lambda}(1-(1-\Delta)\cos\pi\lambda ),\quad \lambda\in\Lambda(\Delta),\end{cases} \] and \(L(r,a,f)=\max _{|z|=r} \log ^+ \frac{1}{|f(z)-a|}\) be an approximation function in the uniform metric, introduced by V. Petrenko. Denote also \[ E(\gamma)=\left\{r: \sum_{k=1}^q L(r,a_k,f)<2B(\gamma,\Delta (0,f')) T(r,f)\right\},\;\Delta(a,f)>0, \] \[ E(\gamma)=\left\{r: \sum_{k=1}^q L(r,a_k,f)<\epsilon T(r,f)\right\},\;\Delta (a,f)=0, \] where \(\Delta (a,f)\) is the Valiron defect of \(f\). The author's main result is that for a transcendental meromorphic function \(f\) of order \(\rho\) and finite lower order \(\lambda\), for arbitrary numbers \(\epsilon >0\) and \(\gamma >0\), and for any finite set of different complex numbers \(\{a_k\}\), \[ L_0(E(\gamma))\geq 1-\lambda /\gamma, \quad l_0(E(\gamma)) \geq 1-\rho /\gamma, \] where \(L_0(E)\) (respectively \(l_0(E)\)) is the upper (respectively, lower) logarithmic density of a measurable set \(E\).