Common Borel directions of a meromorphic function of infinite order and its differential polynomial (Q1200708)

Let \(f\) be meromorphic in the plane. Then any continuous and nondecreasing function \(\rho(r)\) is called an order of \(f\) if \[ \varlimsup_{r\to\infty}{\log T(r,f)\over \log U(r)}=1\quad\text{and}\quad\varlimsup_{r\to\infty}{\log U(R)\over \log U(r)}=1, \] where \(U(r)=r^{\rho(r)}\) and \(R=r+{r\over \log U(r)}\). The ray \(\arg z=\theta\) is called a Borel direction of \(f\) of order \(\rho(r)\), if, for any \(\eta>0\), \(\varlimsup_{r\to\infty}{\log n(r,\theta,\eta,w)\over \log U(r)}=1\) with at most two exceptions \(w=a,b\), where \(n(r,\theta,\eta,w)\) denotes the number of \(w\)-points of \(f\) in \(| z|<r\), \(|\arg z-\theta|<\eta\). The following is proved: If \(\arg z=\theta\) is any Borel direction of order \(\rho(r)\) of a given differential polynomial in \(f\), then \(\arg z=\theta\) is also a Borel direction of order \(\rho(r)\) of \(f\), provided \(\rho(r)\to\infty\) as \(r\to\infty\).