Unicity of meromorphic functions of class \(\mathcal{A}\) (Q946847)

The authors treat a meromorphic function \(f\) which belongs to class \({\mathcal A}\), namely \(f\) satisfies \[ \overline N(r,f)+\overline N\left(r,\frac 1f\right)=S(r,f). \] Let \(f_1,f_2,\dots,f_q\) be \(q\) nonconstant meromorphic functions and \(a\) a complex number. Define \(\overline N_0 (r,a,f_1,f_2,\dots,f_q)\) to be the reduced counting function of common zeros of \(f_j-a\), \(1\leq j\leq q\). Below we write it simply as \(\overline N(r,a)\). For given meromorphic functions \(f_1,f_2,\dots,f_q\) of class \({\mathcal A}\) we define \[ \tau=\lim_{r\to\infty,r\notin E}\frac{\overline N(r,1)}{\sum^q_{j=1}T(r,f_j)}. \] It is known that for two meromorphic functions of class \({\mathcal A}\) we have \(\tau\leq 1/3\) and for three meromorphic functions of class \({\mathcal A}\) we have \(\tau\leq 1/4\). It can be expected that \(\tau\leq 1/(q+1)\) for \(q\) meromorphic functions of class \({\mathcal A}\). The authors show that even the following better conclusion holds. For \(q\geq 3\), it holds \(\tau\leq 2/3q\) if \(q\) is even and \(\tau\leq 2/(3q-1)\) if \(q\) is odd.