Relative defects corresponding to the common roots of two meromorphic functions (Q788145)

Let \(f_1(z)\), \(f_2(z)\) be two non-constant meromorphic functions and let \(a\) be any complex number. Let \(n_0(r,a)\) denote the number of common roots in the disk \(| z| \leq r\) of the two equations \(f_1(z)=a\) and \(f_2(z)=a\), and let \(\bar n_0(r,a)\) denote the number of common roots in the disk \(| z| \leq r\) of the two equations \(f_1(z)=a\) and \(f_2(z)=a\), where multiplicity is disregarded. Set \[ \begin{aligned} \bar N_ 0(r,a) &= \int^{r}_{0} \frac{\bar n_0(t,a)-\bar n_0(0,a)}{t} dt + \bar n_0(0,a) \log r,\\ \bar N_{1,2}(r,a) &= \bar N\left(r,\frac{1}{f_ 1-a}\right) + \bar N\left(r,\frac{1}{f_ 2-a}\right) - 2 \bar N_0(r,a).\end{aligned} \] Let \(\bar n_ 0^{(k)}(r,a)\), \(\bar N^{(k)}_{1,2}(r,a)\) etc. denote the corresponding quantities with respect to \(f_1^{(k)}\) and \(f_2^{(k)}\). Set \[ \begin{aligned} \theta_{1,2}(a) &= 1 - \limsup_{r\to\infty} \frac{\bar N_{1,2}(r,a)}{T(r,f_1)+T(r,f_2)}, \\ \theta^{(k)}_{1,2}(a) &= 1 - \limsup_{r\to\infty} \frac{\bar N^{(k)}_{1,2}(r,a)}{T(r,f_1)+T(r,f_2)}, \\ \delta_{1,2}(a) &= 1 - \limsup_{r\to\infty} \frac{N_{1,2}(r,a)}{T(r,f_1)+T(r,f_2)}.\end{aligned} \] \(\theta_0(a)\) and \(\theta_0^{(k)}(a)\) are similarly defined. The term \(S(r,f)\) will denote any quantity satisfying \(S(r,f) = o(T(r,f))\) as \(r\to\infty\) except possibly for a set of finite measure. The author proves a number of theorems concerning the quantities \(\theta^{(k)}_{1,2}\), \(\theta_0\) and \(\theta_{1,2}\). A typical result follows: Theorem. Let \(f_1(z)\), \(f_2(z)\) be two meromorphic functions such that \(N\left(r,\frac{1}{f_ 1}\right) = S(r,f_1)\) and \(N\left(r,\frac{1}{f_2}\right) = S(r,f_2).\) Then for any \(a\neq 0,\infty \) \[ \theta^{(k)}_{1,2}(a) + 2\theta_0^{(k)}(a) \leq 5 - \left(\theta_{1,2}(\infty) +2\theta_0(\infty)\right). \]