The argument distribution of infinite order meromorphic functions (Q2774173)

Theorem 1 has the following form: Let \(f\) be an entire function of infinite order with ``type function'' \(u(r)=r^{\rho(r)}\) for \(r>0\). If \(k\) is a positive integer and \(\alpha\) and \(\beta\) are distinct complex numbers with \(\beta\neq 0\), then there is a ray arg \(z=\theta\) with \(0\leq\theta <2\pi\) such that for any \(\varepsilon\) with \(0<\varepsilon <\pi/2\), NEWLINE\[NEWLINE\limsup_{r\to\infty} {\log\bigl \{n(r,\theta, \varepsilon,f= \alpha) +n(r,\theta, \varepsilon, f^{(k)}= \beta)\bigr\} \over\rho (r)\log r}=1,NEWLINE\]NEWLINE where the notation \(n(r,\theta \varepsilon,g= \gamma)\) counts the number of \(\gamma\) points of \(g\) taken on with multiplicity in the sector centered at the origin symmetric about the ray arg \(z=\theta\) and with opening \(\varepsilon\). Theorem 2 concerning meromorphic functions in the plane is similar in form.