Meromorphic functions sharing values in an angular domain (Q952151)

This paper offers several results on shared values in an angular domain and the corresponding uniqueness of meromorphic functions. Typical results in this paper are as follows: (1) Let \(f,g\) be two meromorphic functions of finite order, \(a_{1},a_{2},a_{3}\) be distinct extended complex values, \(\Delta_{\delta}=\{ z; |\arg z-\theta_{0}|\leq\delta\}\), \(0<\delta <\pi\), be a sectorial domain, \(\omega =\pi /2\delta\). Suppose that \(f,g\) share \(a_{1},a_{2},a_{3}\) ignoring multiplicity. If now \[ \limsup_{\varepsilon\rightarrow 0+}\limsup_{r\rightarrow\infty}\frac{T(r,\Delta_{\delta -\varepsilon},f)}{\log r}=\lambda >\omega , \] for the sectorial Nevanlinna characteristic \(T(r,\Delta_{\delta -\varepsilon},f)\), then the same limit result holds for \(T(r,\Delta_{\delta -\varepsilon},g)\) as well. (2) Introducing the notion of precise order \(\rho (r)\) for \(T(r):=\max (T(r,f),\) \(T(r,g))\), the same conclusion follows with \(\log r\) replaced with \(\rho (r)\log r\) in the denominator of the limiting quantity. As for the notion of precise order introduced by Hiong in the 30's, both results above imply a corresponding uniqueness result: If under the previous situation, \[ \limsup_{\varepsilon\rightarrow 0+}\limsup_{r\rightarrow\infty}\frac{T(r,\Delta_{\delta -\varepsilon},f)}{\log r}>\omega \] (in the sectorial domain!), then \(f=g\). The rather technical proofs in this paper make extensive use of the sectorial characteristics of meromorphic functions, both in the Ahlfors as well as in the Nevanlinna form.