Let and be transcendental meromorphic functions in the complex plane. A value is called an IM (ignoring multiplicities) shared value in of and if in , if and only if . A value is called an CM (counting multiplicities) shared value in of and if in , and assume at the same points in with the same multiplicities. In the case , there are several sharing conditions for uniqueness, see e.g., G. G. Gundersen [J. Lond. Math. Soc., II. Ser. 20, 456–466 (1979; Zbl 0413.30025)]. For example, (C1) If and share five values IM, then , which is due to R. Nevanlinna. (C2) If and share four values IM, and if for another value , then . (C3) If and share two values IM and if and share two values CM, then and share four values of CM. The author considers the uniqueness problem with the sharing conditions in some angular domains.
In this paper three theorems are obtained generalizing the (C1), (C2) and (C3). We state one of the results. Given an angular domain with and for some positive number and for some , , where is the number of the roots of in , and . If and share five distinct values IM, then .