On a question of Gary G. Gundersen concerning meromorphic functions sharing three distinct values IM and a fourth value CM

Abstract

In 1992, Gundersen (Complex Var. Elliptic Equ.20 (1992), no. 1-4, 99-106.) proposed the following famous open question: if two non-constant meromorphic functions share three values IM and share a fourth value CM, then do the functions necessarily share all four values CM? The open question is a long-standing question in the studies of the Nevanlinna$'$s value distribution theory of meromorphic functions, and has not been completely resolved by now. In this paper, we prove that if two distinct non-constant meromorphic functions $f$ and $g$ of finite order share $0,$ $1,$ $c$ IM and $\infty$ CM, where $c$ is a finite complex value such that $c\not\in\{0,1\},$ then $f$ and $g$ share $0,$ $1,$ $c,$ $\infty$ CM. Applying the main result obtained in this paper, we completely resolve a question proposed by Gary G. Gundersen on Page 458 of his paper (J. London Math. Soc. 20(1979), no. 2, 457-466.)concerning the nonexistence of two distinct non-constant meromorphic functions sharing three distinct values DM and a fourth value CM. The obtained result also improves the corresponding result on Pages 109-117 in (E. Mues, Bemerkungen zum vier-punkte-satz, Complex Methods on Partial Diferential Equations, 109-117, Math. Res. 53, Akademie-Verlag, Berlin, 1989.) concerning the nonexistence of two distinct non-constant entire functions that share three distinct finite values DM. Examples are provided to show that the main results obtained in this paper, in a sense, are best possible.