A New Characterization of Gundersen’s Example of Two Meromorphic Functions Sharing Four Values

Abstract

Let ƒ and g be nonconstant meromophic functions sharing four values IM and satisfying ƒ⁻¹({a}) ⊂ ⁻¹({b}) for two values a, b not shared by ƒ and g. Then either ƒ = T o g with a Möbius transformation T or $f=L\ {\rm o}\ \hat{f}\ {\rm o}\ h$ and $g=L\ {\rm o}\ \hat{g}\ {\rm o}\ h$, where $\hat{f}(z)=({\rm exp}\ z+1)/({\rm exp}\ z-1)^2$ and $\hat{g}(z)=({\rm exp}\ z+1)^2/(8({\rm exp}\ z-1))$ are the functions in Gundersen’s example [1], L is a Möbius transformation and h is an entire function.