Meromorphic functions that share two values (Q1359632)

This paper gives a couple of uniqueness theorems for certain meromorphic functions that share two values counting multiplicity (CM). Moreover, a positive answer to a question offered by \textit{F. Gross} [Complex analysis, Theory Appl. 599, 51-69 (1977; Zbl 0357.30007)] will be given. The problem by Gross reads as follows: Does there exist two finite sets \(S_1\), \(S_2\) such that two entire functions \(f\), \(g\) which satisfy \(f^{-1}(S_j)=g^{-1}(S_j)\), \(j=1,2\), counting multiplicities must be identical? Theorem 1: Let \(f\), \(g\) be two meromorphic functions such that \(f^n+a\) and \(g^n+a\) share \(0\) and \(\infty\) CM, where \(a\in \mathcal C \setminus \{0\}\) and \(n>5\) is an integer. Then \(f^n=g^n\) of \(f^ng^n=a^2\), either \(f=cg\) of \(fg=d\) for some constants \(c\), \(d\). Theorem 2 (answers the Gross' problem): Let \(f\) and \(g\) be two nonconstant entire functions, \(n>4\) be an integer, and \(a\), \(b\) be finite non-zero complex numbers such that \(a^{2n+2}\neq b^{2n}\). Set \(S_1=\{ \omega\mid\omega^n+a=0 \}\) and \(S_2=\{ \omega\mid\omega^{n+1}+b=0 \}\). If \(f^{-1}(S_j)=g^{-1}(S_j)\) for \(j=1,2\), counting multiplicities, then \(f=g\). Finally, we remark that in the proof of Theorem 1, reference to Lemma 3 should be given in a crucial phase, instead of Lemma 1.