Unicity theorems for meromorphic functions. (English) Zbl 1026.30033

For a non-constant meromorphic function in the plane, denote by \(T(r,f)\) the Nevanlinna characteristic of \(f\), and by \(S(r,f)\) any quantity which satisfies \(S(r,f)=o(T(r,f))\) as \(r\to\infty\) except possibly a set of positive numbers \(r\) of finite linear measure. Furthermore, for \(a \in \widehat{\mathbb C}\), let \(N_1(r,a,f)\) be the counting function of simple \(a\)-points of \(f\). If \(a=\infty\), this is the set of simple poles of \(f\). Finally, let \(m(r,a,f)\) be the proximity function of the \(a\)-points of \(f\).
If \(f\) and \(g\) are non-constant meromorphic functions, we say that \(f\) and \(g\) share the value \(a \in \widehat{\mathbb C}\) CM, if \(f\) and \(g\) have the same \(a\)-points with the same multiplicities. The well-known four-point-theorem of R. Nevanlinna states that if \(f\) and \(g\) are non-constant meromorphic functions which share four distinct values CM, then \(f\) is a Möbius transformation of \(g\). Many authors dealt with the problem of uniqueness of meromorphic functions which share three distinct values. Without loss of generality we may assume that these values are \(0\), \(1\), \(\infty\). In this paper, the authors prove the following result.
Theorem. Let \(f\) and \(g\) be two distinct non-constant meromorphic functions which share the values \(0\), \(1\) and \(\infty\) CM. If there exists a set \(I\) of infinite linear measure such that \[ \limsup_{\substack{ r\to\infty \\ r \in I }} \frac{N_1(r,\infty,f)+N_1(r,0,f)-m(r,1,g)}{T(r,f)} < 1 , \tag{1} \] then \(f\) and \(g\) satisfy one of the following relations: \[ \begin{alignedat}{2} f &= \frac{e^{s\gamma}-1}{e^{-(k+1-s)\gamma}-1} , &\quad g &= \frac{e^{-s\gamma}-1}{e^{(k+1-s)\gamma}-1} , \tag{i} \\ f &= \frac{e^{(k+1)\gamma}-1}{e^{s\gamma}-1} , &\quad g &= \frac{e^{-(k+1)\gamma}-1}{e^{-s\gamma}-1} , \tag{ii} \\ f &= \frac{e^{s\gamma}-1}{e^{(k+1)\gamma}-1} , &\quad g &= \frac{e^{-s\gamma}-1}{e^{-(k+1)\gamma}-1} , \tag{iii} \end{alignedat} \] where \(s\) and \(k\) are positive integers such that \(1 \leq s \leq k\), \(s\) and \(k+1\) are relatively prime, and \(\gamma\) is a non-constant entire function. Furthermore, there holds \[ N_1(r,\infty,f)+N_1(r,0,f)-m(r,1,g) = \left(1-\frac{1}{k}\right) T(r,f) + S(r,f) . \] The authors also give examples which show that the assumption (1) is sharp.
This theorem is an improvement and extension of results of M. Ozawa [J. Anal. Math. 30, 411-420 (1976; Zbl 0337.30020)], H. Ueda [Kodai Math. J. 6, 26-36 (1983; Zbl 0518.30029)] and G. Brosch [Aachen: RWTH Aachen, Math.-Naturwiss. Fak., Diss. 77 S. (1989; Zbl 0694.30027)].


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)
30D20 Entire functions of one complex variable (general theory)
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