Uniqueness of meromorphic functions sharing three values.

*(English)*Zbl 1085.30027For a meromorphic function \(f\) in the complex plane, let \(T(r,f)\) denote the Nevanlinna characteristic, and let \(S(r,f)\) be any quantity that satisfies \(S(r,f)=o(T(r,f))\) as \(r\to\infty\) except possibly on a set of finite linear measure. For \(a \in \widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}\) and a positive integer \(k\) or \(k=\infty\), let \(N(r,a;f| \leq k)\) denote the counting function of the \(a\)-points (poles if \(a=\infty\)) of \(f\) with multiplicity at most \(k\). Then the usual counting function is given by \(N(r,a;f)=N(r,a;f| \leq\infty)\), where every \(a\)-point is counted according to its multiplicity. If every \(a\)-point is counted only once, the corresponding function is denoted by \(\overline{N}(r,a;f)\). Furthermore, let \(E_k(a;f)\) be the set of \(a\)-points of \(f\), where an \(a\)-point of multiplicity \(m\) is counted \(m\) times if \(m \leq k\) and \(k+1\) times if \(m>k\). It is said that two meromorphic functions \(f\) and \(g\) share a value \(a \in \widehat{\mathbb{C}}\) with weight \(k\), or shortly they share \((a,k)\), if \(E_k(a;f)=E_k(a;g)\). Obviously, \(f\) and \(g\) share \(a\) IM (ignoring multiplicities) or CM (counting multiplicities) if and only if \(f\) and \(g\) share \((a,0)\) or \((a,\infty)\), respectively.

A theorem of H. Ueda [Kodai Math. J. 6, 26–36 (1983; Zbl 0518.30029)] states that if two non-constant meromorphic functions share the values \(0\), \(1\) and \(\infty\) CM, and if \[ \limsup_{r\to\infty}{\frac{N(r,0;f)+N(r,\infty;f)}{T(r,f)}} < \frac{1}{2}\,, \tag{\(*\)} \] then either \(f \equiv g\) of \(fg \equiv 1\). The special case for entire functions of finite order was already done by M. Ozawa [J. Anal. Math. 30, 411–420 (1976; Zbl 0337.30020)]. An improvement was achieved by H. X. Yi [Kodai Math. J. 13, 363–372 (1990; Zbl 0712.30029)] by replacing \((*)\) by the weaker condition \[ \limsup_{r\to\infty}{\frac{N(r,0;f| \leq 1)+N(r,\infty;f| \leq 1)}{T(r,f)}} < \frac{1}{2}\,. \tag{\(**\)} \]

W. R. Lü and H. X. Yi replaced the bound \(\frac{1}{2}\) in \((**)\) by \(1\) and obtained that then \[ f \equiv \frac{e^{s\gamma}-1}{e^{-(k+1-s)\gamma}-1} \quad \text{and} \quad g \equiv \frac{e^{-s\gamma}-1}{e^{(k+1-s)\gamma}-1}\,, \] where \(s\) and \(k\) are relatively prime positive integers with \(1 \leq s \leq k\) and \(\gamma\) is a non-constant entire function. The functions \(f=(e^\gamma-1)^2\) and \(g=e^\gamma-1\) show that this result is not true in general if \(f\) and \(g\) share the value \(0\) only IM. In this paper, the authors consider the question whether it is possible to relax the nature of sharing the value \(0\). Their result reads as follows.

Theorem. Let \(f\) and \(g\) be two distinct non-constant meromorphic functions sharing \((0,1)\), \((1,m)\) and \((\infty,k)\), where \((m-1)(mk-1)>(m+1)^2\). If \((**)\) holds, then \(f\) and \(g\) satisfy the relations \[ \left(1+\frac{\alpha}{f}-\frac{1}{f}\right)^s \equiv \alpha^{s+t} \quad \text{and} \quad \left(1+\frac{1}{g\alpha}-\frac{1}{g}\right)^s \equiv \alpha^{-(s+t)}\,, \] where \(\alpha\) is a non-constant meromorphic function such that \(\overline{N}(r,0;\alpha)+\overline{N}(r,\infty;\alpha) = S(r,f)\), and \(s\), \(t\) are relatively prime non-zero integers with \(s>0\) and \(s+t \neq 0\). In particular, the above result of W. R. Lü and H. X. Yi remains valid if \(f\) and \(g\) share \((0,1)\), \((1,\infty)\) and \((\infty,\infty)\).

A theorem of H. Ueda [Kodai Math. J. 6, 26–36 (1983; Zbl 0518.30029)] states that if two non-constant meromorphic functions share the values \(0\), \(1\) and \(\infty\) CM, and if \[ \limsup_{r\to\infty}{\frac{N(r,0;f)+N(r,\infty;f)}{T(r,f)}} < \frac{1}{2}\,, \tag{\(*\)} \] then either \(f \equiv g\) of \(fg \equiv 1\). The special case for entire functions of finite order was already done by M. Ozawa [J. Anal. Math. 30, 411–420 (1976; Zbl 0337.30020)]. An improvement was achieved by H. X. Yi [Kodai Math. J. 13, 363–372 (1990; Zbl 0712.30029)] by replacing \((*)\) by the weaker condition \[ \limsup_{r\to\infty}{\frac{N(r,0;f| \leq 1)+N(r,\infty;f| \leq 1)}{T(r,f)}} < \frac{1}{2}\,. \tag{\(**\)} \]

W. R. Lü and H. X. Yi replaced the bound \(\frac{1}{2}\) in \((**)\) by \(1\) and obtained that then \[ f \equiv \frac{e^{s\gamma}-1}{e^{-(k+1-s)\gamma}-1} \quad \text{and} \quad g \equiv \frac{e^{-s\gamma}-1}{e^{(k+1-s)\gamma}-1}\,, \] where \(s\) and \(k\) are relatively prime positive integers with \(1 \leq s \leq k\) and \(\gamma\) is a non-constant entire function. The functions \(f=(e^\gamma-1)^2\) and \(g=e^\gamma-1\) show that this result is not true in general if \(f\) and \(g\) share the value \(0\) only IM. In this paper, the authors consider the question whether it is possible to relax the nature of sharing the value \(0\). Their result reads as follows.

Theorem. Let \(f\) and \(g\) be two distinct non-constant meromorphic functions sharing \((0,1)\), \((1,m)\) and \((\infty,k)\), where \((m-1)(mk-1)>(m+1)^2\). If \((**)\) holds, then \(f\) and \(g\) satisfy the relations \[ \left(1+\frac{\alpha}{f}-\frac{1}{f}\right)^s \equiv \alpha^{s+t} \quad \text{and} \quad \left(1+\frac{1}{g\alpha}-\frac{1}{g}\right)^s \equiv \alpha^{-(s+t)}\,, \] where \(\alpha\) is a non-constant meromorphic function such that \(\overline{N}(r,0;\alpha)+\overline{N}(r,\infty;\alpha) = S(r,f)\), and \(s\), \(t\) are relatively prime non-zero integers with \(s>0\) and \(s+t \neq 0\). In particular, the above result of W. R. Lü and H. X. Yi remains valid if \(f\) and \(g\) share \((0,1)\), \((1,\infty)\) and \((\infty,\infty)\).

Reviewer: Rainer Brück (Dortmund)

##### MSC:

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |