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Uniqueness of meromorphic functions sharing three values. (English) Zbl 1085.30027
For 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 $$(\infty,\infty)$$.
##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
##### Keywords:
meromorphic function; uniqueness; shared value
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