## 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)].

### MSC:

 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)

### Keywords:

Nevanlinna theory; shared value; unicity theorem

### Citations:

Zbl 0337.30020; Zbl 0518.30029; Zbl 0694.30027
Full Text:

### References:

 [1] Hayman, W. K.: Meromorphic Functions. Clarendon Press, Oxford (1964). · Zbl 0115.06203 [2] Yi, H.-X., and Yang, C.-C.: Uniqueness Theory of Meromorphic Functions. Pure and Applied Math. Monographs, no. 32, Science Press, Beijing (1995). [3] Gundensen, G. G.: Meromorphic functions that share three or four values. J. London Math. Soc., 20 , 457-466 (1979). · Zbl 0413.30025 [4] Nevanlinna, R.: Le Th$$\acute{e}$$or$$\grave{e}$$me de Picard-Borel et la Th$$\acute{e}$$orie des Fonctions M$$\acute{e}$$romorphes. Gauthiers-Villars, Paris (1929). · JFM 55.0773.03 [5] Ozawa, M.: Unicity theorems for entire functions. J. d’Anal. Math., 30 , 411-420 (1976). · Zbl 0337.30020 [6] Ueda, H.: Unicity theorems for meromorphic or entire functions II. Kodai Math. J., 3 , 26-36 (1983). · Zbl 0518.30029 [7] Ueda, H.: On the zero-one-pole set of a meromorphic function II. Kodai Math. J., 13 , 134-142 (1990). · Zbl 0707.30024 [8] Brosch, G.: Eindeutigkeitssätze für meromorphe funktionen. Thesis, Technical University of Aachen (1989). · Zbl 0694.30027 [9] Mues, E.: Shared value problems for meromorphic functions. Value Distribution Theory and Complex Differential Equations (Joensuu, 1994), Joensuun Yliop. Luonnont. Julk., vol. 35, pp. 17-43 (1995). · Zbl 0951.30027 [10] Yi, H.-X.: Meromorphic functions that share three values. Chin. Ann. Math., 9A , 433-440 (1988). · Zbl 0699.30024 [11] Yi, H.-X.: Meromorphic functions that share two or three values. Kodai Math. J., 13 , 363-372 (1990). · Zbl 0712.30029 [12] Yi, H.-X.: Unicity theorems for meromorphic functions that share three values. Kodai Math. J., 18 , 300-314 (1995). · Zbl 0868.30032 [13] Yi, H.-X.: Meromorphic functions that share three values. BHKMS, 2 , 55-64 (1998). · Zbl 0919.30024 [14] Li, P., and Yang, C.-C.: On the characteristics of meromorphic functions that share three values CM. J. Math. Anal. Appl., 220 , 132-145 (1998). · Zbl 0909.30021 [15] Zhang, Q.-C.: Meromorphic functions sharing three values. Indian J. Pure Appl. Math., 30 , 667-682 (1999). \end{} · Zbl 0934.30025
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