Uniqueness of meromorphic functions with three common values. (Chinese. English summary) Zbl 0756.30030
By applying Nevanlinna’s second fundamental theorem and Borel type inequalities the author has improved some earlier results on unicity of meromorphic functions and obtained the following main result:
Let \(f(z)\) and \(g(z)\) be two distinct non-constant meromorphic functions satisfying for all \(z\in\mathbb{C}\): \[ f(z)=0\Leftrightarrow g(z)=0,\quad f(z)=1\Leftrightarrow g(z)=1,\text{ and }f(z)=\infty\Leftrightarrow g(z)=\infty. \] Assume further that \(\delta(a,f)+\delta(b,f)+\delta(\infty,f)>3/2\) for some \(a\neq 0,1\), \(b\neq 0,1\).
Then either (i) \(a\) and \(1-a\) are deficient values of \(f\) and \(g\), respectively, and \(f\) and \(g\) must satisfy one of the following three relations: 1) \((f-a)(g+a-1)=a(1-a)\), 2) \(f-(1-a)g=a\), 3) \(f=ag\),
or (ii) \(b\) and \((1-b)\) are deficient values of \(f\) and \(g\), respectively, and \(f\) and \(g\) must satisfy one of the following three relations: 1) \((f-b)(g+b-1)=b(1-b)\), 2) \(f-(1-b)g=b\), 3) \(f=bg\).
Let \(f(z)\) and \(g(z)\) be two distinct non-constant meromorphic functions satisfying for all \(z\in\mathbb{C}\): \[ f(z)=0\Leftrightarrow g(z)=0,\quad f(z)=1\Leftrightarrow g(z)=1,\text{ and }f(z)=\infty\Leftrightarrow g(z)=\infty. \] Assume further that \(\delta(a,f)+\delta(b,f)+\delta(\infty,f)>3/2\) for some \(a\neq 0,1\), \(b\neq 0,1\).
Then either (i) \(a\) and \(1-a\) are deficient values of \(f\) and \(g\), respectively, and \(f\) and \(g\) must satisfy one of the following three relations: 1) \((f-a)(g+a-1)=a(1-a)\), 2) \(f-(1-a)g=a\), 3) \(f=ag\),
or (ii) \(b\) and \((1-b)\) are deficient values of \(f\) and \(g\), respectively, and \(f\) and \(g\) must satisfy one of the following three relations: 1) \((f-b)(g+b-1)=b(1-b)\), 2) \(f-(1-b)g=b\), 3) \(f=bg\).
Reviewer: C.-C.Yang (Hongkong)
MSC:
30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
30D30 | Meromorphic functions of one complex variable (general theory) |
30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |