对复平面C的非空有限子集S1和S2, 记在复平面区域D内满足{z ∈ D : f(z) ∈ S1}={z ∈ D : f'(z) ∈ S2}的全体亚纯函数f形成的函数族为〈S1,S2〉D, 那么当S1和S2共有至少12个元素时函数族〈S1,S2〉D正规. 特别地, 当S1具有至少三个复数时, 我们得到了准确的结果.
Abstract
For nonempty finite sets S1 and S2 and a domain D on C, denote by 〈S1, S2〉 D the family of meromorphic functions f which satisfy {z ∈ D : f(z) ∈ S1} = {z ∈ D : f'(z) ∈ S2}. We prove that if S1 and S2 have totally at least 12 elements, then 〈S1, S2〉 D is normal. In particular, when S1 has at least three elements, we get the sharp results.
关键词
亚纯函数 /
正规族 /
分担集合
{{custom_keyword}} /
Key words
meromorphic functions /
normal family /
shared sets
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Yang L., Value Distribution Theory, Berlin: Spring-Verlag, 1993.
[2] Hayman W. K., Meromorphic Functions, Oxford: Clarendon Press, 1964.
[3] Schwick W., Sharing values and normality, Arch Math., 1992, 59: 50-54.
[4] Pang X. C., Zalcman L. Normality and shared values, Ark. Mat., 2000, 38: 171-182.
[5] Chang J. M., Fang M. L., Zalcman L., Normality and fixed-points of meromorphic functions, Ark. Mat., 2005, 43: 307-321.
[6] Liu X. J., Pang X. C., Shared values and normal function, Acta Mathematica Sinica, Chinese Series, 2007, 50(2): 409-412.
[7] Chang J. M., Wang Y. F., Shared values, Picard values and normality, Tohoku Math. J., 2011, 63: 149-162.
[8] Hayman W. K., Miles J., On the growth of a meromorphic function and its derivatives, Complex Vari., 1989, 12: 245-260.
[9] Pang X. C., Zalcman L., Normal families and shared values, Bull. London Math. Soc., 2000, 32: 325-331.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
PDF(457 KB)
