涉及复合亚纯函数和不动点的亚纯函数的正规族

假设f是一个超越整函数,G是定义在区域DC上的全纯函数族. 如果对G中每个元素g,f(g)-α₁在区域D上的每个零点重数≥2,f(g)-α₂和g-α₂在区域D上IM分担0, 这里α₁和α₂是2个判别的有穷复数,则G在区域D上是正规的,该结果推广了Bergweiler 2004年的一个结果.同时还证明了: 假设R 是一个次数满足deg R≥2(deg R≥3,并且R在在复平面上有3个判别的有限的不动点)的有理函数,F是一个定义在区域DC上的全纯函数(亚纯函数),并且对F中每个元素f,Rf(z)-z 和f(z)-z在区域D上IM分担0,则F是区域D上的正规族, 该结果推广了方明亮与袁文俊2000年的一个结果, 也推广了常建明、方明亮与L.Zalcman 2005年的一个结果, 并举例说明本文结果从某种意义上来讲是最佳的.

Normal Families of Meromorphic Functions Concerning Composite Meromorphic Functions and Fixed Points

Then following result is proved: If f is a transcendental entire function, such that the family G of all functions g holomorphic in the domain DC for which every zero of f(g)-α₁ is of multiplicity ≥2,f(g)-α₂ and g-α₂ share 0 IM in D, where α₁ and α₂ are two distinct finite values, then G is normal in D. This result extends Theorem 1 of paper in Bergweiler.The following result is also proved: If R is a rational function with degR≥2(respectively ≥3, and R has three distinct finite fixed points in the complex plane)such that the family F of all functions f holomorphic(respectively meromorphic)in the domain DC for which R f(z)-z and f(z)-z share 0 IM in D, then F is normal in D. The results extend the corresponding results due to Fang-Yuan and Chang-Fang-Zalcman respectively. Examples are provided to show that the main results in this paper, in a sense, are the best possible.