整函数与其二次迭代周期性之间关系的Gross问题 献给杨乐教授80华诞

The Gross’ problem concerning periodicity relation between the entire function and its second iteration

1972年, Gross提出了下述有趣的问题:如果f(z)是非常数整函数,而且f(f(z))是周期函数,那么f(z)一定也是周期函数吗?关于这一问题,本文证明了如下结果:如果f(z)是非常数整函数,f(f(z))是以τ为周期的周期函数,且f′(z+nτ)(n=0, 1, 2,...)在复平面C上不局部一致收敛于0,则f(z)也是周期函数.

In 1972,Gross posed the following interesting open problem:if f(z)is an entire function,and f?2(z)is periodic,must f(z)also be periodic?In this paper,we prove the following result concerning this problem:if f(z)is a nonconstant entire function,f?2(z)is a periodic function with periodτ,and f′(z+nτ)(n=1,2,3,...)dose not converge locally uniformly to 0 in the complex plane,then f(z)is also periodic.