半純函數之值分佈與其相關問題

Abstract

Hayman 猜測對於任意的超越半純函數f (z)， f ` (z) f (z)n 取任意非零的值無窮多次，因此，很自然的問題是f (k ) (z) f (z)n，其中k 2，具有相同的結果；同時，在文獻上我們也知道f(z)af (z)n 取任意非零的值無窮多次，所以，我們也將探討 f (k ) (z)af (z)n的值分佈，這些是本計畫研究的一部分。另外，本計畫也將著重於兩個半純函數共值或共小函數的問題進行研究。

A famous conjecture of Hayman says that if f (z) is a transcendental meromorphic function, then f ` (z) f (z)n assumes all finite values except possibly zero infinitely often. The conjecture was solved in most cases. Therefore, it is natural to ask what is the value distribution of f (k ) (z) f (z)n ? Also, we know that f(z)af (z)n assumes all finite values except possibly zero infinitely often. So, we can ask the same question for f (k ) (z)af (z)n . This is a part of the project which we will study. Another part is to study the problems of sharing values (small functions) of two meromorphic functions.