关于C.C.Yang的涉及亚纯函数增长性的几个问题

ON SOME PROBLEMS ON THE GROWTH OFMEROMORPHIC FUNCTIONS PROPOSED BY C.C.Yang

得到如下一些结果：（ｉ）设ｆ是任一亚纯函数，若ｌｉｍｒ→∞Ｔ（ｒ，ｆ（ｚ＋１））Ｔ（ｒ，ｆ（ｚ））＝∞，则级ρｆ＝∞．（ｉ）设ｆ、ｇ１、ｇ２皆为非常数整函数，且Ｔ（ｒ，ｆ）＝Ｏ＊（（ｌｏｇｒ）α）（α＞１），ｇ２的级为有穷，ａ≠∞δ（ａ，ｇ２）＝１，Ｔ（ｒ，ｇ１）＝ｏ（Ｔ（ｒ，ｇ２））（ｒ→∞）．则Ｔ（ｒ，ｆ（ｇ１））＝ｏ（Ｔ（ｒ，ｆ（ｇ２）））（ｒ→∞），其中Ｔ（ｒ，ｆ）＝Ｏ＊（（ｌｏｇｒ）α）表示：当ｒ→∞时，Ｔ（ｒ，ｆ）与（ｌｏｇｒ）α是同阶无穷大．从而解决了Ｃ．Ｃ．Ｙａｎｇ提出的关于亚纯函数增长性的几个问题．

In this paper, we obtain the following results:(i) Let f be a meromorphic function, iflimr→∞T(r,f(z+1))T(r,f(z))=∞, then the order of g is infinity.(ii) Letf、g1 and g2 be nonconstant entire functions, with T(r,f)=O*((logr)α)(α>1), order of g2 is ρg<∞ and a≠∞δ(a,g2)=1. If T(r,g1)=o(T(r,g2)) (r→∞). ThenT(r,f(g1))=o(T(r,f(g2))) (r→∞),where T(r,f)=O*((logr)α) (α>1) means that as r→∞, T(r,f) and (logr)α are of the same order. These results solved some problems proposed by C.C.Yang.