ON THE GROWTH OF LOGARITHMIC DIFFERENCES, DIFFERENCE QUOTIENTS AND LOGARITHMIC DERIVATIVES OF MEROMORPHIC FUNCTIONS

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	ON THE GROWTH OF LOGARITHMIC DIFFERENCES, DIFFERENCE QUOTIENTS AND LOGARITHMIC DERIVATIVES OF MEROMORPHIC FUNCTIONS
	Chiang, Yik-Man 1; Feng, Shao-Ji2
刊名	TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
	2009
卷号	361 期号:7 页码:3767-3791
关键词	Difference operators Poisson-Jensen formula Wiman-Valiron theory integrable difference equations
ISSN号	0002-9947
英文摘要	A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen (2000, 2007) that a connection exists between discrete Painleve equations and (finite order) Nevanlinna theory is an estimate of the integrated average of log(+) vertical bar f(z + 1)/f(z)vertical bar on vertical bar z vertical bar = r. We obtained essentially the same estimate in our previous paper (2008) independent of Halburd et al. (2006). We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. In addition to the potential applications of our new estimates in integrable systems, they are also of independent interest. In particular, our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order less than and greater than one. We have established a "difference" analogue of the classical Wiman-Valiron type estimates for meromorphic functions with order less than one, which allow us to prove that all entire solutions of linear difference equations (with polynomial coefficients) of order less than one must have positive rational order of growth. We have also established that any entire solution to a first order algebraic difference equation (with polynomial coefficients) must have a positive order of growth, which is a "difference" analogue of a classical result of Polya.
WOS研究方向	Mathematics
语种	英语
出版者	AMER MATHEMATICAL SOC
WOS记录号	WOS:000264881700015
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/8871]
专题	系统科学研究所
通讯作者	Chiang, Yik-Man
作者单位	1.Hong Kong Univ Sci & Technol, Dept Math, Kowloon, Hong Kong, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100080, Peoples R China
推荐引用方式 GB/T 7714	Chiang, Yik-Man,Feng, Shao-Ji. ON THE GROWTH OF LOGARITHMIC DIFFERENCES, DIFFERENCE QUOTIENTS AND LOGARITHMIC DERIVATIVES OF MEROMORPHIC FUNCTIONS[J]. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY,2009,361(7):3767-3791.
APA	Chiang, Yik-Man,&Feng, Shao-Ji.(2009).ON THE GROWTH OF LOGARITHMIC DIFFERENCES, DIFFERENCE QUOTIENTS AND LOGARITHMIC DERIVATIVES OF MEROMORPHIC FUNCTIONS.TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY,361(7),3767-3791.
MLA	Chiang, Yik-Man,et al."ON THE GROWTH OF LOGARITHMIC DIFFERENCES, DIFFERENCE QUOTIENTS AND LOGARITHMIC DERIVATIVES OF MEROMORPHIC FUNCTIONS".TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 361.7(2009):3767-3791.