Please use this identifier to cite or link to this item: http://hdl.handle.net/1783.1/6617
On the Growth of Logarithmic Differences, Difference Quotients and Logarithmic Derivatives of Meromorphic Functions
Authors  Chiang, YikMan
Feng, ShaoJi 


Issue Date  2009  
Source  Transactions of the American Mathematical Society , v. 361, (7), 2009, JUL, p. 37673791  
Summary  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 WimanValiron 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.  
Subjects  
ISSN  00029947  
Rights  First published in Transactions of the American Mathematical Society in v. 361, no. 7, July 2009, published by the American Mathematical Society  
Language  English 

Format  Article  
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