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

摘要

A crucial ingredient in the recent discovery by Ablowitz, Halburd,Herbst and Korhonen (2000, 2007) that a connection exists between discretePainleve equations and (finite order) Nevanlinna theory is an estimate of theintegrated average of log+ f (z 1) f (z)|on |z|= r. We obtained essentiallythe same estimate in our previous paper (2008) independent of Halburd et al.(2006). We continue our study in this paper by establishing complete asymp-totic relations amongst the logarithmic differences, difference quotients andlogarithmic derivatives for finite order meromorphic functions. In addition tothe potential applications of our new estimates in integrable systems, they arealso of independent interest. In particular, our findings show that there aremarked differences between the growth of meromorphic functions with Nevan-linna order less than and greater than one. We have established a "difference"analogue of the classical Wiman-Valiron type estimates for meromorphic func-tions with order less than one, which allow us to prove that all entire solutionsof linear difference equations (with polynomial coefficients) of order less thanone must have positive rational order of growth. We have also established thatany entire solution to a first order algebraic difference equation (with polyno-mial coefficients) must have a positive order of growth, which is a "difference"analogue of a classical result of Pdlya.