On the Growth of Logarithmic Differences, Difference Quotients and Logarithmic Derivatives of Meromorphic Functions

Abstract. We obtain growth comparison results of logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. Our results are both generalizations and extensions of previous results. We construct examples showing that the results obtained are best possible in certain sense. Our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order smaller 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 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 Pólya.