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On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. (English) Zbl 1172.30009

For a meromorphic function f of order σ, the logarithmic derivative f ' /f satisfies the estimate |f ' (z)/f(z)||z| σ-1+ε outside a small exceptional set. This result has many applications, in particular to complex differential equations. In the study of difference equation, a similar role is played by the estimate |f(z+η)/f(z)|exp(|z| σ-1+ε ) which was obtained independently by R. G. Halburd and R. J. Korhonen [J. Math. Anal. Appl. 314, No. 2, 477–487 (2006; Zbl 1085.30026)] and by Y.-M. Chiang and S.-J. Feng [Ramanujan J. 16, No. 1, 105–129 (2008; Zbl 1152.30024)].

In the present paper the authors establish a connection between logarithmic derivatives and differences by showing that

f(z+η) f(z)=expηf ' (z) f(z)+O(r β+ε )

for |z| outside a set of finite logarithmic measure, where β is defined as follows: denoting by λ the maximum of the exponents of convergence of the zeros and poles of f, we have β=max{σ-2,2λ-2} if λ<1 and β=max{σ-2,λ-1} if λ1.

The above result is used to show that

f(z+η)-f(z) f(z)=ηf ' (z) f(z)+Or 2σ-2+ε

outside the exceptional set. Extensions to higher order difference quotients are also included.

Finally the paper contains a difference version of Wiman-Valiron theory which is used to show that entire solutions of first order algebraic difference equations have positive order.

30D30General theory of meromorphic functions
30D35Distribution of values (one complex variable); Nevanlinna theory
39A05General theory of difference equations
46E25Rings and algebras of continuous, differentiable or analytic functions