Estimates for the Logarithmic Derivative of a Meromorphic Function, Plus Similar Estimates

Abstract. We investigate the growth of the Nevanlinna Characteristic of f (z + η) for a fixed η ∈ C in this paper. In particular, we obtain a precise asymptotic relation between T`r, f (z + η)´and T (r, f ), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f (z + η)/f (z) which is a discrete version of the classical logarithmic derivative estimates of f (z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker [40] concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst [1] concerning integrable difference equations.

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“…[14]). In particular, the following estimate of Gundersen [13,Cor. 2] gives a sharp upper bound of logarithm derivatives.…”

Section: Pointwise Estimatesmentioning

confidence: 99%

“…We recall that Gundersen [13,Cor. 1] has given a precise pointwise estimate for the logarithmic derivative for a meromorphic function f (z) of order σ to be…”

Section: Introductionmentioning

confidence: 99%

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

2008

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“…We begin with recalling a lemma on logarithmic derivatives due to Gundersen [9], which plays an important role in proving our results. …”

Section: Auxiliary Resultsmentioning

confidence: 99%

Growth of solutions of second order linear differential equations with extremal functions for Denjoy's conjecture as coefficients

Long¹

2016

Tamkang J. Math.

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“…The function f has finitely many zeros and non-real poles, and k(z) = 2πz. Hence (3.1) and standard estimates for logarithmic derivatives [13] show that…”

Section: Lemma 42 the Functionmentioning

confidence: 99%

Non-real zeros of derivatives of meromorphic functions

Langley

2017

JAMA

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Estimates for the Logarithmic Derivative of a Meromorphic Function, Plus Similar Estimates

Cited by 300 publications

References 7 publications

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

Growth of solutions of second order linear differential equations with extremal functions for Denjoy's conjecture as coefficients

Non-real zeros of derivatives of meromorphic functions

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