On the Derivatives of Composite Functions

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New Zealand Journal of Mathematics

Vol. 36, (2007), Pages 57-61

J.K. Langley

School of Mathematical Sciences

University of Nottingham

NG7 2RD

E.F. Lingham

Department of Mathematical Sciences

Loughborough University

LE11 3TU



Abstract Let g be a non-constant polynomial and let f be transcendental and meromorphic of sub-exponential growth in the plane. Then if k\geq 2 and Q is a polynomial the function (f\circ g)^{(k)}-Q has infinitely many zeros. The same conclusion holds for k \geq 0 and with Q a rational function if f has finitely many poles. We also show by example that this result is sharp.

Keywords composite functions, Nevanlinna theory

Classification (MSC2000) 30D30, 30D35

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