# 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 and *Q* is a polynomial the function has infinitely many zeros. The same conclusion holds for 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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