# On the Derivatives of Composite Functions

### 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