Holomorphic functions having large images under the action of differential operators

We prove in this note that, given a simply connected domain G in the complex plane and a sequence of infinite order linear differential operators generated by entire functions of subexponential type satisfying suitable conditions, then there are holomorphic functions f on G such that the image of any open subset under the action of those operators on f is arbitrarily large. This generalizes an earlier result about images of derivatives. A known statement about close orbits is also strengthened.