Holomorphic functions having large images under the action of differential operators (Q1283027)

The paper deals with a generalization of a theorem from \textit{R. M. Gethner} and \textit{J. H. Shapiro} [Proc. Am. Math. Soc. 100, 281-288 (1987; Zbl 0618.30031)] which states that in the space \(H(G)\), of in a simply connected domain \(G\) holomorphic functions, there exists a dense subset \(M(H(G))\) such that for every function \(f\in M\) and every nonempty set \(V\subset G\) there are infinitely many \(n\) such that \(f^{(n)} (V)\supset D(0,\rho_n)\), where the radii \(\rho_n\) of the covered discs grow to infinity such that \({1\over n}\root n\of {\rho_n}\to 0\). The main concern of the paper is to replace the \(n\)-th derivatives by certain differential operators of infinite order and of subexponential type. This means that the differential operators \(\Phi(D)= \sum^\infty_{j=0} c_jD^j\) are defined by the power series \(\Phi(z)= \sum^\infty_{j=0} c_jz^j\) with \(|\Phi(z) |\leq K_\varepsilon e^{ \varepsilon |z|}\) for all \(\varepsilon>0\) and \(z\) sufficiently large.