A test for an entire function (Q909057)

Let \(F\) be a function analytic in the disc \(| z| <1\) and let \(F^{(n)}(0)\neq 0\), \(n=1,2,\dots\). Put \[ \frac{F^{(n)}(z)- F^{(n)}(0)}{F^{(n+1)}(0)}=z+\sum^{\infty}_{k=2}d_{kn}z^ k,\quad n=0,1,2,... \] The main result of the paper is the following. If for some \(k\geq 2\) we have \(\sup \{| d_{kn}|:n=0,1,\dots\}<\infty\) then \(F\) is an entire function of exponential type. This is a generalization of the theorem due to \textit{S. M. Shah} and \textit{S. Y.Trimble} [Bull. Am. Math. Soc. 75, 153--157 (1969; Zbl 0184.30801)].