On a version of the problem B of Mahler involving derivatives (Q6591989)

In 1902 \textit{P. Stäckel} [Acta Math. 25, 371--384 (1902; JFM 33.0432.02)] proved that there exists a transcendental function \(f(z)= -z+f_2z^2+f_3z^3+\dots\) with rational coefficients, which converges in a neighborhood of the origin and has the property that both \(f(z)\) and its inverse function, as well as all their derivatives are algebraic at all algebraic points in this neighborhood. Based on this result, in [Lectures on transcendental numbers. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0332.10019)] \textit{K. Mahler} questioned whether this result (without involving derivatives) could be extended to transcendental entire functions, and named this question as Problem B. This problem was solved by \textit{D. Marques} and \textit{C. G. Moreira} [Math. Ann. 368, No. 3--4, 1059--1062 (2017; Zbl 1387.11056)]. It is natural to ask whether we have an analogous result for transcendental entire functions, involving derivatives, too. In the paper under review, the author shows that, there are in fact an accountable amount of such functions. In the proof, classical theorems by Rouché and Hurwitz are also used.