On power serves with Gelfond--Leontev derivatives satisfying a special condition

M. N. Sheremeta

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (1996), pp. 423-445.

Résumé

Necessary and sufficient conditions on a function $l$ and an increasing sequence $(n_p)$ of non-negative integers are found in order that $f$ be an entire function whenever for all $p\in z_+$ the Gelfond–Leontev derivative $D_l^{n_p}f$ belongs to the class $A_\lambda(0)$, where the class $A_\lambda(0)$ consists of all functions $g(z)=\sum_{k=0}^\infty g_k(z^k)$ such that $|g_k|\le\lambda_k|g_1|$ ($k\geq1$) and $\lambda=(\lambda_k)$ is a sequence of positive numbers.