On the Gelfond – Leont’ev – Sǎlǎgean and Gelfond – Leont’ev – Ruscheweyh operators and analytic continuation of functions

Abstract

UDC 517.537

Let $A_{\lambda} (0)$ denote the class of power series $g(z) = \sum^{\infty}_{k=0} g^k z_k$ such that $|g_k| \leq \lambda_k| g_1|$ for all $k \geq 1$, where $\lambda = (\lambda_k)$ is a sequence of positive numbers. We obtain necessary and sufficient conditions imposed on a function $l$ and an increasing
sequence $(n_p)$ of non-negative integers ensuring that the assumption that the Gelfond – Leont’ev – Sălăgean derivative $D^{n_p}_{l,[S]}f$ and the Gelfond – Leont’ev – Ruscheweyh derivative $D^{n_p}_{l,[R]}f$ belong to the class $A_{\lambda} (0)$ for all $p \in {\Bbb Z}_+$ implies that f is an entire function.