On the geometric properties of series in systems of functions

Abstract

Let $f(z) = \dsum_{k = 1}^{\infty}f_k z^{k}$ be an entire transcendental function, let $(\lambda_n)$ be a sequence of positive numbers increasing to $+ \infty$, and let the series $A(z) = \dsum_{n = 1}^{\infty}a_nf(\lambda_n z)$ be regularly convergent in ${\mathbb{D}} = \{z\colon |z|<1\}$. The starlikeness and convexity of the function $A$ are studied. For example, if $\dsum_{n = 1}^{\infty}\lambda^{-\tau}_n = T< + \infty$, $\ln |a_n|\le -e\lambda_n$, and $T\dsum_{k = 2}^{\infty}k|f_k| (k + \tau)^{k + \tau}\le \left|f_1\dsum_{n = 1}^{\infty}a_n\lambda_n\right|$, then the function $A$ is starlike. It is proved that, under certain conditions on the parameters, the differential equation $z^2w'' + (\beta_0 z^2 + \beta_1z)w' + (\gamma_0z^2 + \gamma_1 z + \gamma_2) w = 0$ has an entire solution $A$ that is starlike or convex in ${\mathbb{D}}$.