Universally prestarlike functions as convolution multipliers

Abstract

Universally prestarlike functions (of order α ≤ 1) in the slit domain $${\Lambda:=\mathbb{C}{\setminus}[1,\infty]}$$ have recently been introduced in Ruscheweyh et al. (Israel J Math, to appear). This notation generalizes the corresponding one for functions in the unit disk $${\mathbb{D}}$$ (and other circular domains in $${\mathbb{C}}$$ ). In this paper we study the behaviour of universally prestarlike functions under the Hadamard product. In particular it is shown that these function classes (with α fixed), are closed under convolution, and that their members, as Hadamard multipliers, also preserve the prestarlikeness (of the same order) of functions in arbitrary circular domains containing the origin.