Universally prestarlike functions as convolution multipliers

Universally prestarlike functions (of order α ≤ 1) in the slit domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda:=\mathbb{C}{\setminus}[1,\infty]}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{D}}$$\end{document} (and other circular domains in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document}). 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.