One More Note on Neighborhoods of Univalent Functions
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One More Note on Neighborhoods of Univalent Functions Richard Fournier1 Received: 30 May 2020 / Revised: 31 July 2020 / Accepted: 10 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We some open questions linked to the notion of neighbourhood of a univalent function as introduced by Stephan Ruscheweyh in 1981. Keywords Neighborhoods of univalent functions · Starlike functions · Close-to-convex functions Mathematics Subject Classification 30C45 1 My first contact with Stephan Ruscheweyh took place through reading his papers written in the early seventies about the convolution of analytic functions, first two papers in German (including some joint work with Karl–Joachim Wirths) and then of course the famous paper together with Terence Sheil–Small where the Pólya–Schoenberg conjecture was solved. My advisor, the late Q. I. Rahman, was deeply impressed by his work and suggested that I study carefully his papers for my master’s thesis at Université de Montréal. Later in the late seventies, Rahman visited Ruscheweyh in Afghanistan and came back to Montreal with the first draft of what was to become the paper “On neighborhoods of univalent functions” to appear in 1981. This time, Rahman suggested that I write my Ph.D. thesis on some open questions stated by Ruscheweyh at the end of his paper. Then happened what happened: I met Ruscheweyh at the 1982 “Séminaire de Mathématiques Supérieures” at Université de Montréal, I wrote my thesis inspired by his wise ideas and I finally ended up spending a year in Würzburg as a post-doc
Communicated by Filippo Bracci. In Memory of Stephan Ruscheweyh.
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Richard Fournier [email protected] DMS and CRM, Université de Montréal, C.P.6128, succursale Centre-ville, Montréal, QC H3C 3J7, Canada
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R. Fournier
under his guidance. Over the years, we kept visiting each other and collaborated on over a dozen papers. He was the main inspiration for my mathematical career. This short note is intended to be a humble tribute to Stephan as well as a concise review of some open questions related to neighborhoods of univalent functions. The paper [4] was relatively influential (more or less 80 quotes in MathSciNet and more than 300 quotes in Google Scholar): I apologize in advance for my very narrow point of view on the topic! 2 Let A0 denote the class of functions f analytic in the unit disc D = {z |z| < 1} of the complex f (0) = f (0) − 1 = 0. For such a function f with ∞plane satisfying n f (z) = z + n=1 an z , it is well-known that the condition ∞
|an | ≤ 1
(1)
n=2
implies that f is one-to-one (univalent) in D and f (D) is a starlike domain with respect to the origin. This result is generally attributed to Kobori or Noshiro and it is related to the work of Alexander on univalent functions. It has been rediscovered by A.W. Goodman, Clunie and Keogh and many others. An interesting discussion concerning the work of Alexander can be found in [3], together with exact references. We denote by St the subclass of A0 consisting of the univalent
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