Typically real functions and typically real derivatives
type: Item
instance of: scholarly article
title: Typically real functions and typically real derivatives
zbMATH Open document ID: 0638.30012
DOI: 10.1155/S0161171287000577
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46260
review text: Let D denote the unit disk \(\{\) \(z: | z| <1\}\). W. W. Rogosinski defined the class T of typically real functions \(f(z)=z+a_ 2z^ 2+..\). regular in D and such that Im z\(=0\) if, and only if, Im f(z)\(=0\). The author modifies this definition to remove the normalization conditions on f and, motivated by the fact that \(f(z)=z+z^ 3\in T\) while f is not univalent, he gives sufficient conditions in terms of typically real derivatives which ensure that functions are univalent. In particular, he defines \(T'=\{f\in T:\) f' is also typically real in \(D\}\) and shows that if \(f\in T'\) then f is univalent in D. Furthermore, let E denote the class of functions \(f(z)=z+a_ 2z^ 2+..\). such that \(f^{(n)}\) is univalent in D for \(n=0,1,2,..\). and ER those functions \(f(z)=z+a_ 2z^ 2+..\). in E with \(a_ n\) real for \(n=2,3,... \). \(\overline{ER}\) denotes the class of functions which are uniform limits on compact subsets of D of sequences in ER, and ERP those functions in ER such that \(a_ n>0\) for \(n=2,3,... \). The author shows that \(f\in ER\) if, and only if, \(f^{(n)}\) is typically real in D for \(n=0,1,2,..\). and includes various results for the classes ER, \(\overline{ER}\) and \(\overline{ER}\setminus ER\), together with a proof that ERP is convex.
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C55
zbMATH DE Number: 4040237
zbMATH Keywords: typically real functions
author: Selden Y. Trimble
reviewed by: Malcolm T. McGregor
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4040237