Univalent functions. A primer
type: Item
instance of: scholarly article
title: Univalent functions. A primer
zbMATH Open document ID: 1397.30002
DOI: 10.1515/9783110560961
author: Derek K. Thomas
author: Nikola Tuneski
author: A. Vasudevarao
published in: De Gruyter Studies in Mathematics
publication date: 2018-02-19T00:00:00Z
review text: The authors mainly concentrate on different subclasses of univalent functions \(f(z)=z+\sum_{n=2}^{\infty}a_nz^n\) in the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) and primarily address the book to researchers who are new to the field of geometric function theory. They describe the ideas and techniques used in solving extremal problems and update a variety of results on classes of conformal mappings. Some of complicated proofs are omitted. The contents include 16 chapters. The first chapter gives an elementary introduction to the area of univalent functions, the Bieberbach conjecture and growth and distortion theorems. Star-like, convex, close-to-convex and other functions are defined in Chapter 2. All these function sets relate to analytic functions \(p(z)=1+\sum_{n=1}^{\infty}p_nz^n\) with positive real part in \(\mathbb D\) that are described in Chapter 3. Also, the concept of subordination is presented in Chapter 3 together with Littlewood's subordination theorem, Rogosinski's domination results, theorems by Mocanu and Nunokawa and the useful Clunie-Jack lemma. The two largest Chapters 4 and 9 are devoted to the basic classes of star-like and convex functions and close-to-convex functions, respectively. The coefficient theorems of Chapter 4 give estimates for \(|a_n|\), \(|a_3|-|a_2|\), \(||a_{n+1}|-|a_n||\), \(|a_3-\mu a_2^2|\) with \(\mu\in\mathbb C\), and the Hankel second-order and third-order determinants in the class \(S^*\) of star-like functions. Refined growth theorems give estimates for \(\text{Re}\{zf'(z)/f(z)\}\) and \(|zf'(z)/f(z)|\) dependent on \(|f(z)|\). The same is done for the class \(C\) of convex functions. In addition, some subclasses of \(S^*\) are studied in a similar manner. An analytic function \(f\) in \(\mathbb D\) is said to be star-like of order \(\alpha\) or convex of order \(\alpha\), \(0\leq\alpha<1\), if \[ \text{Re}\frac{zf'(z)}{f(z)}>\alpha\quad \text{or}\quad\text{Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>\alpha,\;\;\;z\i…
Mathematics Subject Classification ID: 30-01
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 6840584
zbMATH Keywords: univalent function
zbMATH Keywords: star-like function
zbMATH Keywords: convex function
zbMATH Keywords: close-to-convex function
reviewed by: Dmitri V. Prokhorov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110560961
OpenAlex ID: W4254218790
description: scientific article; zbMATH DE number 6840584