Growth results for a subclass of Bazilevich functions
type: Item
instance of: scholarly article
title: Growth results for a subclass of Bazilevich functions
zbMATH Open document ID: 0588.30016
DOI: 10.1155/S0161171285000862
author: R. M. El-Ashwah
author: Derek K. Thomas
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46041
review text: Let S be the class of regular, normalized, univalent functions with power series expansion \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) for \(z\in D\), where \(D=\{z:\) \(| z| <1\}\). Let B(\(\alpha)\) denote the subclass of S whose functions are Bazilevich of type \(\alpha\). Choosing the associated starlike function \(g(z)=z\) gives the proper subclass \(B_ 1(\alpha)\) of B(\(\alpha)\). For B(\(\alpha)\), correct growth estimates in terms of the area function A(r) are unknown. The authors give several results in this direction for \(B_ 1()\). For example, they prove that if \(f\in B_ 1()\) then \[ n| a_ n| \leq o(1)+O(1)\sqrt{A(1- \frac{1}{n})}\quad as\quad n\to \infty \quad and\quad L(r)=O(1)\{\sqrt{A(r)\log \frac{1}{1-r}}\}\quad as\quad r\to 1. \] For f regular in \(D_ 1\) define for \(\lambda\) real, \[ I_{\lambda}(r,f)=\int^{2\pi}_{0}| f(re^{i\theta})|^{\lambda} d\theta. \] Let \(M(r)=\max_{| z| =r}| f(re^{i\theta})|\). The authors prove that if \(f\in B_ 1()\) and \(\lambda >1\) then \[ I_{\lambda}(r^ 2,zf')\leq C(\lambda)\int^{r}_{0}\frac{M(p)^{\lambda /2}}{(1-p)^{\lambda}}dp \] where C(\(\lambda)\) is a constant depending only on \(\lambda\). Finally, the authors prove that if \(f\in B_ 1()\) then for \(0\leq r\leq 1\), \(I_ 1(r^ 2,f)\leq I_ 1(r^ 2,f_ 0)\) where \[ f_ 0(z^ 2)=(\int^{z}_{0}\frac{1+t^ 2}{1-t^ 2}dt)^ 2. \]
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 3944213
zbMATH Keywords: close-to-convex functions
zbMATH Keywords: length-area estimates
zbMATH Keywords: Bazilevich functions
zbMATH Keywords: starlike function
reviewed by: David J. Hallenbeck
MaRDI profile type: Publication
OpenAlex ID: W2125658430
description: scientific article; zbMATH DE number 3944213