×

On neighbourhoods of univalent starlike functions. (English) Zbl 0631.30008

Let A denote the class of analytic functions f in \(E=\{z:| z| <1\}\) and normalized \(f(0)=f'(0)-1=0\). For \(f(z)=z+a_ 2z^ 2+...\in A\) we define the \(\delta\)-neighbourhood of f as \[ N_{\delta}(f)=\{g(z)=z+b_ 2z^ 2+...\in A: \sum^{\infty}_{k=2}k| a_ k-b_ k| \leq \delta \}. \] Let \(B\subset A\). For \(\delta >0\) we define \[ B^{1,\delta}=\{f\in B: (f(z)+\epsilon z)/(1+\epsilon)\in B,\quad | \epsilon | <\delta \}, \]
\[ B^{n,\delta}=\{f\in B: f(z)+\epsilon z^ n\in B,\quad | \epsilon | <\delta \},\quad n=2,3,.... \] In this paper some various properties of \(N_{\delta}(f)\), \(B^{n,\delta}\) are studied. In particular some problems raised by Ruscheweyh, Rahman and Stankiewicz are settled. For example it is proved that:
Theorem 1. Let \(\delta >0\). Then \(f\in S^{*2,\delta}\Rightarrow\) \(N_{\delta}(f)\subset S^*.\)
Theorem 2. Let \(n>2\). Then there is a function \(f\in A\) and a real number \(\delta\), \(0<\delta <1\), such that \(f\in S^{*n,\delta}\) and \(N_{\delta}(f)\not\subset S^*\). Here \(S^*\subset A\) denotes the class of starlike functions.
Reviewer: J.Stankiewicz

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C55 General theory of univalent and multivalent functions of one complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
PDFBibTeX XMLCite
Full Text: DOI