Geometric properties of the Cesàro means
type: Item
instance of: scholarly article
title: Geometric properties of the Cesàro means
zbMATH Open document ID: 0768.30011
DOI: 10.1007/BF03323120
author: Stephan Ruscheweyh
published in: Results in Mathematics
publication date: 1993-04-01T00:00:00Z
review text: Set \(S_ \gamma^*\) denote the class of functions \(f(z)=z+\dots\) analytic in the unit disk \(\mathbb{D}\) such that \(\text{Re}[zf'(z)/f(z)]>\gamma\). The author shows that, if \(\alpha\geq 1\), \[ {{n+\alpha-1} \choose {n-1}}^{-1} \sum_{k=1}^ n {{n+\alpha-k} \choose {n-k}} {{2\gamma-2} \choose {k-1}}z^ k \in S_ \gamma^* \] for \(\gamma=(3-\alpha)/2\) but not for any smaller \(\gamma\). This result implies that, for any function \(f(z)=z+\dots\) analytic in \(\mathbb{D}\), \[ f\in S_{(3-\alpha)/2}^* \iff \forall n:\quad S_ n^ \alpha (z,f)\in S_{(3-\alpha)/2}^* \] where \(S_ n^ \alpha(z,f)\) is the Cesàro mean. It is conjectured that \[ \sigma_ n^ \alpha (z)={{n+\alpha} \choose n}^{-1} \sum_{k=0}^ n {{n+\alpha-k} \choose {n-k}}z^ k \] satisfies the subordination relation \(\sigma_ n^ \alpha(z)\prec \sigma_ n^ \beta (z)\) for \(2\leq\beta\leq\alpha\).
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 147382
zbMATH Keywords: Cesàro means
zbMATH Keywords: starlike
zbMATH Keywords: Hadamard convolution
zbMATH Keywords: subordination
reviewed by: Christian Pommerenke
MaRDI profile type: Publication
cites work: Jacobi Polynomial Sums and Univalent Cesaro Means
cites work: Applications of a Convolution Theorem to Jacobi Polynomials
cites work: Power series with multiply monotonic coefficients
cites work: Linear operators between classes of prestarlike functions
cites work: Hadamard products of schlicht functions and the Polya-Schoenberg conjecture
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description: scientific article; zbMATH DE number 147382