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On order convolution consistence of the analytic functions. (English) Zbl 1240.30037

Suppose that \(\mathcal {X,Y}\) and \(\mathcal Z\) are subclasses of the space \(\mathcal H(\mathcal D)\) of analytic functions in the unit disc \(\mathcal D=\{z\in\mathbb C:| z| <1\}\).
In this paper, the authors consider the convolution \(f\ast g\) of analytic functions \(f\) and \(g\) belonging to the classes \(\mathcal X\) and \(\mathcal Y\), respectively. They discuss when this convolution \(f\ast g\) belongs to \(\mathcal Z\).
Specifically, given the classes \(\mathcal {X,Y}\) and \(\mathcal Z\), they define the Sălăgean number \(S(\mathcal {X,Y,Z})\) to be \[ \begin{aligned} S(\mathcal{X,Y,Z})&=\min\{s\in\mathbb R:T^s(f\ast g)\in\mathcal Z \text{ for all } f\in\mathcal X \text{ and all } g\in\mathcal Y\}\\ &=\min\{s\in\mathbb R:T^s(\mathcal X\ast\mathcal Y)\subseteq\mathcal Z\}. \end{aligned} \] Here, \(T^s\) is the Sălăgean operator defined by \(T^s(z^n)=\frac 1{n^s}z^n\) for a positive integer \(n\) and \(T^s(1)=1\). It follows that for each \(s\in\mathbb R\), \(T^s:\mathcal H(\mathcal D)\to \mathcal H(\mathcal D)\) is a linear operator, that is, \(T^s(f)=a_0+\sum_{n=1}^\infty \frac {a_n}{n^s} z^n\) for each \(f(z)=a_0+\sum_{n=1}^\infty a_n z^n\) analytic in the unit disc \(\mathcal D\).
The authors calculate the Sălăgean numbers for certain families. In some other cases, they determine the upper and lower bounds for Sălăgean numbers, provided they exist.

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
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