×

Inclusion properties of certain subclass of analytic functions defined by multiplier transformations. (English) Zbl 1190.30012

Summary: Let \(A\) denote the class of analytic functions with the normalization \(f(0)=f'(0)-1=0\) in the open unit disk \(U=\{z:| z|<1\}\). Set
\[ f^m_{\lambda,l}(z)=z+\sum_{k=2}^\infty\Big[\frac{l+1+\lambda(k-1)}{l+1}\Big]^m z^k,\qquad z\in U,\; m\in \mathbb N_0,\; \lambda\geq 0,\; l\geq 0, \] and define \(f^m_{\lambda,l,\mu}\) in terms of the Hadamard product
\[ f^m_{\lambda,l}(z)*f^m_{\lambda,l,\mu}(z)=\frac{z}{(1-z)^\mu},\qquad z\in U,\; \mu>0. \]
In this paper, we introduce several new subclasses of analytic functions defined by means of the operator \(I^m_{\lambda,l,\mu}f(z)=f^m_{\lambda,l,\mu}f(z)*f(z)\), \(f\in A\), \(m\in \mathbb N_0\), \(\lambda\geq 0\), \(l\geq 0\), \(\mu>0\).
Inclusion properties of these classes and the classes involving a generalized Libera integral operator are also considered.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
PDFBibTeX XMLCite
Full Text: DOI Link