Convolution conditions for bounded \(\alpha\)-starlike functions of complex order

A. Y. Lashin

Abstract


Let \(A\) be the class of analytic functions in the unit disc \(U\) of the complex plane \(\mathbb{C}\) with the normalization \(f(0)=f^{^{\prime }}(0)-1=0\). We introduce a subclass \(S_{M}^{\ast }(\alpha ,b)\) of \(A\), which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class \(S_{M}^{\ast }(n,\alpha ,b)\) (\(n\geq 0\)) related to \(S_{M}^{\ast }(\alpha ,b)\) is also considered under the same conditions. Among other things, we find convolution conditions for a function \(f\in A\) to belong to the class \(S_{M}^{\ast }(\alpha ,b)\). Several properties of the class \(S_{M}^{\ast }(n,\alpha ,b)\) are investigated.

Keywords


Univalent functions; bounded starlike functions of complex order; bounded convex functions of complex order; \(\alpha\)-starlike functions

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References


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DOI: http://dx.doi.org/10.17951/a.2017.71.1.65
Data publikacji: 2017-06-30 17:33:56
Data złożenia artykułu: 2017-06-30 13:09:40

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