Sufficient and necessary conditions for the generalized distribution series to be in subclasses of univalent functions

W. Y.  Kota; R. M.  El-Ashwah

doi:10.3842/umzh.v75i10.7267

W. Y. Kota Department of Mathematics, Faculty of Science, Damietta University New Damietta, Egypt
R. M. El-Ashwah Department of Mathematics, Faculty of Science, Damietta University New Damietta, Egypt

DOI: https://doi.org/10.3842/umzh.v75i10.7267

Keywords: Generalized distribution; univalent functions; analytic functions; Hadamard product.

Abstract

UDC 517.5

We establish a relationship between the subclasses of univalent functions and generalized distribution series. The main aim of our investigation is to obtain necessary and sufficient conditions for the generalized distribution series to belong to the classes ${\mathcal{TF}}(\rho,\vartheta),$ $\mathcal{{TH}}(\rho,\vartheta), \,\mathcal{{TJ}}(\rho,\vartheta)$, and $\mathcal{{TX}}(\rho,\vartheta)$. In addition, we obtain some particular cases of our main results.

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