Volterra-type operator on the subclasses of univalent functions

• Date: 2022
• Main Author: Mahboobi, M.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2022
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/1116
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.5 In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order $b$ $(b\neq 0)$ and spirallike functions of type $ \lambda$ $\Big( {-\dfrac{\pi}{2}}<\lambda<\dfrac{\pi}{2} \Big)$ with the complex order $b$ $(b\neq 0).$ We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions belonging to the mentioned classes. In the main part of this paper, we obtain the necessary and sufficient conditions of boundedness for the image of the open unit disk $ \mathbb{D}=\lbrace z\in \mathbb{C}:\vert z\vert<1\rbrace $ under the action of a Volterra-type operator and the product of the composition operator and Volterra-type operator in the space of univalent functions and its subspace. Finally, we obtain an estimate of the Schwartzian norm of the above operators in these spaces.