Article ID: MTJPAM-D-21-00040

Title: Geometric properties of a family of univalent functions whose derivative lies in a half-plane


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00040; Volume 4 / Issue 1 / Year 2022, Pages 120-127

Document Type: Research Paper

Author(s): Faroze Ahmad Malik a , Chitaranjan Sharma b

aDepartment of Mathematics, Government Holkar (Model, Autonomous) Science College, Indore (M.P.)-452017, India

bDepartment of Mathematics, Government Holkar (Model, Autonomous) Science College, Indore (M.P.)-452017, India

Received: 5 June 2021, Accepted: 15 November 2021, Published: 11 January 2022.

Corresponding Author: Faroze Ahmad Malik (Email address: malikferooze@gmail.com)

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Abstract

Let \mathcal{A} consist of analytic functions f defined in the open unit disk \mathbb{U} satisfying f(0)=0 and f'(0)=1. For 0 ≤ β < 1, let \mathcal{R}(\beta) be the family of functions defined as

\mathcal{R}(\beta):=\left\{f\in\mathcal{A}:\mathrm{Re}\left(f'(\zeta)\right)>\beta,\; \zeta\in\mathbb{U}\right\}.

In this paper, we determine sharp estimates on \left|\frac{\zeta f'(\zeta)}{f(\zeta)}\right| for f\in\mathcal{R}(\beta) and find out the arc-length of the boundary curve \partial(f(\mathbb{U})). Further, we study the inclusion properties of the sequences of partial sums of f(\zeta)=\zeta+\sum_{n=2}^\infty a_n\zeta^n\in\mathcal{R}(\beta).

Keywords: Starlike and convex functions, close-to-convex functions, extremal problems, subordination, arc-length problem

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