# 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.

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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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