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arXiv · 2406.13328

Radii for sections of functions convex in one direction

Prachi Prajna Dash·Jugal Kishore Prajapat·Naveen Kumari
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Abstract

Let $\mathcal{G}(\alpha)$ denote the family of functions $ f(z)$ in the open unit disk $\mathbb D :=\{z\in\mathbb{C}: |z|<1\}$ that satisfy $ f(0)=0= f'(0)=1$ and \[\Re \left(1+ \dfrac{z f''(z)}{ f'(z)}\right)<1+\dfrac{\alpha}{2} , \quad z\in \mathbb D.\] We determine the disks $|z|<\rho_n$ in which sections $ s_n(z; f)$ of $ f(z)$ are convex, starlike, and close-to-convex of order $\beta\;(0\le \beta< 1)$. Further, we obtain certain inequalities of sections in the considered class of functions.

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Prachi Prajna Dash, Jugal Kishore Prajapat, Naveen Kumari. 2024-06-19. Radii for sections of functions convex in one direction. https://arxiv.org/abs/2406.13328

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