Radii for sections of functions convex in one direction

TL;DR

This paper investigates the radii within which sections of functions, convex in one direction, are guaranteed to be convex, starlike, or close-to-convex, providing precise bounds and inequalities for these properties.

Contribution

It determines the radii for sections of functions convex in one direction to be convex, starlike, or close-to-convex, and establishes related inequalities within this function class.

Findings

Identified radii for convexity, starlikeness, and close-to-convexity of sections.

Derived inequalities for sections of functions in the class.

Provided explicit bounds for the behavior of these functions' sections.

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.