Radii for sections of functions convex in one direction

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

$$\begin{aligned} \Re \left( 1+ \dfrac{zf''(z)}{f'(z)}\right) <1+\dfrac{\alpha }{2} , \quad z\in {\mathbb {D}}. \end{aligned}$$

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.