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 $f(0)=0=f^{\prime }(0)=1$ and \[\Re \left(1+ \dfrac{z f''(z)}{ f'(z)}\right)<1+\dfracα{2} , \quad z\in \mathbb D.\] 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.