Radii for sections of functions convex in one direction

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.