Naveen Kumari

Radii for sections of functions convex in one direction

Let $\mathcal{G}(α)$ 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 \[\Re \left(1+ \dfrac{z f''(z)}{ f'(z)}\right)<1+\dfracα{2} , \quad z\in \mathbb D.\] We determine the disks $|z|<ρ_n$ in which sections $ s_n(z; f)$ of $ f(z)$ are convex, starlike, and close-to-convex of order $β\;(0\le β< 1)$. Further, we obtain certain inequalities of sections in the considered class of functions.