Uniform close-to-convexity radius of sections of functions in the close-to-convex family

The authors consider the class $\F$ of normalized functions $f$ analytic in the unit disk $\ID$ and satisfying the condition $${\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2},\quad z\in\D. $$ Recently, Ponnusamy et al. \cite{samy-hiroshi-swadesh} have shown that $1/6$ is the uniform sharp bound for the radius of convexity of every section of each function in the class $\F$. They conjectured that $1/3$ is the uniform univalence radius of every section of $f\in \F$. In this paper, we solve this conjecture affirmatively.