Directional convexity of level lines for functions convex in a given direction

Let $K(\varphi )$ be the class of functions $f(z)=z+a_{2}z^{2}+\dots$ which are holomorphic and convex in direction $e^{i\varphi }$ in the unit disk $D$, i.e. the domain $f(D)$ is such that the intersection of $f(D)$ and any straight line $\{w:w=w_{0}+te^{i\varphi },t\in \mathbb{R}\}$ is a connected or empty set. In this note we determine the radius $r_{\psi ,\varphi }$ of the biggest disk $\vert z\vert\leq r_{\psi ,\varphi }$ with the property that each function $f\in K(\psi )$ maps this disk onto the convex domain in the direction $e^{i\varphi }$.