Schwarzian Derivative and Covering Arcs of a Pencil of Circles by Holomorphic Functions

Let $f$ be a holomorphic function in the disk $U=\{z:|z|<1\}$, $|f(z)|<1$ in $U$, let $f(\pm1)=\pm1$ in the sense of angular limits, and let the angular Schwarzian derivatives $S_{f}(\pm1)$ exist. We establish an upper bound for the sum $S_{f}(-1)+S_{f}(1)$ under the assumption that the image $f(U)$ does not contain open arcs of the pencil of circles $\arg[(1+w)/(1-w)]=\theta$, $-\pi/2<\theta<\varphi$, with endpoints $w=\pm1$ and
$$ \operatorname{Re} f''(1)+f'(1)(1-f'(1))=-\operatorname{Re} f''(-1)+f'(-1)(1-f'(-1))=0. $$
This bound depends on $\varphi$ and $f'(\pm1)$ only.