Boundary values of the Schwarzian derivative of a regular function

Regular functions $f$ in the half-plane $\operatorname{Im} z>0$ admitting an asymptotic expansion $f(z)=c_1z+c_2z^2+c_3z^3+\gamma(z)z^3$, where $c_1>0$, $\operatorname{Im} c_2=0$ and the angular limit $\angle\lim_{z\to0}\gamma(z)=0$, are considered. For various conditions on the function $f$ inequalities for the real part of the Schwarzian derivative $S_f(0)=6(c_3/c_1-c_2^2/c_1^2)$ are established. These inequalities complement and refine some known versions of Schwarz's lemma. The results obtained are close to the well-known Burns-Krantz rigidity theorem on regular self-maps and its generalizations due to Tauraso, Vlacci and Shoikhet.
Bibliography: 16 titles.