Two-point distortion theorems and the Schwarzian derivatives of meromorphic functions

For a meromorphic function $f$ in the unit disk $U=\{z:\mathrm{\mid }z\mathrm{\mid }<1\}$ and arbitrary points $z_1,z_2$ in $U$ distinct from the poles of $f$, a sharp upper bound on the product $\mathrm{\mid }{f}^{\mathrm{\prime }}(z_1)f^{\mathrm{\prime }}(z_2)\mathrm{\mid }$ is established. Further, we prove a sharp distortion theorem involving the derivatives $f^{\mathrm{\prime }}(z_1)$, $f^{\mathrm{\prime }}(z_2)$ and the Schwarzian derivatives $S_f(z_1)$, $S_f(z_2)$ for $z_1,z_2\in U$. Both estimates hold true under some geometric restrictions on the image $f(U)$.