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

V. Dubinin

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arXiv · 1803.04619

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

V. Dubinin

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Abstract

For a meromorphic function $f$ in the unit disk $U=\{z:\;|z|<1\}$ and arbitrary points $z_1,z_2$ in $U$ distinct from the poles of $f$, a sharp upper bound on the product $|f'(z_1)f'(z_2)|$ is established. Further, we prove a sharp distortion theorem involving the derivatives $f'(z_1)$, $f'(z_2)$ 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 $f(U)$.

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V. Dubinin. 2018-03-13. Two-point distortion theorems and the Schwarzian derivatives of meromorphic functions. https://arxiv.org/abs/1803.04619

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