Geodesic
The Kraus Inequality for Multivalent Functions
Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 559-564

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For a holomorphic function $f,f'(0)\ne 0$, in the unit disk $U$, we establish a geometric constraint on the image $f(U)$ for which the classical Kraus inequality $|S_{f}(0)|\le 6$ holds; earlier, it was known only in the case of the conformal mapping of $f$. Here $S_{f}(0)$ is the Schwarzian derivative of the function $f$ calculated at the point $z=0$. The proof is based on the strengthened version of Lavrentev's theorem on the extremal decomposition of the Riemann sphere into two disjoint domains.
Keywords: Schwarzian derivative, holomorphic function, condenser capacity. 
@article{MZM_2017_102_4_a7,
     author = {V. N. Dubinin},
     title = {The {Kraus} {Inequality} for {Multivalent} {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {559--564},
     publisher = {mathdoc},
     volume = {102},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a7/}
}
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V. N. Dubinin. The Kraus Inequality for Multivalent Functions. Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 559-564. http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a7/