The Kraus Inequality for Multivalent Functions

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.