The Kraus inequality for multivalent functions

Abstract

For a holomorphic function f, f′(0) ≠ 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)| ≤ 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 Lavrent’ev’s theorem on the extremal decomposition of the Riemann sphere into two disjoint domains.