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The Kraus inequality for multivalent functions

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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.

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Authors and Affiliations

  1. Far-Eastern Federal University, Vladivostok, Russia

    V. N. Dubinin

  2. Institute of Applied Mathematics, Far-Eastern Branch, Russian Academy of Sciences, Vladivostok, Russia

    V. N. Dubinin

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  1. V. N. Dubinin
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Correspondence to V. N. Dubinin.

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Original Russian Text © V. N. Dubinin, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 4, pp. 559–564.

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Dubinin, V.N. The Kraus inequality for multivalent functions. Math Notes 102, 516–520 (2017). https://doi.org/10.1134/S0001434617090231

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  • DOI: https://doi.org/10.1134/S0001434617090231

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