Krzyz猜想的新认识
A new look at Krzyz's conjecture
摘要：
最近，作者提出了一种新的方法来解决几何函数论中的极值问题。它涉及Bers同构定理的Teichmuller空间的穿孔黎曼曲面。在这里，我们表明，这种方法，结合拟共形理论，也可以适用于非零全纯函数从$H^\infty$。特别是这给出了一个旧的开放Krzyz猜想证明这样的功能和其推广。将$H^\infty$的单位球$H_1^\infty$自然地嵌入到泛Teichmuller空间中，并将函数$f \in H_1^\infty$视为单位圆盘上单叶函数的Schwarzian导数.
英文摘要：
Recently the author has presented a new approach to solving extremal problems of geometric function theory. It involves the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. We show here that this approach, combined with quasiconformal theory, can be also applied to nonvanishing holomorphic functions from $H^\infty$. In particular this gives a proof of an old open Krzyz conjecture for such functions and of its generalizations. The unit ball $H_1^\infty$ of $H^\infty$ is naturally embedded into the universal Teichmuller space, and the functions $f \in H_1^\infty$ are regarded as the Schwarzian derivatives of univalent functions in the unit disk.
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