<p>Recently the author presented a new approach to solving the coefficient problems for various classes of holomorphic functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(z) = \sum \limits _0^\infty c_n z^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mn>0</mn> <mi>∞</mi> </munderover> <msub> <mi>c</mi> <mi>n</mi> </msub> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, not necessarily univalent. This approach is based on lifting the given polynomial coefficient functionals <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$J(f) = J(c_{m_1}, \dots , c_{m_s}), \ 2< c_{m_1}< \dots< c_{m_s} < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>J</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <msub> <mi>m</mi> <mn>1</mn> </msub> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>c</mi> <msub> <mi>m</mi> <mi>s</mi> </msub> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mn>2</mn> <mo><</mo> <msub> <mi>c</mi> <msub> <mi>m</mi> <mn>1</mn> </msub> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>c</mi> <msub> <mi>m</mi> <mi>s</mi> </msub> </msub> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, onto the Bers fiber space over universal Teichmüller space and applying the analytic and geometric features of Teichmüller spaces, especially the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. In this paper, we extend this approach to more general classes of functions. In particular, this provides strengthening de Branges’ theorem solving the Bieberbach conjecture.</p>

A LINK BETWEEN COVERING AND COEFFICIENT THEOREMS FOR HOLOMORPHIC FUNCTIONS

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发表日期：

2025年11月1日

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Samuel L. Krushkal

摘要

Recently the author presented a new approach to solving the coefficient problems for various classes of holomorphic functions $f(z) = \sum \limits _0^\infty c_n z^n$ $f (z) = \sum_{0}^{\infty} c_{n} z^{n}$ , not necessarily univalent. This approach is based on lifting the given polynomial coefficient functionals $J(f) = J(c_{m_1}, \dots , c_{m_s}), \ 2< c_{m_1}< \dots< c_{m_s} < \infty $ $J (f) = J (c_{m_{1}}, \dots, c_{m_{s}}), 2 < c_{m_{1}} < \dots < c_{m_{s}} < \infty$ , onto the Bers fiber space over universal Teichmüller space and applying the analytic and geometric features of Teichmüller spaces, especially the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. In this paper, we extend this approach to more general classes of functions. In particular, this provides strengthening de Branges’ theorem solving the Bieberbach conjecture.