Successive coefficients for Cauchy transforms of some Hausdorff measures

Let $F\mu$ be the Cauchy transform of the self-similar measure $\mu$ defined by $\mu \frac{1}{m} \sum^{m-1}_{j=0} \mu \circ S^{-1}_j$, where $S_j z = e^{2 \pi ij/m} + r(z - e^{2 \pi ij/m})$ with $0 \lt r \lt 1$. The Laurent coefficients ${\lbrace a_{nm+1} \rbrace}^\infty_{n=0}$ of $F_\mu$ in ${\lvert z \rvert} \gt 1$ were studied in $\href{https://doi.org/10.1016/S0022-1236(02)00069-1}{[DL1]}$ and $\href{https://doi.org/10.1080/10586458.1998.10504368}{[LSV]}$. In this paper, we study the asymptotic formulation of the difference of the successive coefficients. We prove that the set of accumulation points for ${\lbrace (nm)^{\alpha+1} ( a_{(n+1)m+1} - a_{nm+1} ) \rbrace}^\infty_{n=0}$ is exactly a non-degenerated line segment except the cases $m = 2,4$ with $r =\frac{1}{2}$, where $\alpha$ is the Hausdorff dimension of the support of $\mu$. As a corollary, the criterion for determining the monotonicity of ${\lbrace a_{nm+1} \rbrace}^\infty_{n=0}$ can be established.

Keywords

successive coefficients, asymptotic formula, Cauchy transform, Hausdorff measure, Taylor coefficients

2010 Mathematics Subject Classification

Primary 30C45. Secondary 28A25, 28A80.

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Dedicated to the memory of Professor Ka-Sing Lau

Received 6 August 2022

Accepted 3 August 2023

Published 10 July 2024