Laurent coefficients of the Cauchy transform of the Hausdorff measure of the Sierpinski carpet

Abstract Let $$F_\rho $$ be the Cauchy transform of the self-similar measure $$\mu =\frac{1}{8}\sum _{j=0}^{7}\mu \circ S_j$$ , where $$S_j(z)=z_j+\rho (z-z_j)$$ with $$ z_{{2l}} = e^{{\frac{{2l}}{4}\pi i}} , $$ $$ z_{{2l + 1}} = \frac{{\sqrt 2 }}{2}e^{{\frac{{2l + 1}}{4}\pi i}} , $$ $$ l = 0,1,2,3, $$ $$ \rho \in (0,\frac{1}{3}) $$ , and K be the attractor of $$\{S_j\}_{j=0}^{7}$$ . In this paper, we derive asymptotic formulas for the Laurent coefficients $$\{a_{4k+1}\}_{k=1}^{\infty }$$ of $$F_\rho $$ in $$|z|>1$$ , and precisely determine the growth rate and the set of accumulation points for these Laurent coefficients. In addition, we give a lower bound for the domain of the starlikeness of $$F_\rho $$ .