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

Let F_ρ be the Cauchy transform of the self-similar measure μ =1/8∑ ⱼ₌₀⁷μ ∘ Sⱼ , where Sⱼ(z)=zⱼ+ρ (z-zⱼ) with z_2l = e^2l4π i , z_2l + 1 = √ 2 2e^2l + 14π i , $$ l = 0,1,2,3, $$ ρ ∈ (0,1/3) , and K be the attractor of ⱼ\ⱼ₌₀⁷ . In this paper, we derive asymptotic formulas for the Laurent coefficients ₄ₖ₊₁\ₖ₌₁∞ of F_ρ 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_ρ .