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 \(F_\rho \) .
