bi-Lipschitz versus Analytic equivalence of two variable complex quasihomogeneous function-germs

In this paper we address the problem of classifying complex (non-homogeneous) quasihomogeneous polynomials in two variables under bi-Lipschitz equivalence. We prove that pairs of such polynomials are (right) bi-Lipschitz equivalent as function-germs at $0\in {\mathbb{C}}^2$ iff they are analytically equivalent.