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The main goal of this work is to show that if two weighted homogeneous (butnot homogeneous) function-germs $(\C^2,0)\to(\C,0)$ are bi-Lipschitzequivalent, in the sense that these function-germs can be included in astrongly bi-Lipschitz trivial family of weighted homogeneous function-germs,then they are analytically equivalent.

Rigidity of bi-Lipschitz equivalence of weighted homogeneous function-germs in the plane