The authors prove that for any pair \(C\) and \(\widetilde{C}\) of germs of irreducible complex analytic curves in \((\mathbb C^2, 0)\) with different sequences of characteristic exponents, there exists \(\alpha\in(0,1)\) such that \(C\) and \(\widetilde{C}\) are not \(\alpha\)-Hölder homeomorphic. Similar statements, concerning bi-\(\alpha\)-Hölder and bi-Lipschitz equivalences, are valid for curves with two branches. The authors also remark that their results generalize Pham-Teissier’s theorem from [F. Pham and B. Teissier, Fractions Lipschitziennes d’une algebre analytique complexe et saturation de Zariski. Prépublications Ecole Polytechnique No. M17.0669 (1969), http://hal.archives-ouvertes.fr/hal-00384928/fr/], as well as its other versions studied earlier by the first author (see [A. Fernandes, Mich. Math. J. 51, No. 3, 593–606 (2003; Zbl 1055.14028)]).