Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, let a ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the point a. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that |f(z) - R_n(z)| ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ₁⁺/n^(z) and ρ₁^-/n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on {Mathematical expression}.