On S.L. Tabachnikov's conjecture

S. L. Tabachnikov’s conjecture is proved: for any closed curve $\mathrm{\Gamma }$ lying inside convex closed curve ${\mathrm{\Gamma }}_1$ the mean absolute curvature $T(\mathrm{\Gamma })$ exceeds $T({\mathrm{\Gamma }}_1)$ if $\mathrm{\Gamma }\mathrm{\ne }k{\mathrm{\Gamma }}_1$. An inequality $T(\Gamma)\ge T(\Gamma_1)$ is proved for curves in a hemisphere.