On the Minkowski deficit for surfaces of revolution

RESUMO

By studying the classical Minkowski's inequality, $4\pi A\leq M^{2}$, relating area and mean curvature, in the particular case of convex surfaces of revolution, one is led to a functional inequality that holds for general functions. We state and prove this inequality obtaining a new proof of Minkowski's inequality in this particular case and we establish a sharp inequality between the Minkowski deficit of the surface and the isoperimetric deficit of any meridian curve.