A Note on the Reverse Isoperimetric Inequality

In this paper, we derive an improved sharp version of a reverse isoperimetric inequality for convex planar curves of Pan and Zhang (Beitr?ge Algebra Geom 48:303?C308, 2007), with a simpler Fourier series proof. Moreover our result also confirm a conjecture by Pan et?al. (J Math Inequal (preprint), 2010). Furthermore we also present a stability property of our reverse isoperimetric inequality (near equality implies curve nearly circular).