A sharp reverse Bonnesen-style inequality and generalization

We investigate the isoperimetric deficit of the oval domain in the Euclidean plane. Via the kinematic formulae of Poincaré and Blaschke, and Blaschke’s rolling theorem, we obtain a sharp reverse Bonnesen-style inequality for a plane oval domain, which improves Bottema’s result. Furthermore, we extend the isoperimetric deficit to the symmetric mixed isoperimetric deficit for two plane oval domains, and we obtain two reverse Bonnesen-style symmetric mixed inequalities, which are generalizations of Bottema’s result and its strengthened form.