Bonnesen Type Symmetric Mixed Inequality

Perhaps the classical isoperimetric inequality is the oldest geometric inequality. It relates area to perimeter of a Euclidean plane domain bounded by a simple closed curve. Bonnesen-style inequalities are strengthens of the isoperimetric inequality, that have closely connect with the first eigenvalue of the Laplacian operator, Wulff flows, Sobolev inequality and other applications through the development of Chern, Bonnesen, Hadwiger, Osserman, Santal′o, Ren, Zhou, Zhang, et al. Reverse Bonnesen-style inequalities have been concerned gradually too. The symmetric mixed isoperimetric inequality of two plane convex domains is one of the generalization of the isoperimetric inequality. Bonnesen-style symmetric mixed inequalities of two plane convex domains are strengthens of the symmetric mixed isoperimetric inequality. In this thesis, we investigate mainly Bonnesenstyle symmetric mixed inequalities and reverse Bonnesen-style symmetric mixed inequalities of two plane convex domains.In Chapter 3, we first deal with Bonnesen-style symmetric mixed inequalities of two plane convex domains. We estimate the symmetric mixed isoperimetric deficit ?2(K0, K1) of two plane convex domains K0 and K1via the known kinematic formulae of Blaschke and Poincar′e in integral geometry. We obtain some Bonnesen-style symmetric mixed inequalities, and prove the conditions of equality signs hold in these inequalities. These new Bonnesen-style symmetric mixed inequalities generalize results of Kotlyar and Bonnesen. Then, we investigate reverse Bonnesen-style symmetric mixed inequalities of two plane convex domains.We obtain some new reverse Bonnesen-style symmetric mixed inequalities of two plane oval domains via the kinematic formulae of Blaschke and Poincar′e, and Blaschke rolling theorem. Moreover, we also obtain some reverse Bonnesen-style symmetric mixed inequalities for any plane convex domains that generalize the known Bottema’s result. The last, we generalize Bol-Fujiwara theorem in R2,that is, we obtain generalized Bol-Fujiwara theorem of two plane oval domains.We also describe a application of Bonnesen-style symmetric mixed inequalities on the second kind of complete elliptic integral.In Chapter 4, we investigate the symmetric mixed isoperimetric inequality and Bonnesen-style symmetric mixed inequalities in surface of constant curvature.