Studies On Ros’ Inequality In Euclidean Plane R~2

In this thesis, we study the Ros’Inequality in Euclidean Plane R2. The main content is made up of the parts:Chapter three and Chapter four.In Chapter three, we consider the functional version of the Ros’ Inequality in R2 and its stability problem. Firstly, by the theorems of Fourier seriers, we construct a functional version of Ros’inequality whose geometric equivalent is strengthening Ros’inequality in R2. Secondly, we consider the stability problem for the functional version of Ros inequality and then get the stability properties of the geometric Ros inequality.In Chapter four, we study the integral of the mixed curvature radii in R2. First and foremost, by defining the integral of the mixed curvature radii, we obtain Minkowski style inequality which extents the Ros’Inequality in plane. Besides, we also get Brunn-Minkowski style inequality for the integral of the mixed curvature radii. Last but not least, we conclude the relation for the integrals of the mixed curvature radii of oval domain and its difference body.