An isoperimetric deficit upper bound of the convex domain in a surface of constant curvature

In this paper, we give a reverse analog of the Bonnesen-style inequality of a convex domain in the surface $ mathbb{X} $ _∈ of constant curvature ∈, that is, an isoperimetric deficit upper bound of the convex domain in $ mathbb{X} $ _∈ . The result is an analogue of the known Bottema’s result of 1933 in the Euclidean plane $ mathbb{E} $ ².