Convexity and Variation Diminishing Property of Bernstein Polynomials over Triangles

Let Bn(f;P) denote the Bernstein polynomials over triangle T and \({\hat f_n}\) denote the Bezier net associated with Bn(f;P). A certain type of variations of \({\hat f_n}\) is introduced by GOODMAN quite recently. In the present paper the corresponding variation of Bn(f;P) is defined by integration of the absolute value of the Laplacian of BP(f;P) over T. It is shown that the variation of \({\hat f_n}\) is always greater or equal to the variation of Bn(f;P). The equality holds if and only if \({\hat f_n}\) is either convex (or concave) over T. The convexity of \({\hat f_n}\) implies the convexity of Bn(f;P). As an application we receive a simple proof of a theorem due to Chang and Davis.