Converse theorems of convexity for Bernstein polynomials over triangles (Q1813783)

The authors consider the \(n\)th Bernstein polynomial \(B_ n(f;p)\) of a real function \(f(p)\) whose domain is a given triangle \(T\). They show that if the function \(f(p)\) is continuous on \(T\) and one of the inequalities \(B_ n(f;p)\geq f(p)\) or \(B_ n(f;p)>B_{n+1}(f;p)\) holds of all positive integers \(n\) and for all points in \(T\), then the function \(f\) cannot have a strict local maximum at an interior point of \(T\).