The paper has a methodical content and is addressed to young researchers. Its main goal is to prove how the property of monotonicity can be transferred from the sequence of univariate Bernstein polynomials to those of bivariate Bernstein polynomials.
Let be the set of positive integers, , , , , continuous on . Denote by the Bernstein bivariate operator. This operator associates to each function the bivariate Bernstein polynomial . It is well known that the sequence converges to , uniformly on for each .
In the present paper one investigates the monotonicity of the sequence . One proves that if is convex of order on the sequence is monotonous decreasing and , .