CLASSES OF FUNCTIONS WITH IMPROVED ESTIMATES IN APPROXIMATION BY THE MAX-PRODUCT BERNSTEIN OPERATOR

In this paper, we find large classes of positive functions, others than those in [1], having even a Jackson-type estimate, omega(1)(f; 1/n), in approximation by the nonlinear max-product Bernstein operator. The uniform estimate of the order O[n omega(1)(f; 1/n)(2) + omega(1)( f; 1/n)] is achieved, while near to the endpoints 0 and 1, the better pointwise estimate of the order omega(1)(f, root x(1 - x)/n) is obtained. Finally, we prove that besides the preservation of quasi-convexity found in [1], the nonlinear max-product Bernstein operator preserves the quasi-concavity too.