Uniform Estimates of Monotone and Convex Approximation of Smooth Functions

We obtain uniform estimates for monotone and convex approximation of functions by algebraic polynomials in terms of the weighted Ditzian-Totik moduli of smoothness formula] where ?(x)? formula], for r ? 3 and r ? 5 in monotone and convex cases, respectively. Together with known results in the positive and negative directions for the other r this complements the investigation of the rate of shape preserving approximation in terms of ?k?(f(r), n?1)?r, ∞ in the sense of the orders of these moduli. It is also shown that some extra conditions on the smoothness of f allow direct results in the cases for which the general estimate in terms of ?k?(f(r), n?1)?r, ∞ is not correct.