On monotone and convex approximation by algebraic polynomials

The following results are obtained: If >0, 2, [3, 4], andf is a nondecreasing (convex) function on [–1, 1] such thatE_n (f) n^– for any n>, then E_n⁽¹⁾(f)Cn^– (E_n⁽²⁾(f)Cn^–) for n>, where C=C(), E_n(f) is the best uniform approximation of a continuous function by polynomials of degree (n–1), and E_n⁽¹⁾(f) (E_n⁽²⁾(f)) are the best monotone and convex approximations, respectively. For =2 ( [3, 4]), this result is not true.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1266–1270, September, 1994.