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.