One example in Monotone approximation

For each non-negative integer n a function f=f(n) is constructed such that f has a continuous and non-negative derivative f' on I:=[-1, 1] and 1/200 < e(n+1)((1))(f)/E(n)(f') < 2, where E(n)(f') (E(n+1)((1)) (f)) is the value of the best uniform approximation on I of the function f' (f) by arbitrary (monotone on I) algebraic polynomials of degree less than or equal to n (n + 1). (C) 1996 Academic Press, Inc.