Convex approximation by rational functions

In this paper, the convex approximation to $| x |$ and to convex functions with continuous derivatives are investigated. In the first case, the approximation order $c_1 e^{ - c_2 \sqrt n } $ is achieved by using $H^\infty $ quadrature. In the second case, the estimate $| {f(x) - R_n (x)} | \leq C\frac{1}{{n^{2 - \epsilon } }}$ is proved, where $\epsilon $ is any positive real number.