On the order of approximation of convex functions by rational functions

We show that for arbitrary convex functions the order of approximation (in the metric $C[a,b]) by rational functions of degree no higher than $n$ does not exceed the quantity $C\cdot M\cdot\frac{\ln^2n}n$ ($C$ an absolute constant, $M$ the maximum modulus of the convex function). We prove also the existence of a~convex function whose order of approximation is greater than $\frac1{n\ln^2n}$.