Rational approximations to convex functions with given modulus of continuity

It is shown that for any convex continuous functions $f(x)$ ($x\in[a,b]$, $-\infty<a,b<\infty$) with modulus of continuity $\omega(\delta)$ the order of approximation by rational functions does not exceed
$$ C\cdot\frac{\ln^2n}n\cdot\inf_{0<\lambda<1}\biggl\{\omega(\lambda)+M\cdot\frac{\ln^2n}n\cdot\ln\frac{b-a}\lambda\biggr\}, $$
where $C$ is an absolute constant and $M=\max|f(x)|$.
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