Approximation, by rational functions, of convex functions with given modulus of continuity

We denote by $R_n[f]$ the least deviation of the continuous function $f(x)$, $x\in[a,b]$, from the rational functions of order at most $n$.
We establish the following theorems.
Theorem 1. Let $f(x)$ be convex on $[a,b]$ $(-\infty<a<b<+\infty)$ with modulus of continuity $\omega(\delta,f)$. Then
$$ R_n[f]\leqslant c\frac{\ln^6n}{n^2}\max_{(b-a)e^{-n}\leqslant\theta\leqslant b-a}\biggl\{\omega(\theta)\ln\frac{b-a}{\theta}\biggr\},\qquad n=2,3,\dots, $$
where $c$ is an absolute constant.
\medskip Theorem 2. There exist a convex function $f^*(x)$ and a sequence $n_k\nearrow\infty$ such that 1) $\omega(\delta,f^*)\leqslant(\ln(e/\delta))^{-\gamma}$, $0<\delta\leqslant1$, and 2) $R_{n_k}[f^*]\geqslant c_1\gamma/n^{1-\gamma}_k$, where $c_1$ is an absolute constant.
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