On rational approximations of functions with a convex derivative

Let $R_N[f]$ be the least uniform deviation of a continuous function $f(x)$ ($x\in[a,b]$) from the rational functions of degree not greater than $N$ ($N=2,3,\dots$).
Theorem. \textit{Suppose a function $f(x)$ is given on an interval $[a,b]$ $(-\infty<a<b<\infty)$ and is $p$ times differentiable $(p\geqslant1)$, its $p$th derivative being convex. Then
\begin{equation} R_N[f]\leqslant C_p(b-a)^pM_p\frac{\ln^3N}{N^{p+2}},\qquad N\geqslant2p, \end{equation}
where $C_p$ is a constant depending on $p$ and $M_p=\max\{|f^{(p)}(x)|\}$.}
The estimate is sharp for any $p=1,2,\dots$ and any modulus of continuity of the function $f^{(p)}$ if the factors of form $\ln^\gamma N$ are neglected.
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