The exact order of the best approximation to convex functions by rational functions

We show that the least uniform rational deviations $R_n(f)$ from the function $f(x)$, continuous and convex on the interval $[a,b]$, satisfy the condition $R_n(f)=o(1/n)$ as $n\to\infty$, and that $R_n(f)=O(1/n)$ uniformly for the continuous convex functions $f$ whose absolute values are bounded by unity. These estimates are precise with respect to the rate of decrease of the right-hand sides.
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