Geodesic
On the order of approximation of convex functions by rational functions
Izvestiya. Mathematics , Tome 3 (1969) no. 5, pp. 1067-1080

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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 $\frac1{n\ln^2n}$. 
@article{IM2_1969_3_5_a7,
     author = {A. P. Bulanov},
     title = {On the order of approximation of convex functions by rational functions},
     journal = {Izvestiya. Mathematics },
     pages = {1067--1080},
     publisher = {mathdoc},
     volume = {3},
     number = {5},
     year = {1969},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1969_3_5_a7/}
}
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A. P. Bulanov. On the order of approximation of convex functions by rational functions. Izvestiya. Mathematics , Tome 3 (1969) no. 5, pp. 1067-1080. http://geodesic.mathdoc.fr/item/IM2_1969_3_5_a7/