Geodesic
On the approximation of $x^\alpha$ by rational functions
Izvestiya. Mathematics , Tome 16 (1981) no. 1, pp. 83-101

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Weak equivalence type estimates are obtained for the smallest deviation of $x^\alpha$ ($\alpha$ a proper fraction) from the rational functions of degree not greater than $n=1,2,\dots$ in the metrics of $L_p[0,1]$ ($1\leqslant p\leqslant\infty$). Bibliography: 15 titles. 
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     author = {N. S. Vyacheslavov},
     title = {On the approximation of $x^\alpha$ by rational functions},
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N. S. Vyacheslavov. On the approximation of $x^\alpha$ by rational functions. Izvestiya. Mathematics , Tome 16 (1981) no. 1, pp. 83-101. http://geodesic.mathdoc.fr/item/IM2_1981_16_1_a4/