On the size of the coefficients of rational functions approximating powers. II (Q1813649)

[For part I see ibid. 53, No. 3, 365-366 (1988; Zbl 0669.41012).] In this short paper, the author gives a lower bound for the coefficients of the numerator and denominator polynomials in rational functions approximating \(f(x)=x^ \alpha\) on [0,1]. For example, let \(1/2\leq\alpha<1\), and let \(\{a_ j\}\), \(\{b_ j\}\) be complex numbers with \(a_ 0=0\) or \(| a_ 0|\geq 1\) and where \(| b_ 0|\geq 1\), such that \[ \max_{[0,1]}\left| x^ \alpha-\sum^ n_{k=0}a_ kx^ k/\sum^ n_{k=0} b_ kx^ k\right|<\epsilon, \] where \(0<2\epsilon\leq 4^{-\alpha}\leq 2^{- 1}\). Then \[ \max\{| a_ 1|,| a_ 2|,\dots,| a_ n|,| b_ 0|,| b_ 1|,\dots,| b_ n|\}>4\epsilon/[7(2\epsilon)^{1/\alpha}]. \]