A note on rational approximation to \((1-x)^{\sigma}\) (Q1820943)

The author proves three theorems. The first one reads as follows: For \(0<\sigma \leq 1\), \(0\leq x\leq 1\) and \(n\geq 0\) \[ 0\leq S_ n^{-1}(x)-(1- x)^{\sigma}\leq x^{n+1}/S_ n(x)\leq \left( \begin{matrix} \sigma +n\\ n\end{matrix} \right)^{-1} \] holds, where \[ S_ n(x)=\sum^{n}_{j=0}\left( \begin{matrix} \sigma +j-1\\ j\end{matrix} \right)x^ j. \] In the other two theorems \(\| (1-x)^{\sigma}- (P(x)/Q(x))\|_{L^{\infty}[0,1]}\) is estimated from below under the assumptions that P and Q, or only P have only real nonnegative coefficients.