Sharp inequalities for the Padé approximant errors in the Stieltjes case (Q1178585)

Let \([m/n]\) be a Padé approximant to the nonrational Stieltjes function \(f=\int_ 0^{1/R}{d\mu(t) \over 1-tz}\). The authors improve the knowing inequalities between the contiguous errors, i.e. the differences \(f(x)-[m/n](x)\), by introducing a factor in \(x\), for example: \(0<f(x)- [n+1/n](x)<(x/R)\{f(x)-[n/n](x)\}\). This type of inequalities is obtained for all directions (rows, columns, diagonals) in the one half of the Padé table (i.e. for \(m\geq n\)).