Laszkiewicz et al.'s convergence conjecture for the scalar Padé iteration

Let $p$ be a positive integer, let $P_{km}$ and $Q_{km}$ be the numerator and denominator polynomials of the Padé approximant $[k/m]$ for the relevant scalar function, and define

$$x_{l+1}=h_{km}(x_l):=x_l\frac{P_{km}(1-x_l^p)}{Q_{km}(1-x_l^p)},\qquad x_0=\lambda.$$

Let $L^{(\operatorname{Pad\acute{e}})}_p=\{z\in\mathbb{C}:|1-z^p|<1\}$.

Laszkiewicz et al.'s convergence conjecture. If $k\ge m-1$ and $x_0$ lies in $L^{(\operatorname{Pad\acute{e}})}_p$, then

$$|1-x_l^p|\le |1-x_0^p|^{(k+m+1)^l}.$$

This conjecture proposes a convergence estimate for the scalar Padé family of iterations associated with the matrix sector function and the computation of matrix $p$th roots. The case $k=0$, $m=1$ is known from the stated theorem, while the general range $k\ge m-1$ is presented as an open conjecture.