Convergence of repeated-pole rational approximation of the exponential on infinite sectors

For $h>0$, degree $m\ge0$, $0\leβ0$, let $Σ_β=\{re^{iθ}:r\ge0,\ |θ|\leβ/2\}$ and let $E_{m,β}(q)$ denote the best uniform error for approximating $e^{-hz}$ on $Σ_β$ by rational functions whose only possible finite pole is $s=-q/h$, with multiplicity at most $m$. For the linear scaling $q