Maximal Absolute Monotonicity of Balanced Rational Functions with a Single Repeated Pole

Abstract

We study the radius of absolute monotonicity of rational approximations to the exponential of the form \[ ϕ(z)=\frac{P(z)}{(1-z/b)^s}, \qquad b>0,\qquad °P\le s, \] with approximation orders two and three. For second-order approximation, we prove that the maximal radius in degree $s=5$ is $10$ and determine the unique extremizer. Hence the conjectured value $2s$ by Lóczi and Ketcheson (LMS Journal of Computation and Mathematics, 17 (2014), no.~1, 159--205) remains valid in this case. In contrast, for every integer $s\ge6$ we construct explicitly a rational function with rational coefficients and exact radius \[ 2s+\frac{4(s-3)}{s^2-3s+4}>2s. \] For third-order approximation, we construct, for every $s\ge5$, an explicit rational family with exact radius \[ 2s-1>s-1+\sqrt{s^2-1}. \] These constructions disprove the conjectures of Lóczi and Ketcheson concerning rational functions with a single repeated pole.