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)=P(z)/((1−z/b)s), b>0, °P≤ 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≥6 we construct explicitly a rational function with rational coefficients and exact radius 2s+(4(s−3))/(s2−3s+4)>2s. For third-order approximation, we construct, for every s≥5, an explicit rational family with exact radius 2s−1>s−1+√(s2−1). These constructions disprove the conjectures of Lóczi and Ketcheson concerning rational functions with a single repeated pole.
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BibTeX
@misc{aithaddou2026maximal,
title = {{Maximal Absolute Monotonicity of Balanced Rational Functions with a Single Repeated Pole}},
author = {Rachid Ait-Haddou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06941},
eprint = {2611.06941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06941},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximal Absolute Monotonicity of Balanced Rational Functions with a Single Repeated Pole
AU - Rachid Ait-Haddou
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.06941
UR - https://arxiv.org/abs/2611.06941
AB - We study the radius of absolute monotonicity of rational approximations to the exponential of the form ϕ(z)=P(z)/((1−z/b)^s), b>0, °P≤ 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≥6 we construct explicitly a rational function with rational coefficients and exact radius 2s+(4(s−3))/(s^2−3s+4)>2s. For third-order approximation, we construct, for every s≥5, an explicit rational family with exact radius 2s−1>s−1+√(s^2−1). These constructions disprove the conjectures of Lóczi and Ketcheson concerning rational functions with a single repeated pole.
ER -