PreprintCC BY 4.0arXiv
A sharp Mittag-Leffler bound for series with separated exponents
Abstract
Let η≥ 0 and δ>0, and let (rj)j≥ 0 be a sequence of real numbers satisfying η≤ r0 < r1 < r2 < ⋯ and rj+1−rj ≥ δ for every j≥ 0. We prove that, for δ≤ 1 and every x>0, ∑j=0∞ xrj/(G(1+rj)) ≤ ∑j=0∞ (xη+jδ)/(G(1+η+jδ)) = xηEδ,1+η(xδ), where Eδ,1+η denotes the two-parameter Mittag-Leffler function. Moreover, equality holds for some x>0 if and only if rj = η+ jδ for every j. The proof combines a finite compression argument for unimodal functions with a shift inequality for the Mittag-Leffler function. The upper bound δ≤ 1 is sharp, since counterexamples exist for every 1 < δ< 2.
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BibTeX
@misc{molteni2026sharp,
title = {{A sharp Mittag-Leffler bound for series with separated exponents}},
author = {Giuseppe Molteni},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04540},
eprint = {2611.04540},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04540},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A sharp Mittag-Leffler bound for series with separated exponents
AU - Giuseppe Molteni
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04540
UR - https://arxiv.org/abs/2611.04540
AB - Let η≥ 0 and δ>0, and let (r_j)_j≥ 0 be a sequence of real numbers satisfying η≤ r_0 < r_1 < r_2 < ⋯ and r_j+1−r_j ≥ δ for every j≥ 0. We prove that, for δ≤ 1 and every x>0, ∑_j=0^∞ x^r_j/(G(1+r_j)) ≤ ∑_j=0^∞ (x^η+jδ)/(G(1+η+jδ)) = x^ηE_δ,1+η(x^δ), where E_δ,1+η denotes the two-parameter Mittag-Leffler function. Moreover, equality holds for some x>0 if and only if r_j = η+ jδ for every j. The proof combines a finite compression argument for unimodal functions with a shift inequality for the Mittag-Leffler function. The upper bound δ≤ 1 is sharp, since counterexamples exist for every 1 < δ< 2.
ER -