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The sharp constant in the Mashreghi-Ransford inequality
Abstract
Let (an)n≥0 be a sequence of complex numbers, and define bn=∑k=0n binomnk ak, cn=∑k=0n binomnk(−1)n−kak. Let β>1, put α=√(β2−1), and suppose that bn,cn=O(βn). Mashreghi and Ransford proved that limsupn→∞|an|/αn ≤ κ(limsupn→∞|bn|/βn) 1/2 (limsupn→∞|cn|/βn) 1/2 with a universal constant satisfying 2/√3 ≤ κ≤2. We prove that the optimal constant is indeed κ=2/√3. The proof, which uses exponential generating functions, Phragmén-Lindelöf estimates, and Cauchy's formula, is compared to the classical one of Mashreghi and Ransford.
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BibTeX
@misc{bouthat2026sharp,
title = {{The sharp constant in the Mashreghi-Ransford inequality}},
author = {Ludovick Bouthat},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08852},
eprint = {2609.08852},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08852},
}
RIS
TY - UNPB
TI - The sharp constant in the Mashreghi-Ransford inequality
AU - Ludovick Bouthat
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08852
UR - https://arxiv.org/abs/2609.08852
AB - Let (a_n)_n≥0 be a sequence of complex numbers, and define b_n=∑_k=0^n binomnk a_k, c_n=∑_k=0^n binomnk(−1)^n−ka_k. Let β>1, put α=√(β^2−1), and suppose that b_n,c_n=O(β^n). Mashreghi and Ransford proved that limsup_n→∞|a_n|/α^n ≤ κ(limsup_n→∞|b_n|/β^n)^ 1/2 (limsup_n→∞|c_n|/β^n)^ 1/2 with a universal constant satisfying 2/√3 ≤ κ≤2. We prove that the optimal constant is indeed κ=2/√3. The proof, which uses exponential generating functions, Phragmén-Lindelöf estimates, and Cauchy's formula, is compared to the classical one of Mashreghi and Ransford.
ER -