PreprintCC BY 4.0arXiv
A functional inequality related to Domar's uniform boundedness theorem
Abstract
We study the functional inequality f(r+s)≤ g(r)+αf(s) (r,s>0). Here g:(0,∞)→[0,∞) is a given decreasing function, α is a constant such that 0<α<1, and the problem is to determine whether the family of decreasing functions f:(0,∞)→[0,∞) that satisfy this inequality is bounded above by some finite function on (0,∞) and, if so, to find bounds for this function. We present a solution to this problem, and use it to give a new proof of a theorem of Domar on the uniform boundedness of certain families of subharmonic functions, in addition obtaining explicit bounds.
Related
- Uniform Kreiss boundedness does not imply strong Cesàro boundedness
- Dynamical uniform boundedness for unicritical polynomials
- Global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent qr=(p−1)(s+1)
- Uniform L∞-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
- Uniform-in-N Log-Sobolev Inequality for the Mean-Field Langevin Dynamics with Convex Energy
BibTeX
@misc{ransford2026functional,
title = {{A functional inequality related to Domar's uniform boundedness theorem}},
author = {Thomas Ransford},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08718},
eprint = {2607.08718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08718},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A functional inequality related to Domar's uniform boundedness theorem
AU - Thomas Ransford
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2607.08718
UR - https://arxiv.org/abs/2607.08718
AB - We study the functional inequality f(r+s)≤ g(r)+αf(s) (r,s>0). Here g:(0,∞)→[0,∞) is a given decreasing function, α is a constant such that 0<α<1, and the problem is to determine whether the family of decreasing functions f:(0,∞)→[0,∞) that satisfy this inequality is bounded above by some finite function on (0,∞) and, if so, to find bounds for this function. We present a solution to this problem, and use it to give a new proof of a theorem of Domar on the uniform boundedness of certain families of subharmonic functions, in addition obtaining explicit bounds.
ER -