Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions
Pith reviewed 2026-05-21 01:58 UTC · model grok-4.3
The pith
The integral from 0 to x of e^{-gamma t} I_nu(t) t^{-nu} dt is bounded uniformly by C times e^{-gamma x} I_{nu+1}(x) x^{-nu} for all x>0 when nu>-1/2 and 0<gamma<1.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For every nu > -1/2 and 0 < gamma < 1 the integral from 0 to x of e^{-gamma t} I_nu(t) t^{-nu} dt is bounded above by an explicit constant multiple of e^{-gamma x} I_{nu+1}(x) x^{-nu}, uniformly in x > 0. More generally, when mu > -1, q > -1, 0 < gamma < 1, theta in (gamma,1), and w(x) x^{-q} is nondecreasing, the integral from 0 to x of e^{-gamma t} w(t) t^{-mu} I_mu(t) dt is controlled by an explicit multiple of e^{-gamma x} w(x) x^{-mu} I_{mu+1}(x). The case w identically 1, q=0, mu=nu recovers the solution to Gaunt's problem.
What carries the argument
The weighted uniform endpoint majorant theorem that bounds the integral by its endpoint expression whenever the scaled weight w(x) x^{-q} is nondecreasing.
If this is right
- The same comparison yields two-sided bounds once the weight comparison is reversed.
- Shifted-order and moment estimates for the integrals follow by choosing appropriate powers and weights.
- The result applies directly to approximate power weights and to monotone regularly varying amplitudes.
- The sharp power-weighted quotient can be analyzed through endpoint expansions, a stationary differential equation, and monotonicity in the parameters.
Where Pith is reading between the lines
- The explicit constant supplied by the proof could be used to obtain rigorous a-priori error bounds in numerical quadrature routines that evaluate Bessel integrals.
- The monotonicity condition on the scaled weight suggests a natural way to extend the majorant technique to other special functions that obey similar recurrence or differential relations.
- Because the bound is uniform down to x=0 and out to infinity, it supplies a convenient comparison function for studying the decay rates of related Laplace transforms or generating functions.
Load-bearing premise
The assumption that w(x) x^{-q} is nondecreasing on the whole positive half-line for the general weighted theorem.
What would settle it
Numerical evaluation of the ratio between the left-hand integral and the proposed right-hand majorant, for gamma approaching 1 from below and nu slightly larger than -1/2, to check whether the ratio stays below the claimed explicit constant for both very small and very large x.
read the original abstract
We give an affirmative full-range solution to Gaunt's 2019 Open Problem~2.10. The problem asks whether, for every \(\nu>-1/2\) and \(0<\gamma<1\), the reciprocal-power integral \(\int_0^x e^{-\gamma t}I_\nu(t)t^{-\nu}\,\dd t\) is bounded by a constant multiple of \(e^{-\gamma x}I_{\nu+1}(x)x^{-\nu}\), uniformly for all \(x>0\). Earlier exponential-tilt estimates proved such endpoint majorants only under an additional smallness condition on \(\gamma\). We prove the estimate throughout the natural range \(0<\gamma<1\), with an explicit admissible constant. More generally, if \(\mu>-1\), \(q>-1\), \(0<\gamma<1\), and \(w(x)x^{-q}\) is nondecreasing on \((0,\infty)\), then for every \(\theta\in(\gamma,1)\), \(\int_0^x e^{-\gamma t}w(t)t^{-\mu}I_\mu(t)\,\dd t\) is controlled by an explicit multiple of \(e^{-\gamma x}w(x)x^{-\mu}I_{\mu+1}(x)\). The case \(w\equiv1\), \(q=0\), and \(\mu=\nu\) resolves Gaunt's problem. The weighted theorem also yields shifted-order and moment estimates, applies to approximate power weights and monotone regularly varying amplitudes, and provides two-sided estimates under a reversed comparison. We further analyze the sharp power-weighted quotient via endpoint expansions, a stationary equation, and parameter monotonicity.
Editorial analysis