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.
The paper gives a full-range affirmative answer to an open question posed by Gaunt in 2019. It shows that an exponentially tilted integral of a modified Bessel function against a reciprocal power remains controlled by the value of a similar expression evaluated only at the upper limit x. The control holds with an explicit constant for every positive x and throughout the interval 0
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
These are editorial extensions of the paper, not claims the
author makes directly.
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
A structured set of objections, weighed in public.
Desk editor's note, referee report, simulated authors' rebuttal, and a
circularity audit. Tearing a paper down is the easy half of reading it; the
pith above is the substance, this is the friction.