Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions

Abstract

We give an affirmative full-range solution to Gaunt's 2019 Open Problem~2.10. The problem asks whether, for every \(ν>-1/2\) and \(0<γ<1\), the reciprocal-power integral \(\int_0^x e^{-γt}I_ν(t)t^{-ν}\,\dd t\) is bounded by a constant multiple of \(e^{-γx}I_{ν+1}(x)x^{-ν}\), uniformly for all \(x>0\). Earlier exponential-tilt estimates proved such endpoint majorants only under an additional smallness condition on \(γ\). We prove the estimate throughout the natural range \(0<γ<1\), with an explicit admissible constant. More generally, if \(μ>-1\), \(q>-1\), \(0<γ<1\), and \(w(x)x^{-q}\) is nondecreasing on \((0,\infty)\), then for every \(θ\in(γ,1)\), \(\int_0^x e^{-γt}w(t)t^{-μ}I_μ(t)\,\dd t\) is controlled by an explicit multiple of \(e^{-γx}w(x)x^{-μ}I_{μ+1}(x)\). The case \(w\equiv1\), \(q=0\), and \(μ=ν\) 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.