Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions

TL;DR

This paper solves a longstanding open problem by establishing uniform bounds for integrals involving modified Bessel functions across the full parameter range, with explicit constants and generalizations.

Contribution

It provides a complete solution to Gaunt's 2019 open problem, extending bounds to the entire natural parameter range with explicit constants and broader weighted estimates.

Findings

Established uniform bounds for integrals involving modified Bessel functions for all \\( u>-1/2\") and \\(\\gamma\") in (0,1).

Generalized bounds to weighted integrals with monotone weights and shifted orders.

Derived explicit constants and analyzed the quotient behavior via endpoint expansions and monotonicity.

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.