Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions

We give an affirmative full-range solution to Gaunt’s 2019 Open Problem~2.10. The problem asks whether, for every $\nu >-1\mathrm{/}2$ and $0<\gamma <1$, the reciprocal-power integral ${\int }_0^x{e}^{-\gamma t}{I}_{\nu }(t)t^{-\nu }\text{dd}t$ 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 $(0,\mathrm{\infty })$, then for every $\theta \in (\gamma ,1)$, ${\int }_0^x{e}^{-\gamma t}w(t)t^{-\mu }{I}_{\mu }(t)\text{dd}t$ is controlled by an explicit multiple of $e^{-\gamma x}w(x)x^{-\mu }{I}_{\mu +1}(x)$. The case $w\equiv 1$, $q=0$, 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.