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 $\nu>-1/2$ 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.