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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.20983 |
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| _version_ | 1866914608331096064 |
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| author | Yang, Yaoran Zhang, Yutong |
| author_facet | Yang, Yaoran Zhang, Yutong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions Yang, Yaoran Zhang, Yutong Classical Analysis and ODEs 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. |
| title | Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2605.20983 |