Inequalities for an integral involving the modified Bessel function of the first kind

2025/01/21 by Robert E. Gaunt, Gaunt, Robert E.
Mathematics · #26D15 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Primary 33C20

paper · pdf · doi:10.48550/arxiv.2501.12197

openalex publication_date 2025/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Simple bounds are obtained for the integral ∫₀^xe^-γtt^νI_ν(t) dt, x>0, ν>-1/2, 0≤γ<1, together with a natural generalisation of this integral. In particular, we obtain an upper bound that holds for all x>0, ν>-1/2, 0≤γ<1, is of the correct asymptotic order as x→0 and x→∞, and possesses a constant factor that is optimal for ν≥0 and close to optimal for ν>-1/2. We complement this upper bound with several other upper and lower bounds that are tight as x→0 or as x→∞, and apply our results to derive sharper bounds for some expressions that appear in Stein's method for variance-gamma approximation.

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