2021/05/06 by Javier Segura, Segura, Javier · 3 citations
Mathematics · #26D07 #26D15 #33C10 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials #math.CA #msc:26D07 #msc:26D15 #msc:33C10
paper · pdf · doi:10.48550/arxiv.2105.02524
arxiv created 2021/05/06 · openalex publication_date 2021/05/06 · arxiv updated 2021/05/07 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
Let Iν(x) and Kν(x) be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by xα Φi,ν(x), i=1,2, with Φ1,ν=Iν-1(x)/Iν(x) and Φ2,ν(x)=-Kν-1(x)/Kν(x), are bounds for xα Φi,ν(x), which are solutions with unique monotonicity properties; these bounds hold at least for ± α∉ (0,1) and ν≥ 1/2. Properties for the product Pν(x)=Iν(x)Kν(x) can be obtained as a consequence; for instance, it is shown that Pν(x) is decreasing if ν≥ -1 (extending the known range of this result) and that xPν(x) is increasing for ν≥ 1/2. We also show that the double ratios Wi,ν(x)=Φi,ν+1(x)/Φi,ν(x) are monotonic and that these monotonicity properties are exclusive of the first and second kind modified Bessel functions. Sharp trigonometric bounds can be extracted from the monotonicity of the double ratios. The trigonometric bounds for the ratios and the product are very accurate as x→ 0+, x→ +∞ and ν→ +∞ in the sense that the first two terms in the power series expansions in these limits are exact.