Riccati Reductions for Modified Bessel Ratios: Bernstein Positivity, Exact Certificates, and Transfer Obstructions

Abstract

Several open inequalities for ratios and logarithmic derivatives of the modified Bessel functions $I_ν$ of the first kind and $K_ν$ of the second kind reduce to sign questions for quadratic Riccati expressions. We isolate this reduction and use it in two directions. First, for the quotient $W_ν(z)=zI_ν(z)/I_{ν+1}(z)$, the canonical product for $I_{ν+1}$ yields the partial fraction $W_ν(\sqrt{s})=2(ν+1)+2\sum_{n\ge1}s/(s+j_{ν+1,n}^2)$, where $j_{ν+1,n}$ is the $n$-th positive zero of $J_{ν+1}$. Consequently $x\mapsto W_ν(x^τ)$ is a Bernstein function for $ν>-1$ and $0<τ\le1/2$, and this positive exponent range is sharp. Second, an exact rational certificate at $(ν,u)=(0,10)$ places $I_1(10)/I_0(10)$ below 0.949. This refutes the log-concavity question of Baricz, Ponnusamy, and Vuorinen for $u\mapsto \sqrt{u} I_ν(u)$ and its displayed Riccati reformulations. The same framework completes the monotonicity classification of $K_ν'/K_ν^2$, refutes Baricz--Ponnusamy--Vuorinen Question 7 at $ν=1/2$, and gives an entire counterexample to Baricz's coefficient-ratio complete-monotonicity transfer problem.

Disclosure

“ceived no specific fund- ing for this work. No empirical data were used. Every numerical claim reduces to finite exact rational arithmetic, reproducible from the formulas in Appendix Appendix A. During manuscript preparation, the author used ChatGPT and Codex to support literature triage, organization, language re- vision, and consistency checks; the author reviewed the content, verified the mathematical claims and cited sources, and takes full responsibility for the manuscript. Refer”

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2026-08-24T00:34:09.244Z

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