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 $K_ν&#39;/K_ν^2$, refutes Baricz--Ponnusamy--Vuorinen Question 7 at $ν=1/2$, and gives an entire counterexample to Baricz&#39;s coefficient-ratio complete-monotonicity transfer problem.