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

Several open inequalities for ratios and logarithmic derivatives of the modified Bessel functions $I_\nu$ of the first kind and $K_\nu$ 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 $W_\nu(z)=zI_\nu(z)/I_{\nu+1}(z)$, the canonical product for $I_{\nu+1}$ yields the partial fraction $W_\nu(\sqrt{s})=2(\nu+1)+2\sum_{n\ge1}s/(s+j_{\nu+1,n}^2)$, where $j_{\nu+1,n}$ is the $n$-th positive zero of $J_{\nu+1}$.

Consequently $x\mapsto W_\nu(x^\tau)$ is a Bernstein function for $\nu>-1$ and $0<\tau\le1/2$, and this positive exponent range is sharp.

Second, an exact rational certificate at $(\nu,u)=(0,10)$ places $I_1(10)/I_0(10)$ below 0.949.

This refutes the log-concavity question of Baricz, Ponnusamy, and Vuorinen for $u\mapsto \sqrt{u} I_\nu(u)$ and its displayed Riccati reformulations.

The same framework completes the monotonicity classification of $K_\nu'/K_\nu^2$, refutes Baricz--Ponnusamy--Vuorinen Question 7 at $\nu=1/2$, and gives an entire counterexample to Baricz's coefficient-ratio complete-monotonicity transfer problem.