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 $$W_ν(\sqrt{s})=2(ν+1)+2\sum_{n\ge1}s/(s+j_{ν+1,n}<sup>2)$$, where $j_{ν+1,n}$ is the $n$-th positive zero of $J_{ν+1}$. Consequently $x\mapsto W_ν(x<sup>τ)$ is a Bernstein function for $ν&gt;-1$ and $0&lt;τ\le1/2$, 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 $K_ν&#39;/K_ν<sup>2$, refutes Baricz--Ponnusamy--Vuorinen Question 7 at $ν=1/2$, and gives an entire counterexample to Baricz's coefficient-ratio complete-monotonicity transfer problem.

• The paper uses Riccati reduction methods to transform complex Bessel ratio inequalities into quadratic sign conditions, resolving longstanding conjectures.
• It employs canonical product expansions and explicit rational certificates to establish Bernstein positivity for specific Bessel function transforms.
• The study provides counterexamples that refute general monotonicity and transfer conjectures, precisely delineating parameter regimes.

Riccati Reductions for Modified Bessel Ratios: Summary and Analysis

Overview and Principal Contributions

This work addresses a set of open monotonicity and positivity problems concerning ratios and logarithmic derivatives of modified Bessel functions $I_\nu$ and $K_\nu$. By systematically reducing these analytic questions to sign conditions on quadratic Riccati-type expressions, the author creates a uniform framework for both establishing and refuting a series of conjectures, particularly those collected by Baricz, Ponnusamy, and Vuorinen (BPV), and Yang and Tian. Key techniques include a partial fraction/canonical product expansion over Bessel zeros, explicit rational certificates for pointwise counterexamples, and sharp endpoint and transfer analyses.

The study settles or refutes several prominent conjectures and questions:

• Affirmatively, the conjectured Bernstein positivity and precise exponent range for functions of the form $x \mapsto W_\nu(x^\tau)$, where $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$.
• Negatively, the strict log-concavity of $u \mapsto \sqrt{u} I_\nu(u)$ for all $\nu$, BPV's monotonicity questions about $K_\nu'/K_\nu^2$ outside specific parameter regimes, and Baricz's coefficient-ratio transfer conjecture for complete monotonicity.
• Completes the monotonicity classification for $K_\nu'/K_\nu^2$.

Numerical bounds and explicit rational constructions play a central role, clarifying the delicate boundary between true and false parameter regimes.

Core Methods and Reductions

Riccati Reductions

For a large class of analytic inequalities concerning Bessel function ratios or their logarithmic derivatives, the author demonstrates that applying the governing ODE in logarithmic derivative (Riccati) form reduces each functional inequality to a quadratic sign condition in the logarithmic derivative variable.

For instance, strict log-concavity and convexity problems for $f(u) = \sqrt{u} I_\nu(u)$ reduce via Lemma 2.3 (Riccati $I_\nu$ form) to questions of the sign of

$K_\nu$0

where $K_\nu$1. An equivalent restatement in terms of the ratio $K_\nu$2 offers alternative algebraic perspectives. This explicit reduction streamlines the verification of both positive and negative results: entire classes of inequalities can be resolved by computing the sign of these quadratic forms at a single parameter value.

Partial Fraction Expansion and Bernstein Class

For the function $K_\nu$3, the author leverages a canonical product representation of $K_\nu$4 to establish a partial fraction expansion in terms of the positive zeros $K_\nu$5 of $K_\nu$6,

$K_\nu$7

This positive superposition ensures that $K_\nu$8 is a Bernstein function for $K_\nu$9, and that the exponent range is sharp; for exponents $x \mapsto W_\nu(x^\tau)$0, the Bernstein property is lost due to the structure of the resulting derivatives at $x \mapsto W_\nu(x^\tau)$1. Composition-closure properties of the Bernstein function class are invoked to extend the result from $x \mapsto W_\nu(x^\tau)$2 to $x \mapsto W_\nu(x^\tau)$3.

Rational Certificates and Counterexamples

Negative results are established by constructing explicit rational bounds (certificates) at single points which contradict the required sign condition for the Riccati quadratic. For example, at $x \mapsto W_\nu(x^\tau)$4, it is demonstrated (without numerical approximation) that $x \mapsto W_\nu(x^\tau)$5, which leads to the failure of the log-concavity, Riccati, and lower-bound quadratic inequalities for $x \mapsto W_\nu(x^\tau)$6 at that parameter set. An explicit series truncation with rational estimates is provided.

Endpoint and Half-Order Classification

Monotonicity properties of $x \mapsto W_\nu(x^\tau)$7 are completely classified by analyzing the endpoint sign of the associated Riccati quadratic as $x \mapsto W_\nu(x^\tau)$8 and $x \mapsto W_\nu(x^\tau)$9. Explicit half-order closed forms allow for transparent sign analysis, especially at $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$0, where all critical values can be written in closed form as functions of square roots.

Coefficient-Ratio Transfer Obstruction

The paper constructs analytic counterexamples showing that even strictly completely monotone coefficient ratio sequences do not guarantee that the quotient of their generating functions is completely monotone. By explicit power series expansion and recursive monotonicity arguments, it locates the precise obstruction in the second Taylor coefficient, thereby decisively refuting Baricz's transfer conjecture.

Key Numerical and Structural Results

• Bernstein function regime: For all $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$1, $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$2 is Bernstein if and only if $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$3.
• Log-concavity failure: There exists an explicit neighborhood around $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$4 where $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$5 is not log-concave; the Riccati quadratic is strictly positive at this point.
• Monotonicity of $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$6: Strictly decreasing if and only if $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$7. For $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$8, the function increases near $W_\nu(z) = zI_\nu(z)/I_{\nu+1}(z)$9 and decreases for large $u \mapsto \sqrt{u} I_\nu(u)$0 (nonmonotone). At $u \mapsto \sqrt{u} I_\nu(u)$1, the unique critical point is $u \mapsto \sqrt{u} I_\nu(u)$2 (maximum).
• Coefficient-ratio transfer: Explicit entire functions $u \mapsto \sqrt{u} I_\nu(u)$3 are constructed with strictly completely monotone ratio sequences $u \mapsto \sqrt{u} I_\nu(u)$4 such that $u \mapsto \sqrt{u} I_\nu(u)$5 fails to be completely monotone on any interval $u \mapsto \sqrt{u} I_\nu(u)$6.

Implications and Future Directions

The systematic Riccati reduction methodology enables the rapid resolution of a wide class of inequalities in Bessel analysis, and suggests a general program for attacking ratio and derivative monotonicity questions elsewhere in special function theory. The use of explicit rational certificates illustrates a path to truly rigorous computer-assisted (or fully constructive) proofs without reliance on floating-point numerics, which is crucial near delicate regime boundaries.

Practical implications extend to statistical distribution theory, stochastic processes, and mathematical physics, wherever modified Bessel functions encode key transition kernels, normalization constants, or condition numbers—especially given the probabilistic interpretation of Bernstein functions as Laplace exponents of subordinators.

The transfer obstruction result highlights the necessity of structure beyond sequence positivity—global analytic properties, such as canonical product representations, cannot be bypassed for monotonicity transfer via power series alone.

Potential avenues for further research include:

• Classification of the precise log-concavity regime for $u \mapsto \sqrt{u} I_\nu(u)$7 in $u \mapsto \sqrt{u} I_\nu(u)$8-space by mapping the zero set of the Riccati quadratic.
• Extension of endpoint classification to additional Bessel function ratios and related Turán-type inequalities.
• Establishing necessary and sufficient conditions under which completely monotone coefficient sequences induce completely monotone generating function quotients.

Conclusion

Through a unified approach employing Riccati reduction, canonical product expansions, and explicit certificates, this work resolves a sequence of open inequalities for Bessel function ratios and their derivatives, precisely delimiting the scope of validity for several prominent conjectures and exposing obstructions to monotonicity transfer. The techniques exemplify an effective paradigm for both theory-driven and computer-assisted special function analysis, blending sharp functional analysis with concrete arithmetic verification.