bibkey: karpvishnyakovazhang2025unimodality
authors: D. Karp, A. Vishnyakova, and Y. Zhang
year: 2025
title: Unimodality preservation by ratios of functional series and integral transforms
doi: 10.1007/s00025-025-02431-4
url: https://arxiv.org/abs/2408.01755
claim: "Conjecture 1 asserts unimodality of modified Bessel ratios and log-concavity for positive ordered orders."
strata_touched:
  - D5/S3/Analytic/BesselRatioLogConcavity/Result
license: citation-only
triage: anchor

Modified Bessel ratios

Verified locator

https://arxiv.org/abs/2408.01755

https://arxiv.org/abs/2408.01755v2

https://doi.org/10.1007/s00025-025-02431-4

The supplied text is arXiv:2408.01755v2. Conjecture 1 is on printed page 12. Its statement, with line breaks joined, is:

Conjecture 1. Suppose b ≥ 0, ν1 ≥ ν2 > −1 and 0 < a1 ≤ a2. Then the ratio x → Iν1(a1 x)/Iν2(a2 x) is unimodal on (0, ∞). Moreover, if ν1 ≥ ν2 > 0 it is log-concave.

The parameter $b$ does not occur in the ratio. The log-concavity clause is encoded by concavity of the logarithm on the positive real half-line.

Standard series and mathematical correspondence

Printed page 11 states verbatim:

with a > 0, b ≥ 0, ν > −1, µ > 0 and Iν standing for the modified Bessel function of the first kind.

It then states:

Using the standard power series expansion of Iν [28, p.217] we have

The displayed ratio expansion on printed page 11 appears verbatim in the supplied text extraction as follows (line order and detached superscripts retained):

P∞ 2k 2k k
Iν1 (a1 x)
k=0 a1 x /[4 k!Γ(k + ν1 + 1)]
2 ν1 −ν2
P
.
= 2ν2 −ν2 aν11 a−ν
x
∞
2
2k 2k
k
Iν2 (a2 x)
k=0 a2 x /[4 k!Γ(k + ν2 + 1)]

The text extraction separates superscripts in the prefactor. The normalization correspondence below is derived from the series; it is not a verbatim quotation of a standalone formula in the paper. Factoring $(z/2{)}^{\nu }$ gives

$$I_{\nu }(z)=(z/2{)}^{\nu }\sum _{k=0}^{\infty }\frac{z^{2k}}{4^{k}k!\Gamma (k+\nu +1)}.$$

Thus the ratio has prefactor $2^{{\nu }_2-{\nu }_1}{a}_1^{{\nu }_1}{a}_2^{-{\nu }_2}{x}^{{\nu }_1-{\nu }_2}$. For equal orders and scales $1,2$ this is $2^{-\nu }$, and the common Gamma factor cancels after $\Gamma (\nu +1+k)=(\nu +1{)}_k\Gamma (\nu +1)$.

Result.besselI is the corresponding real infinite sum. Result.claim records exactly the positive-order log-concavity clause, and Result.result refutes it. The order is $\nu =1/10$ at both numerator and denominator, with scales $1,2$ and arguments $3/2,9/2,15/2$. The series converges for every positive argument and every $\nu >-1$ in the proof. Sixty-term partial sums and a geometric tail bound give a strict failure of midpoint log-concavity. The unimodality clause is a separate question.