bibkey: bariczmehrez2018redheffer
authors: Á. Baricz and K. Mehrez
year: 2018
title: Redheffer type bounds for Bessel and modified Bessel functions of the first kind
doi: 10.1007/s00010-018-0545-8
url: https://arxiv.org/abs/1701.08446
claim: "Conjecture 1 in section 3 asserts the lower bound in inequality (3.2) for the normalized Bessel ratio before its first positive zero."
strata_touched:
  - D5/S3/Analytic/BesselRedheffer/Defs
  - D5/S3/Analytic/BesselRedheffer/Result
license: citation-only
triage: anchor

The normalized Bessel Redheffer lower bound

Verified locator

DOI: 10.1007/s00010-018-0545-8

Source: https://arxiv.org/abs/1701.08446

Pinned version: https://arxiv.org/abs/1701.08446v1, §3, Conjecture 1, inequality (3.2). Journal publication: Aequationes Mathematicae 92 (2018), 425–439.

Mathematical correspondence

The normalized function is

$${\mathcal{J}}_{\nu }(x)=\sum _{k\ge 0}\frac{(-x^2/4{)}^k}{k!(\nu +1{)}_k},\nu >-1.$$

Here $(\nu +1{)}_k$ is the ascending Pochhammer symbol and $j_{\nu ,1}$ is the first positive zero. For $|x|<j_{\nu ,1}$ the proposed lower bound is

$${\left(\frac{j_{\nu ,1}^2}{j_{\nu ,1}^2-x^2}\right)}^{{\vartheta }_{\nu }}\le \frac{{\mathcal{J}}_{\nu +1}(x)}{{\mathcal{J}}_{\nu }(x)},{\vartheta }_{\nu }=\frac{j_{\nu ,1}^2}{4(\nu +1)(\nu +2)}.$$

The source treats the left-hand inequality as the conjecture. The right-hand inequality in (3.2), with exponent ${\eta }_{\nu }=1$, is a known result. The source proves the conjectured lower bound for $\nu =-1/2$.

Defs.nbessel is exactly the displayed series; Defs.firstZero is the infimum of its positive zeros; Defs.claim is the universally quantified lower bound with the real power. Result.result refutes this claim. Result.failure_family supplies a strict counterexample for every real $\nu \ge 30$. Expansion.failure_of_energy_bound, together with the first-zero estimates, gives failure at every sufficiently small positive argument in this range. First-zero existence and attainment are proved here for $\nu \ge 30$; no such formal result for the remaining orders is asserted by these modules.

The bounded literature scope and its limitations are stated in Problems/baricz-mehrez-2018-bessel-redheffer-lower-bound.md. Exhaustive literature completeness and priority are ASSUMED-UNVERIFIED.