For nu>0, let J_nu denote the Bessel function of the first kind, let j_nu,n be its nth positive zero, and define the normalized Turán expression by Phi_nu(x)=fracJ_nu(x)^2-J_nu-1(x)J_nu+1(x)J_nu(x)^2. For every n≥ 1, the branch of Phi_nu on (j_nu,n,j_nu,n+1) has a local minimum at some alpha_nu,n∈(j_nu,n,j_nu,n+1). If beta_nu,n=Phi_nu(alpha_nu,n) denotes the corresponding minimum value, then beta_nu,n beta_nu_2,n.

Baricz’s conjecture on Bessel-function Turán inequalities

For $\nu>0$ , let $J_\nu$ denote the Bessel function of the first kind, let $j_{\nu,n}$ be its $n$ th positive zero, and define the normalized Turán expression by $\Phi_\nu(x)=\frac{J_\nu(x)^2-J_{\nu-1}(x)J_{\nu+1}(x)}{J_\nu(x)^2}$ . For every $n\geq 1$ , the branch of $\Phi_\nu$ on $(j_{\nu,n},j_{\nu,n+1})$ has a local minimum at some $\alpha_{\nu,n}\in(j_{\nu,n},j_{\nu,n+1})$ . If $\beta_{\nu,n}=\Phi_\nu(\alpha_{\nu,n})$ denotes the corresponding minimum value, then $\beta_{\nu,n}<\beta_{\nu,n+1}$ for every $n\geq1$ , the sequence $(\beta_{\nu,n})_{n\geq1}$ converges, and for every fixed $n$ , $\beta_{\nu,n}$ is strictly decreasing as a function of $\nu$ : if $0<\nu_1<\nu_2$ , then $\beta_{\nu_1,n}>\beta_{\nu_2,n}$ .

References

Primary source

arXiv

Additional references

Turán type inequalities for oscillatory special functions — arXiv — Ibrahim Aktaş, Árpád Baricz

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed solved

An unrefereed preprint claims to prove Baricz’s conjecture, but the result has not received independent mathematical verification.

Baricz’s conjecture concerns successive local minima of a normalized Bessel-function Turán expression and predicts that their values increase from one zero interval to the next.

Known results

A 2011 paper formulated the conjecture for $\nu>0$ , with critical points $\alpha_{\nu,n}\in(j_{\nu,n},j_{\nu,n+1})$ and strictly increasing values $\beta_{\nu,n}=\Phi_\nu(\alpha_{\nu,n})$ , based on numerical evidence.

September 2026 preprint

Ibrahim Aktaş and Árpád Baricz claim to prove the conjectured behavior and monotonicity of the local minima, while extending the method to other oscillatory special-function families. The claim is supported only by an unrefereed preprint.

Current status (as of October 2026): The conjecture is claimed solved by Aktaş and Baricz, but the proof remains independently unverified.

Sources

No solutions have been posted yet.