International Journal of Open Problems in Computer Science and Mathematics

Sharp inequalities for sine and cosine on complex circles and applications

Mohamed Bouali — 2026-05-30

We investigate sharp inequalities for the functions | sin z| and | cos z| along circles in the complex plane. Building on recent results of Qi, we establish precise bounds for the quantities | sin(reiθ)| − | cos(reiθ)| and | sin(reiθ) − cos(reiθ)|.
We show that their behavior undergoes a phase transition governed by a unique critical parameter r0 defined by cos(2r0) =2r0. As an application, we completely resolve several open problems posed by Bagul and Chesneau concerning doublesided inequalities for trigonometric and hyperbolic functions.
In particular, we prove that
sin(kx)/kx+ ksin x/x> 1 + k cos x, holds if and only if k ∈ (0, 2), and that sinh(qx)/qx+ qsinh x/x> 1 + q cosh x
holds if and only if q ≥ 2. We further obtain weighted extensions of these inequalities. Our approach combines complexanalytic techniques with sharp real-variable inequalities and reveals new connections between classical inequalities and the geometry of analytic functions on circles.

Two new extensions of Hardy-Hilbert-type integral inequalities

Christophe Chesneau — 2026-05-23

In this paper, we investigate two new Hardy-Hilbert-type integral inequalities that incorporate both the maximum and the sum of the variables within the integrands. The second inequality is presented as a three-dimensional analogue of the first. Our proofs rely on the H\"older integral inequality and various integral identities. Finally, the paper concludes by posing a related open problem.

On Bishop-Phelps-Bollobas Property in Banach Spaces

Orina Moraa — 2026-05-23

In this paper, we establish Bishop-Phelps-Bollobas Property (BPBp) for finite rank operators (fro) between Banach spaces (BS). We prove that $BPBp$ for $fro$ holds in several settings including when a Banach space $X$ is of finite dimension or uniformly convex. We also extend these results and show that this property also holds on $BS$ with geometrical properties.
Moreover, we characterize the numerical radius (nr) of $fro$ via the $BPBp$. We establish the extent to which $fro$ satisfy $BPBp$ with respect to $nr$. We show that this property holds in $BS$ settings which include when $X$ is reflexive.

Solution to an Open Problem on Riemann Zeta Function

Kwara Nantomah — 2026-05-23

In this note, we provide two solutions to an open problem concerning the Riemann zeta function. The first solution relies on some properties of the Riemann zeta function whilst the second solution relies on the positivity of a certain function associated with the polygamma function.