Let $a, b, c$ be the side lengths of an acute triangle in the Euclidean plane, and let $\Delta$ denote its area. The classical inequality of Ono asserts that\[27 (b^{2}+c^{2}-a^{2})^{2}(c^{2}+a^{2}-b^{2})^{2}(a^{2}+b^{2}-c^{2})^{2} \le (4\Delta)^{6},\]with equality if and only if the triangle is equilateral. The traditional proof relies on trigonometric identities and the AM--GM inequality. In this paper, we provide a new analytic proof based on a Cartesian coordinate reduction and multivariable optimization techniques. By transforming the geometric inequality into a two-variable polynomial extremal problem, we establish the inequality using elementary tools from multivariable calculus. Furthermore, we derive a nontrivial two-parameter generalization valid for all acute triangles: for any $\alpha > 0$ and $\beta > 0$,\[(4\Delta)^{2\alpha+\beta} \ge C_{\alpha,\beta}(b^{2}+c^{2}-a^{2})^{\alpha}(c^{2}+a^{2}-b^{2})^{\alpha}(a^{2}+b^{2}-c^{2})^{\beta},\]where the optimal constant $C_{\alpha,\beta}$ is computed explicitly. The classical Ono inequality is recovered as the special case $\alpha=\beta=2$. We also present an algebraic reformulation of the result and discuss the geometric structure underlying the equality cases.

Performing a thought study of fractional inequalities by means of convexity and fractional operators has conspicuous work in the field of analysis. The main object of this paper is to discuss the coordinated convexity, pre-invexity, and also establish fractional double integral operators (FDIO) having generalized Bessel-Maitland function as used its kernel. We develop a new generation of Hermite-Hadamard (H-H) and trapezoid-type inequalities through different types of coordinated convexities and pre-invexities with successfully implementation of newly designed fractional double integral operators. Moreover, we extract some corollaries, which are generalization of well-known inequalities for different coordinated convexities that show a strong consolidation of our main results.

This study focuses on the analysis of $C$-class functions, with particular attention given to the development of fixed-point theorems for mappings that satisfy $H$-$(\psi, \phi)$-contractive conditions. The principal aim is to extend fixed-point results to the broader framework of $G_{\mathcal{F}}$-complete metric spaces. This generalized setting provides greater flexibility of contractive mappings, covering cases not addressed by traditional fixed-point theory.

In this article, the research examines whether Riemann-Liouville-type fractional derivatives can be used to solve an initial value problem under weak topology conditions. As a way to prove the existence of integrable description and a new type of iteration of a Leray–Schauder nonlinear alternative for the weak topology, it will be demonstrated, and the given problem is first transformed into the sum of two integral operators, and then the modified version of Krasnoselskii’s fixed point hypothesis in weak topology is employed. Finally, an example is given to illustrate the effectiveness of our main findings.

Lemma 3 in the paper [1] [Communications in Algebra, 49(6), 2722–2727, 2021] is revisited. In this paper, we provide an improvised rudimentary proof of the same.

We present the notion of $n$-anti-homoderivations, a novel class of additive mappings on rings. Within this framework, we establish strong necessary conditions for rings' commutativity and the presence of nontrivial central ideals. Our approach not only generalizes several classical results as special cases, but also addresses open problems concerning the interaction between such mappings and the structural properties of rings.

Conditional distribution estimation (CDE) is central in nonparametric forecasting and risk analysis. While considerable progress has been made for finite-dimensional and stationary settings, functional data and nonstationary settings pose new challenges. We propose a Nadaraya-Watson (NW) conditional quantile estimator for regularly mixing locally stationary functional time series (LSFTS). It incorporates three kernel functions: one for time rescaling, another for the functional covariates, and an integrated kernel to act as a cumulative distribution function (CDF) of the response variable. A theoretical framework and the uniform convergence of the estimator were provided. To demonstrate the consistency of the estimator, a numerical experiment was conducted. Finally, we apply the method to financial data, specifically the Nikkei 225.

Fuzzy group theory has evolved beyond single-valued memberships to account for dual polarity and uncertainty. Building on Dib’s fuzzy space and bipolar-valued fuzzy sets, we develop a unified algebraic theory of bipolar-valued fuzzy (BVF) subgroups, including BVF normal subgroups and BVF homomorphisms, via a BVF binary operation (BVFBO) on a BVF-space. We establish necessary and sufficient subgroup criteria, characterize normality through coset symmetry in BVF-space, and prove homomorphism properties that align BVF structures with their classical counterparts through correspondence theorems. The framework clarifies when associativity holds between subgroup elements and ambient BVF-group elements and provides constructive examples. This generalization resolves limitations tied to the absence of a bipolar fuzzy universal set and supports applications in polarity-sensitive decision systems and network analysis.

The main purpose of this study is to demonstrate that the fractional Hardy operators are bounded on the weighted variable central Morrey space. When the symbol functions belong to the λ-central BMO space with a variable exponent, the estimates for their commutators are similar.

Engineers and scientists adopt numerical integration to achieve an approximate solution for definite integrals that have no analytic solution. This research focuses on developing some new derivative based quadrature schemes for numerically integrating the integral of Riemann-Stieltjes (Rs-integral) by proposing new schemes which are based on trapezoid-type quadrature. The process of undetermined coefficients has been used for the derivation of proposed schemes. The theoretical derivation and numerical verification of orders of accuracy have been addressed in line with the degrees of precision, and a sufficient improvement has been demonstrated over the existing schemes. The theorems regarding single and mul-tiple use of the suggested schemes in a finite interval have been proved along with the theoretical results on residual terms, both locally and globally. All suggested schemes have been verified to reduce to corresponding variants for the Riemann integral in the case when integrator g(t) = t. Numerical experiments have been performed on all the discussed schemes with the help of MATLAB coding. The experimental results assure the smaller numerical errors by the proposed schemes in the comparison of existing schemes. The obtained results show the efficiency of proposed schemes in light of computational burden and CPU time (in seconds) with reference to the existing quadrature. The proposed work substantially advances the existing knowledge with an addition of derivative-based correction terms in the usual quadrature in a way that the consequent computational costs and execution times are also minimized.