In recent decades, a wide range of Hardy-Hilbert-type integral inequalities have been established. This article focuses on a one-parameter result introduced by Waadallah Tawfeeq Sulaiman in 2010, which has a unique structure: the double integral involves a power-sum of the variables, as well as a technical power-minimum. The sharp constant factor is also elegantly expressed in terms of the beta function. However, the parameter involved is subject to restrictions on its values. In this article, we refine the inequality by removing this restriction and addressing a theoretical gap in the original proof to yield a sharper result. We provide a thorough, step-by-step proof and demonstrate how this new result can be used to derive additional variants and extensions.

We study the Abel-type family \(y'=C\,y^r(1-y)^s\) under a parity-driven mapping of \((r,s)\), which yields symmetric dynamics for odd \(k\) and asymmetric, potentially stiff dynamics for even \(k\). We correct the normalization by peaking at the true maximizer \(y^\star=r/(r+s)\) and provide the analytic Jacobian \(g'(y)\) for implicit solvers. A matched-accuracy benchmarking protocol sweeps rtol/atol and reports global errors against ultra-tight references (separable/explicit for odd \(k\), Radau for even \(k\)), alongside wall time, \(nfev\), \(njev\), linear-solve counts, rejected steps, and step-size histories. Stiffness is quantified through the proxy \(\tau(t)=1/\lvert g'(y(t))\rvert\) and correlated with step-size adaptation; trajectories are constrained to \(y\in[0,1]\) via terminal events. Across tolerances, DOP853 and LSODA are strong all-rounders in non-stiff regimes, while Radau/BDF dominate when asymmetry and proximity to multiple roots induce stiffness; observed orders align with nominal ones under matched error. The study clarifies how parity and nonlinearity govern solver efficiency for polynomial nonlinearities and provides full environment details and code for reproducibility.

Let \(G\) be a graph of order \(n\) and size \(m\), with adjacency matrix eigenvalues \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_n\). The energy of \(G\), denoted by \(\mathcal{E}(G)\), is defined as the sum of the absolute values of its eigenvalues. A classical upper bound on the energy, originally established by McClelland [1], states that \(\mathcal{E}(G) \leq \sqrt{2mn}\,.\) In this paper, we refine the spectral analysis of graph energy by deriving an exact analytical expression relating \(\mathcal{E}(G)\) to the variance of the vector of absolute eigenvalues \(x = (|\lambda_1|, |\lambda_2|, \dots, |\lambda_n|)\,.\) Specifically, we prove that \(\mathcal{E}(G) = \sqrt{2mn – n^2 \operatorname{Var}(x)},\) providing a more precise and quantitative spectral characterization of graph energy. As an application, this identity allows us to derive improved lower bounds for \(\mathcal{E}(G)\), thereby strengthening and generalizing previously known inequalities. Furthermore we conjecture that for any non-singular graph \(G\) of order \(n\), \(\mathcal{E}(G) \geq 2 \sqrt{\langle d \rangle (n-1)},\) where \(\langle d \rangle = 2m/n\) is the average vertex degree of \(G\). Equality holds if and only if \(G \cong K_n\).

First, this paper provides some approximation and estimation type results for some moments of the Gauss function, motivated by the fact that the moments of even orders \(n=2l,\ l\in \mathbb{N}\mathrm{=}\mathrm{\{}0,1,\dots \}\) of the function \(exp\left(-t^2\right)\) on bounded intervals . Second, the problem of asymptotic behavior of the sequence of all orders for the same function on any interval \(\left[0,b\right]\subseteq \left[0,{1}/{\sqrt{2}}\right]\) is studied and solved. Here the point is using Jensen inequality. Third, the problem of asymptotic behavior of the sequence of all orders for the same function on any interval \(\left[0,b\right]\subset \left[0,+\infty \right)\) is deduced, via elements of complex analysis (Vitali’s theorem). The convergence holds uniformly on compact subsets of the complex plane. Fourth, the asymptotic behavior of the sequence of all moments on \(\left[0,1\right],\ \)as \(n\to \infty ,\) for an arbitrary function \(f\in C\left(\left[0,1\right]\right)\) is determined precisely, by means of Korovkin’s approximation theorem. Consequently, a similar result for complex analytic functions is deduced, using Vitali’s theorem. This is the fifth aim of the paper.

A discrete-time prey-predator system of Leslie-Gower type with harvesting is considered in this paper. The system is first discretized using the Forward-Euler method. The topology and stability of the fixed points of this method are discussed using period-doubling and Neimark-Sacker bifurcation analysis. Secondly, a non-standard finite difference scheme of the same system is presented. We have shown the permanence and dynamical consistency of this scheme. It has been shown that our non-standard finite difference scheme is the best scheme for this system, according to Mickens. Using the center manifold theorem, the normal form of the Neimark-Sacker bifurcation has been derived. Numerical simulations are provided, using a computer package, to illustrate the consistency of the theoretical results. Finally, chaos control techniques have been applied to control the chaotic dynamics of the system.