We establish a sharp positivity threshold for the sixth-order symmetric Toeplitz determinant T_6(1) over the Janowski family of odd starlike functions with real coefficients. Writing Φ (A,B) for an explicit nonnegative quantity built from the Janowski parameters, we prove that Φ (A,B)^2≤ T_6(1)≤ 1 for every f∈𝒮^*_ℝ[A,B], with both bounds sharp, and that Φ (A,B) vanishes precisely at the full odd starlike endpoint (A,B)=(1,-1). Consequently every proper Janowski subclass imposes a strictly positive floor on T_6(1), while that endpoint is the unique class in which T_6(1) may vanish. The bounds rest on a structural device of independent interest: for an odd function the Toeplitz matrix splits, under a parity permutation, into two equal reduced blocks, so that the sixth-order determinant becomes the square of a determinant in only the two coefficients a_3 and a_5. For the exponential class we obtain the sharp two-sided bound between 81/256 and 1, and we show that this real bound does not persist in the complex exponential class, where the sharp estimate rises to 625/256 and is attained by a purely imaginary linear Schwarz function; the real-coefficient hypothesis thus changes the extremal value, not merely the method. The factorisation underlying the threshold is established by an explicit algebraic identity. Such a sharp positivity threshold for T_6(1) across the full Janowski parameter range does not appear to have been previously recorded.
In the paper, the authors identify and correct several errors and methodological shortcomings in the work of Qaisar, He, and Hussain, “A generalizations of Simpson’s type inequality for differentiable functions using (α ,m)-convex functions and applications,” J. Inequal. Appl. 2013, Paper No. 158, 13 pp.; doi:https://doi.org/10.1186/1029-242X-2013-158. These issues include improperly specified domains, the incorrect conflation of α-convexity with (α ,m)-convexity, and invalid applications to special means. By presenting explicit counterexamples and establishing corrected theorems under appropriate hypotheses, the authors rectify these mathematical inaccuracies and strengthen the theoretical foundations of research on (α ,m)-convex functions and their associated integral inequalities.
We establish new generalized Reid-type inequalities for bounded linear operators on complex Hilbert spaces by exploiting the positivity of 2 × 2 block operator matrices and commutation relations. This approach yields refined mixed Schwarz-type inequalities and Hölder-type bounds for finite sums of operators, replacing classical norm estimates with sharper bounds involving the spectral radius. As applications, we derive new upper bounds for the numerical radius of n × n operator matrices, which generalize and improve several existing results in the literature, including recent estimates involving the Moore-Penrose inverse.
In this paper, we present a new conformable fractional-order discrete SIS-epidemic model to analyze the dynamics of infectious diseases. The model incorporates the effects of interspecific competition and memory, providing a novel approach to understanding disease spread. We analyze the stability of the model’s fixed points and calculate the basic reproduction number (R_0), which is critical for understanding the potential for epidemic outbreaks. We explore the chaos and bifurcations of the model through both theoretical analysis and numerical simulations. Our findings reveal complex dynamic behaviors, including stable equilibria, attracting invariant circles, periodic orbits, and chaotic attractors, which are affected by the discretization parameter and the fractional-order parameter.
Abstract ℓ 1 − ℓ 2 $\ell _{1}-\ell _{2}$ minimization problem finds extensive applications across various domains, including signal processing, face recognition, and image restoration. This study begins by reformulating ℓ 1 − ℓ 2 $\ell _{1}-\ell _{2}$ minimization model into an equivalent fixed-point formulation via a projection-based residual method. Based on this, a new second-order continuous-time projection (2-CTP) algorithm is constructed by leveraging the theory of second-order dynamical systems. Subsequently, the existence and uniqueness of the solution to the proposed algorithm are established, and the convergence of its trajectories is further analyzed. To illustrate the effectiveness and performance of the developed method, numerical simulations covering applications in signal processing and image restoration are provided in the end.
This article focuses on characterizing lattice-order structures arising from majorization within the space L^1[0,b). Particular attention is given to the analysis of suprema and infima in the majorization order on continuous strictly unimodal functions.
This paper investigates the exponential mean-square stability of both the continuous solution and the split-step θ-Milstein (SSTM) numerical scheme for neutral stochastic delay integro-differential equations (NSDIDEs) with Poisson jumps. Under a set of reasonable conditions, the trivial solution of the NSDIDE is shown to be exponentially mean-square stable. Subsequently, it is demonstrated that this stability property is preserved by the SSTM method under explicit stepsize constraints. Specifically, for θ∈ [0, 12), exponential mean-square stability of the numerical solution is guaranteed provided the stepsize is sufficiently small; for θ∈ (12, 1], the method is stable under a less restrictive stepsize condition, thereby enhancing computational efficiency while maintaining stability. The theoretical findings are further validated through two numerical examples, which confirm the validity of the derived stability conditions.
In this article, we propose a relaxed inertial projection algorithm for solving inverse quasi-variational inequality problems (IQVIPs) in real Hilbert spaces. The method integrates inertial extrapolation and relaxation techniques to accelerate the convergence of the classical projection scheme. Strong convergence to the unique solution is established under standard monotonicity and Lipschitz continuity assumptions on the operator, covering both the moving-set case and the fully general set-valued constraint setting. Numerical comparisons with an existing algorithm show that the proposed method converges significantly faster, requires fewer iterations, and incurs lower computational cost over a wide range of initial points. The practical effectiveness of the approach is further demonstrated through an application to a realistic continuous-time road pricing problem on a four-bridge traffic network, where the algorithm efficiently computes equilibrium tolls that achieve the desired flow redistribution while respecting toll-dependent capacity constraints. These results confirm the robustness of the proposed method and its potential for solving structured equilibrium models arising in transportation and related fields
The notion of a field is a basic algebraic structure widely used in different areas of mathematics. A complex neutrosophic soft set is a specialised hybrid model that merges the features of soft sets and neutrosophic sets within a complex framework, employing sets as a key component. By merging the properties of complex neutrosophic soft sets with foundational field theory concepts, this paper introduces fields in uncertain environments. In contrast to existing neutrosophic soft fields, the proposed model incorporates complex-valued truth, indeterminacy and falsity membership grades, enabling the representation of both magnitude and phase information in uncertain environments. Therefore, the proposed structure can be regarded as a natural extension and generalisation of neutrosophic soft field which is considered as a special case of a complex neutrosophic soft field when the complex-valued membership functions are reduced to real values (i.e., with zero phase components). Throughout this detailed study, we aim to deepen our understanding of complex neutrosophic soft fields and their characteristics, thereby advancing algebraic analysis and its real-world applications for managing uncertainties and imprecision.
Abstract In this work, we give some topological properties of the sequence spaces $\ell _{p}(\widehat{B}_{q})$ ℓ p ( B ˆ q ) (with $0< p<1$ 0 < p < 1 ), $c_{0}(\widehat{B}_{q})$ c 0 ( B ˆ q ) , $c(\widehat{B}_{q})$ c ( B ˆ q ) , and $\ell _{\infty }(\widehat{B}_{q})$ ℓ ∞ ( B ˆ q ) defined by the matrix $\widehat{B}_{q}$ B ˆ q . Furthermore, we construct bases for the space $c_{0}(\widehat{B}_{q})$ c 0 ( B ˆ q ) and $c(\widehat{B}_{q})$ c ( B ˆ q ) , determine α -, β -, γ -duals of the above defined spaces, classify some matrix classes and provide several results related to compactness of certain matrix operators on the space $c_{0}(\widehat{B}_{q})$ c 0 ( B ˆ q ) .
This paper is devoted to continuous Opial-type inequalities involving the Riemann-Liouville fractional derivatives. The starting point was the inequality in which the integral of the product of the derivatives of one function is evaluated from above. This inequality is traditionally called the multiple Opial-type inequality. Here, we give a few generalizations of it. In the first type of generalization, the product of a finite number of factors is replaced by an expression involving infinitely many factors or even continuously many. Results involving L_p norm and sup norm in the estimations are obtained. Also, a reverse inequality is discussed. The second part of the paper concerns the inequality in which the integral of the product of successive derivatives of one function is bounded from above by the sum of integrals. Here, we make a similar generalization in which we transform the product of finitely many functions into its continuous form, which allows the result not only for finitely many functions but even for infinitely many functions.
This paper presents a unified framework for fixed point theory in complete fuzzy metric spaces by synthesizing fuzzy Hermite-Hadamard inequalities with various contraction types, including rational contractions. We introduce innovative fuzzy transforms M_ψ and M̃_ψ constructed via fuzzy Riemann integrals of convex and concave fuzzy-number-valued mappings. Our central results establish existence and uniqueness theorems under conditions involving the function class Φ _k, with particular emphasis on rational-type contraction conditions that extend classical results to the fuzzy setting. The incorporation of fuzzy Hermite-Hadamard inequalities provides crucial geometric insights that enhance convergence analysis. Each theorem is accompanied by carefully constructed illustrative examples that demonstrate its applicability, and all examples are supported by graphical representations that provide visual verification of the theoretical results. The framework’s robustness is demonstrated through extensive examples encompassing both trigonometric mappings and rational contractions, revealing its adaptability to diverse operator classes. As a significant application, we establish the existence and uniqueness of fuzzy solutions to nonlinear partial differential equations, specifically fuzzy transport equations, and provide numerical examples with graphical verification of the convergence behavior. This research bridges fuzzy convex analysis with fixed point theory, offering powerful new methodologies for analyzing nonlinear operators in fuzzy environments. The integration of partial differential equations applications with graphical validation creates pathways for further applications in fuzzy variational problems and fuzzy differential equations.
In this paper, we introduce a class of convexity known as ℳ_(Ψ ,m)-convexity. First, we investigate multi-term refinements of the inequality associated with ℳ_(Ψ ,m)-convexity. Then, our results are further enhanced and generalized through the application of weak sub-majorization theory. As applications, we derive novel refinements of the classical Hermite-Hadamard inequality for these notions. These findings expand upon and generalize recent work in the field, including the studies presented in [2, 19]. Our work contributes to the ongoing development of mathematical inequalities and their various applications.
In this paper, we prove the existence and local attractivity results for a nonlinear functional integral equation. The equation is considered in the Banach space of real valued functions that are continuous and bounded on ℝ_+. In our considerations, we apply the measure of noncompactness and the Darbo fixed point theorem. The equation we consider is more general in form and the conditions are less stringent than some existing results.
The recent contribution by Gogoladze and Meskhia (Proc. A. Razmadze Math. Inst. 141:29-40, 2006) presents a generalization of classical results by Bernstein, Sz & aacute;sz, Zygmund, and others on the absolute convergence of Fourier series with coefficients raised to a power xi, analyzed via moduli of smoothness. In this work, we extend their findings to Fourier-Laplace series in the weighted function spaces L-p(sigma(m-1)), 1 < p <= 2, and we also establish the sharpness of our conditions when p = 2.
This paper establishes the quantitative vector-valued multi-indexed inequalities for multilinear fractional integral operators under the full range of the multilinear Muckenhoupt class. By constructing an explicit discretization of the multi-indexed potential kernel, the continuous operator is structurally circumscribed by localized dyadic averages over sparse families. This localized estimate facilitates a deterministic decoupling scheme that aligns with contemporary limited-range, off-diagonal extrapolation mechanisms. Consequently, we bypass the topological obstructions inherent in direct sequence-space dualization regimes and obtain quantitative vector-valued norm inequalities with an explicitly tracked dependence on the weight characteristic. The resulting estimates systematically isolate the geometric properties of the underlying operators near the integrability boundaries, providing structural transparency for potential-type scaling dynamics without relying on qualitative abstract limiting procedures.
In this study, we establish existence and Ulam-Hyers results for nonlinear sequential fractional differential equations involving the deformable derivative. We employ the Banach contraction principle and Krasnoselskii’s theorem to achieve these results. A key novelty of this work lies in the use of a sequential application of the deformable derivative, which yields a richer mathematical structure compared to single-order formulations. The theoretical findings are illustrated through a concrete example, and the practical relevance of the results is discussed in the context of stability analysis for dynamical systems modeled by deformable fractional equations.
