Mathematical Inequalities Research Articles

Preface to the Special Issue “A Themed Issue on Mathematical Inequalities, Analytic Combinatorics and Related Topics in Honor of Professor Feng Qi”

This Special Issue of the journal Axioms pays tribute to Professor Feng Qi’s significant contributions and provides some important recent advances in mathematics [...]

Parameterized monogamy and polygamy relations of multipartite entanglement

Monogamy and polygamy relations are important properties of entanglement, which characterize the entanglement distribution of multipartite systems. We explore monogamy and polygamy relations of entanglement in multipartite systems by using two newly derived parameterized mathematical inequalities, and establish classes of parameterized monogamy and polygamy relations of multiqubit entanglement in terms of concurrence and entanglement of formation. We show that these new parameterized monogamy and poelygamy inequalities are tighter than the existing ones by detailed examples.

Some Refinements of Selberg Inequality and Related Results

This paper introduces several refinements of the classical Selberg inequality, which is considered a significant result in the study of the spectral theory of symmetric spaces, a central topic in the field of symmetry studies. By utilizing the contraction property of the Selberg operator, we derive improved versions of the classical Selberg inequality. Additionally, we demonstrate the interdependence among well-known inequalities such as Cauchy–Schwarz, Bessel, and the Selberg inequality, revealing that these inequalities can be deduced from one another. This study showcases the enhancements made to the classical Selberg inequality and establishes the interconnectedness of various mathematical inequalities.

Development and Validation of Self-Learning Packets in Linear Equation and Inequalities in Mathematics 8

The shift of modality brought on by the COVID-19 pandemic, the Department of Education in the Philippines implemented the Distance Learning Modality, which averted the New Normal. Self-Learning Packets (SLPs), satisfy the challenge of distance learning. This descriptive-developmental study aimed to develop and validate SLPs that is aligned with the Most Essential Learning Competency in Mathematics 8. This study underwent five phases guided by the ADDIE Model (Analysis, Design, Development, Implementation, and Evaluation). The developed SLPs was very acceptable as validated by five expert validators invited from Saint Columban College and the Division of Zamboanga Sibugay. Selected Grade 8 students were also asked to evaluate the pretest and posttest. Results shows that the developed SLPs in grade 8 Mathematics were generally valid and acceptable as validated by the experts. The SLPs obtained 96.92% compliance in Content and 95.93% in Layout and Design. Paired t-test was conducted to check if there were significant changes of the student’s performance from pretest and posttest. Based on the results there was a significant difference in the scores in pretest and posttest. The mean of pretest and posttest of SLP 1 were 5.10 and 7.30 with mean difference of 2.2 and the mean of pretest and posttest of SLP 2 were 4.60 and 7.20 with mean difference of 2.6, and p-value is lower than 0.05 for both SLPs. The expert validators strongly agreed that the SLPs possesses adequacy, coherence, usefulness, and reliability in equipping and enhancing student skills in linear Equations and Linear Inequalities.

Study of The Application of Inequalities in Mathematics

Inequalities have a very important place in the study of mathematics, and their use can be very convenient in solving many practical problems in our daily lives. Inequalities have become an important part of the basic theory of mathematics and have attracted many professionals to study them. The use of inequalities provides an intuitive mathematical model for understanding inequalities in the real world, making it easier for us to learn mathematics and solve many practical problems in our lives. In this paper, we explore the use of mathematical inequalities in everyday learning.

Advances in Mathematical Inequalities and Applications

Why do we study inequalities [...]

New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the (m,tgs)-type s-preinvex function, Godunova–Levin (s,m)-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings.

A general result on complete f-moment convergence with its application to nonparametric regression models

In this paper, the complete f-moment convergence for weighted sums of sequences of extended negatively dependent random variables is investigated. As a result, we not only extend and generalize some results on complete convergence obtained by Wu and Zhai (Journal of Mathematical Inequalities, 13(1): 251-260, 2019), but also get some results on the complete consistency for the estimator of nonparametric regression models and provide some simulations to assess the validity of our theoretical results.

Theoretical Advancements on a Few New Dependence Models Based on Copulas with an Original Ratio Form

Copulas are well-known tools for describing the relationship between two or more quantitative variables. They have recently received a lot of attention, owing to the variable dependence complexity that appears in heterogeneous modern problems. In this paper, we offer five new copulas based on a common original ratio form. All of them are defined with a single tuning parameter, and all reduce to the independence copula when this parameter is equal to zero. Wide admissible domains for this parameter are established, and the mathematical developments primarily rely on non-trivial limits, two-dimensional differentiations, suitable factorizations, and mathematical inequalities. The corresponding functions and characteristics of the proposed copulas are looked at in some important details. In particular, as common features, it is shown that they are diagonally symmetric, but not Archimedean, not radially symmetric, and without tail dependence. The theory is illustrated with numerical tables and graphics. A final part discusses the multi-dimensional variation of our original ratio form. The contributions are primarily theoretical, but they provide the framework for cutting-edge dependence models that have potential applications across a wide range of fields. Some established two-dimensional inequalities may be of interest beyond the purposes of this paper.

Higher-Order Nabla Difference Equations of Arbitrary Order with Forcing, Positive and Negative Terms: Non-Oscillatory Solutions

This work provides new adequate conditions for difference equations with forcing, positive and negative terms to have non-oscillatory solutions. A few mathematical inequalities and the properties of discrete fractional calculus serve as the fundamental foundation to our approach. To help establish the main results, an analogous representation for the main equation, called a Volterra-type summation equation, is constructed. Two numerical examples are provided to demonstrate the validity of the theoretical findings; no earlier publications have been able to comment on their solutions’ non-oscillatory behavior.

Ideas and Tools for Error Detection in Opacity Databases

In this article, we propose several ideas and tools in order to check the reliability of radiative opacity and atomic physics databases. We first emphasize that it can be useful to verify that mathematical inequalities, which impose lower and upper bounds on the Rosseland and/or Planck mean opacities, are satisfied, either for pure elements or mixtures. In the second part, we discuss the intriguing law of anomalous numbers, also named Benford’s law, which enables one to detect errors in line-strength collections, required in order to perform fine-structure calculations. Finally, we point out and illustrate the importance of quantifying the uncertainties due to interpolations in the density-temperature opacity (or more generally atomic-data) tables and performing convergence checks, which are crucial in the accuracy-completeness compromise inherent in opacity computations.

Introduction to the Special Issue in Axioms Titled Current Research on Mathematical Inequalities

The importance of inequalities in Mathematics is beautifully summarized in a citation attributed to Respected Professor Andrey Nikolaevich Kolmogorov: [...]

Theoretical Contributions to Three Generalized Versions of the Celebioglu–Cuadras Copula

Copulas are probabilistic functions that are being used more and more frequently to describe, examine, and model the interdependence of continuous random variables. Among the numerous proposed copulas, renewed interest has recently been shown in the so-called Celebioglu–Cuadras copula. It is mainly because of its simplicity, exploitable dependence properties, and potential for applicability. In this article, we contribute to the development of this copula by proposing three generalized versions of it, each involving three tuning parameters. The main results are theoretical: they consist of determining wide and manageable intervals of admissible values for the involved parameters. The proofs are mainly based on limit, differentiation, and factorization techniques as well as mathematical inequalities. Some of the configuration parameters are new in the literature, and original phenomena are revealed. Subsequently, the basic properties of the proposed copulas are studied, such as symmetry, quadrant dependence, various expansions, concordance ordering, tail dependences, medial correlation, and Spearman correlation. Detailed examples, numerical tables, and graphics are used to support the theory.

Jensen–Mercer and Hermite–Hadamard–Mercer Type Inequalities for GA-h-Convex Functions and Its Subclasses with Applications

Many researchers have been attracted to the study of convex analysis theory due to both facts, theoretical significance, and the applications in optimization, economics, and other fields, which has led to numerous improvements and extensions of the subject over the years. An essential part of the theory of mathematical inequalities is the convex function and its extensions. In the recent past, the study of Jensen–Mercer inequality and Hermite–Hadamard–Mercer type inequalities has remained a topic of interest in mathematical inequalities. In this paper, we study several inequalities for GA-h-convex functions and its subclasses, including GA-convex functions, GA-s-convex functions, GA-Q-convex functions, and GA-P-convex functions. We prove the Jensen–Mercer inequality for GA-h-convex functions and give weighted Hermite–Hadamard inequalities by applying the newly established Jensen–Mercer inequality. We also establish inequalities of Hermite–Hadamard–Mercer type. Thus, we give new insights and variants of Jensen–Mercer and related inequalities for GA-h-convex functions. Furthermore, we apply our main results along with Hadamard fractional integrals to prove weighted Hermite–Hadamard–Mercer inequalities for GA-h-convex functions and its subclasses. As special cases of the proven results, we capture several well-known results from the relevant literature.

Adaptive Fixed-Time Synchronization of Delayed Memristor-Based Neural Networks with Discontinuous Activations

Fixed-time synchronization (FTS) of delayed memristor-based neural networks (MNNs) with discontinuous activations is studied in this paper. Both continuous and discontinuous activations are considered for MNNs. And the mixed delays which are closer to reality are taken into the system. Besides, two kinds of control schemes are proposed, including feedback and adaptive control strategies. Based on some lemmas, mathematical inequalities and the designed controllers, a few synchronization criteria are acquired. Moreover, the upper bound of settling time (ST) which is independent of the initial values is given. Finally, the feasibility of our theory is attested by simulation examples.

Mathematical inequalities on some weighted means

Mathematical inequalities on some weighted means

Fast PCB Stack-Up Optimization Using Integer Programming

This paper presents a flexible and efficient methodology to optimize stack-up for multi-layer printed circuit boards (PCBs) with enormous search space and various design constraints. PCB stack-up optimization is crucial in high-speed system design to achieve the desired electrical performance while reducing system costs. The stack-up optimization process is labor-intensive and time-consuming for a large number of layers. Moreover, after the optimization process, the electrical performance of a real design, such as the impedance and loss, may deviate from the target design due to manufacturing variations. Estimating the worst cases due to the manufacturing variations, referred to as “corner cases” in this paper, is essential for a confident PCB design but challenging since the number of related parameters is large. In this paper, PCB stack-up optimization and corner-case searching are addressed and greatly accelerated using the integer programming technique. All constraints are converted to mathematical equalities and inequalities that can be solved rapidly by an integer programming solver to obtain feasible stack-up solutions. After the cross-sections of the transmission lines are optimized based on the stack-up design to achieve a target electrical performance, the upper and lower bound of impedance and loss are acquired using integer programming when the design parameters vary in a particular range. The proposed method is verified using multi-layer PCB designs with practical constraints and demonstrates its effectiveness and high efficiency.

Can room-temperature data for tunneling molecular junctions be analyzed within a theoretical framework assuming zero temperature?

Routinely, experiments on tunneling molecular junctions report values of conductances (GRT) and currents (IRT) measured at room temperature. On the other hand, theoretical approaches based on simplified models provide analytic formulas for the conductance (G0K) and current (I0K) valid at zero temperature. Therefore, interrogating the applicability of the theoretical results deduced in the zero-temperature limit to real experimental situations at room temperature (i.e., GRT ≈ G0K and IRT ≈ I0K) is a relevant aspect. Quantifying the pertaining temperature impact on the transport properties computed within the ubiquitous single-level model with Lorentzian transmission is the specific aim of the present work. Comprehensive results are presented for broad ranges of the relevant parameters (level's energy offset ε0 and width Γa, and applied bias V) that safely cover values characterizing currently fabricated junctions. They demonstrate that the strongest thermal effects occur at biases below resonance (2|ε0| - δε0 - 0.3 ≲ |eV| - 0.3 ≲ 2|ε0|). At fixed V, they affect an ε0-range whose largest width δε0 is about nine times larger than the thermal energy (δε0 ≈ 3πkBT) at Γa → 0. The numerous figures included aim to convey a quick overview on the applicability of the zero-temperature limit to a specific real junction. In quantitative terms, the conditions of applicability are expressed as mathematical inequalities involving elementary functions. They constitute the basis of a proposed interactive data-fitting procedure, which aims to guide experimentalists interested in data processing in a specific case.

Novel non-fragile extended dissipative synchronization of T-S fuzzy complex dynamical networks with interval hybrid coupling delays

<abstract><p>This paper presents the first investigation of extended dissipative synchronization in a specific type of Takagi-Sugeno (T-S) fuzzy complex dynamical networks with interval hybrid coupling delays. First, the decoupling method is employed to reorganize the multiple communication dynamical system, which comprises discrete-time, partial and distributed coupling delays. Second, the non-fragile control, which allows for uncertainty management within predefined norm bounds, has been applied to networks. Moreover, it becomes possible to derive a less conservative condition by utilizing multiple integral Lyapunov functionals, a decoupling strategy, Jensen's inequality, Wirtinger's inequality, and mathematical inequality techniques. This condition ensures that the T-S fuzzy complex dynamical networks, with interval hybrid coupling delays, can attain asymptotic synchronization with the assistance of a non-fragile feedback controller. Additionally, we extended this system to the extended dissipativity analysis, including passivity, $ L_2-L_\infty, H_{\infty} $ and dissipativity performance in a unified formulation. A set of strict linear matrix inequalities (LMIs) conditions is a sufficient criterion. Finally, two simulation examples are proposed to verify the merit of the obtained results.</p></abstract>

Further Insight into Bifurcation and Hybrid Control Tactics of a Chlorine Dioxide-Iodine-Malonic Acid Chemical Reaction Model Incorporating Delays

Delayed differential equation is an important tool to describe the interaction of different chemical substance in chemistry. In this present research, we set up a novel chlorine dioxide-iodine-malonic acid chemical reaction model incorporating delays. The peculiarity of solution and Hopf bifurcation of the formulated delayed chlorine dioxide-iodine-malonic acid chemical reaction model are explored. Firstly, the existence and uniqueness is investigated via fixed point theorem. Secondly, the non-negativeness of solution is studied via some proper mathematical inequality shills. Thirdly, the stability and bifurcation of the formulated delayed chlorine dioxide-iodine-malonic acid chemical reaction model are analyzed. The influence of delay on the delayed chlorine dioxide-iodine-malonic acid chemical reaction model is uncovered. Fourthly, Hopf bifurcation control issue of the formulated delayed chlorine dioxide-iodine-malonic acid chemical reaction model is studied via two hybrid controllers. To check the soundness of acquired key assertions, Matlab simulations are executed. The gained assertions of this research are completely novel and possess tremendous theoretical value in maintaining the balance of the concentrations of chlorine dioxide, iodine in chemistry.