Mathematical Inequalities Research Articles

General Trapezoidal-Type Inequalities in Fuzzy Settings

In this study, trapezoidal-type inequalities in fuzzy settings have been investigated. The theory of fuzzy analysis has been discussed in detail. The integration by parts formula of analysis of fuzzy mathematics has been employed to establish an equality. Trapezoidal-type inequality for functions with values in the fuzzy number-valued space is proven by applying the proven equality together with the properties of a metric defined on the set of fuzzy number-valued space and Höler’s inequality. The results proved in this research provide generalizations of the results from earlier existing results in the field of mathematical inequalities. An example is designed by defining a function that has values in fuzzy number-valued space and validated the results numerically using the software Mathematica (latest v. 14.1). The p-levels of the defined fuzzy number-valued mapping have been shown graphically for different values of p∈0,1.

Schrdinger equation for various quantum systems based on Heisenberg's uncertainty principle

This article establishes the proof of the Schrdinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrdinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrdinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.

Some Applications of Cauchy-Schwarz Inequality on Lightlike Warped Product Submanifolds

The well-known Cauchy-Schwarz inequality is a fundamental tool of analysis in Hilbert spaces. It is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is the most important and widely used inequalities in mathematics. We explore that inequality in case of lightlike warped product hypersurface of lorentzian space form to reach to the existence of minimal screen conformal lightlike hypersurface.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

Mathematical Inequalities in Fractional Calculus and Applications

Mathematical Inequalities in Fractional Calculus and Applications

Stability Results for a Class of Fractional Itô–Doob Stochastic Integral Equations

In this paper, we study the Hyers–Ulam stability of Hadamard fractional Itô–Doob stochastic integral equations by using the Banach fixed point method and some mathematical inequalities. Finally, we exhibit three theoretical examples to apply our theory.

ON THE INEQUALITIES CONCERNING POLYNOMIAL-EXPONENTIAL BOUNDS FOR INVERSE TRIGONOMETRIC FUNCTION

Recent research works on mathematical inequalities shows the importance bounds of polynomial-exponential type for various functions.

Association of Body Mass Index and Abdominal Obesity with Myocardial Infarction: We Reveal Confounding Factors that Historically Distorted Causal Inferences

Cardiovascular diseases, mainly myocardial infarction and stroke, are the leading cause of death globally. Therefore, epidemiological research seems necessary to prevent cardiovascular events and mortality. However, real-world data from obesity metrics has intrinsic limitations for the assessment of causality. Despite of historical studies showing that the body mass index (BMI), the waist-to-hip ratio (WHR) and the waist circumference (WC) have been associated with increased risk of myocardial infarction, they might not be accurate from a causal inference. Our aim was to summarize historical and novel findings about obesity metrics and myocardial infarction to evidence causal association biases. Method: an epidemiological review study was conducted while being original research when adding new anthropometrics in study design. Mathematical inequalities between the simple body measurements in anthropometrically healthy adults were described. Mean values and cut-offs for classic and several newer anthropometric variables were established. Classic metrics, ratios between the means of the simple measurements, a modulus |x| as a result of subtracting some measurement means from others (e.g., mean fat free mass minus fat mass) and somatotype ratings were collated. Mathematically, a non-zero difference for each modulus |x| in any population study would indicate an unbalanced distribution of the measurements between groups being compared, and therefore, the risk exposure levels differing. Thus, when between-groups the high-risk body compositions and somatotype ratings differ, any metric-associated risk is biased from a causal inference. After investigating large epidemiological studies, the historical omission of key anthropometric variables is stated, and as being uncontrolled confounding factors distorted causal inferences. Therefore, a protective overestimate of fat free mass and hip circumference over fat mass and WC, respectively, always occurred. Similarly, when the waist-to-height ratio values of >0.5 are associated; a protective underestimate of height over WC occurs. Any metric-associated risk is biased if prediction is made from WC or technologically measured body compositions without accounting for relative risk volume measures. In conclusion, summarizing the historical and novel findings regarding risk prediction, BMI, WHR and WC alone show evidence of causal association biases because of high-risk body compositions and risk exposure levels always differ between the groups being compared.

Some New Variants of Hermite–Hadamard and Fejér‐Type Inequalities for Godunova–Levin Preinvex Class of Interval‐Valued Functions

The theory of inequalities is greatly influenced by interval‐valued concepts, and this contribution is explored from several perspectives and domains. The aim of this note is to develop several mathematical inequalities such as Hermite–Hadamard, Fejér, and the product version based on center radius ‐order relations. Furthermore, we develop several nontrivial examples and remarks to support the main findings.

ПРОЦЕСС ПОГРУЖНОЙ ОЧИСТКИ МЕТАЛЛИЧЕСКИХ ПОВЕРХНОСТЕЙ ДЕТАЛЕЙ МАШИН

During the operation of agricultural machinery, contamination of various types and properties is deposited on the metal surfaces of parts, which can lead to increased wear of parts and a decrease in the operational reliability of machines as a whole. (Research purpose) The research purpose is studying the influence of the mechanical action of the cleaning solution on the efficiency of removing contaminations from the metal surfaces of machine parts. (Materials and methods) It was determined that the efficiency and quality of cleaning can be improved by increasing the intensity of the mechanical action of the washing liquid flow on the contamination of parts of agricultural machinery and equipment. We have obtained a mathematical inequality describing the condition under which the contamination is destroyed and removed from the surface. (Results and discussion) It has been shown that mixing of the cleaning solution, the creation of mechanical vibrations of the parts to be cleaned and their movement, the creation of cavitation phenomena in the flow of the cleaning solution, where acoustic currents play an important role, which form vortex hydrodynamic flows that facilitate the dissolution and movement of components in a liquid medium, are used as mechanical effects when cleaning metal surfaces of parts and assemblies of agricultural machinery. It was found that the criteria for cleaning efficiency are the hydromechanical action of the washing liquid on the surface to be cleaned and the energy intensity of the intensifying submersible cleaning. (Conclusions) The cleaning efficiency is determined by the physical and chemical properties of detergents, the flow rate of the cleaning solution, heating and softening of contaminants under high accelerations, the temperature of the cleaning solution, the concentration of detergent in water, the exposure of the solution to the surface to be cleaned.

Diverse copulas through Durante’s method. Exploring parametric functions

This article unveils the often underestimated potential of a copula methodology introduced by Durante in 2009. It highlights the remarkable ability of the method to generate a broad spectrum of copulas by exploiting various parametric functions. Our exploration encompasses a collection of power-like, exponential-like, trigonometric-like, logarithmic-like, hyperbolic-like and error-like functions, each dependent on one, two, or three parameters, effectively satisfying the necessary assumptions of Durante’s method. The proofs provided rely on suitable differentiation, comprehensive factorizations, and judicious application of mathematical inequalities. In the vast repertoire of copulas derived from this methodology, we present three distinct series of eight new copulas, supported by a graphical analysis of their respective densities. This theoretical study not only expands the understanding of copula generation but also introduces a new perspective on their construction in various contexts.

Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes

<abstract><p>Mathematical inequalities supporting interval-valued stochastic processes are rarely addressed. Recently, Afzal et al. introduced the notion of $ \mathtt{h} $-Godunova-Levin stochastic processes and developed Hermite-Hadamard and Jensen type inequalities in the setting of interval-valued functions. This note introduces a more generalized class of Godunova-Levin stochastic process that unifies several previously published results through the use of Kulisch-Miranker type order relations that are rarely discussed in relation to stochastic processes. Further, it is the first time that fractional version of Hermite-Hadamard inequality has been developed by using interval-valued stochastic processes in conjunction with a classical operator. Moreover, we give new modified forms for Ostrowski type results and present a new way to treat Jensen type inclusions under interval stochastic processes by using a discrete sequential form. We end with an open problem regarding Milne type results and discuss the importance of different types of order relations related to inequality terms in interval-valued settings.</p></abstract>

Improvements of Integral Majorization Inequality with Applications to Divergences

Within the recent wave of research advancements, mathematical inequalities and their practical applications play a notably significant role across various domains. In this regard, inequalities offer a captivating arena for scholarly endeavors and investigational pursuits. This research work aims to present new improvements for the integral majorization inequalities using an interesting aproach. Certain previous improvements have been achieved for the Jensen inequality as direct outcomes of the main results. Additionally, estimates for the Csiszár divergence and its cases are provided as applications of the main results. The circumstances under which the principal outcomes offer enhanced estimations for majorization differences are also underscored and emphasized.

Some Applications of the Principle of Virtual Displacements in Physics and Mathematics Courses

Key words: problem, force, statics, work, analogy, inequality, proof The work is devoted to the school application of the principle of virtual displacements. The latter is significant since it is related to the importance of energy approaches in physics with which this principle is associated. Only in university mechanics courses can the idea and its many applications be rigorously and completely proven. However, if the essence of the principle is simply interpreted, then even in the confines of a school physics course it can be effectively used in the process of solving statics problems. It should be noted that it can be challenging to answer some statics problems by taking standard equilibrium conditions of forces and torques into account. In such cases, the only viable method for the solution of these problems is to apply the virtual displacements idea. Although this principle may have interesting applications, it has not been widely used in physics teaching yet. The work's scientific and methodological novelty lies in its clear presentation of the principle of virtual displacements within the context of school physics. These concepts are then applied to the solution of mathematical inequalities as well as statics problems. Using a straightforward example, the article first explains the fundamentals of the idea of the virtual displacement, then examins a sequence of progressively intricate statics. In the article's last section, with the help of some analogies, the concepts of the principle are applied to solve mathematical inequalities.

Semiosis of conceptual learning of mathematical inequalities through semiotic meaning triads

The process of semiosis for the conceptual learning of inequalities allows the student to revitalize the interpretation, understanding, and solution of problems both in mathematical contexts and in everyday contexts. This research designed and applied a didactic strategy based on the semiotic theory of semiotic treatments and conversions to develop the semiosis processes of conceptual learning of inequalities through the methodology of the semiotic meaning triad and its three phases applied in class sessions focused on the solution of inequalities, defining the domain and range of functions and interpreting the lipid profile of a person.

Complex-valued neural networks with time delays in the [formula omitted] sense: Numerical simulations and finite time stability

A discrete fractional-order complex-valued neural network is taken into consideration in the present study. For the existence of the solution of the considered model to be stable in finite time, certain requirements are specified. Our strategy focuses on the use of the recently formulated discrete fractional calculus, mathematical inequalities, Krasnoselskii’s fixed point theorem, and the Arzelà–Ascoli theorem. We present afew numerical examples that demonstrate the theoretical results’ implementation.

Arithmetic Mean-Geometric Mean Inequality for Convex Fuzzy Sets

Convex analysis is a discipline of mathematics dedicated to the explication of the properties of convex sets and convex functions. Convex functions are extremely useful in proving many famous inequalities in mathematics. It is widely known that inequalities defined by convex functions originated with the works done by Holder and Jensen among others and applied to modeling a variety of problems both in hard and soft sciences. It is said that the arsenal of an analyst is heavily stocked with inequalities. Studies related to convexity have kept occupying a central position in almost all areas of mathematics, especially in functional analysis and operations research. Following the emergence of fuzzy set theory, fuzzy convexity alongside a number of related concepts has been explicated. However, not much has been done regarding fuzzy inequalities defined by fuzzy convexity. In this paper, a novel attempt to study arithmetic mean-geometric mean is proposed. In this short note, we provide two proofs of the arithmetic mean-geometric mean inequality for convex fuzzy sets. Mathematics Subject Classification 2020: 03E72, and 94D05.

Sharpening celebrated convex inequalities with applications to operators and entropies

The Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities have been key in advancing convex inequalities for scalars and operators, with several applications in fields like mathematical physics, operator theory and mathematical inequalities. This paper presents a new argument leading to interesting refinements of these celebrated inequalities. Applications that include entropy and operator inequalities will be presented.

Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

In the recent era of research developments, mathematical inequalities and their applications perform a very consequential role in different aspects, and they provide an engaging area for research activities. In this paper, we propose a new approach for the improvement of the classical majorization inequality and its weighted versions in a discrete sense. The proposed improvements give several estimates for the majorization differences. Some earlier improvements of the Jensen and Slater inequalities are deduced as direct consequences of the obtained results. We also discuss the conditions under which the main results give better estimates for the majorization differences. Applications of the acquired results are also presented in information theory.

Bidirectional field-steering and atomic steering induced by a magnon mode in a qubit-photon system

This paper investigates the cavity–magnon steering and qubit–qubit steering of a hybrid quantum system consisting of a single-mode magnon, a two-qubit state, and a single-mode cavity field in the presence of their damping rates. The temporal wave vector of the system is obtained for the initial maximally entangled two-qubit state and initial vacuum state of the magnon and cavity modes. Additionally, the mathematical inequalities for obtaining the cavity–magnon steering and qubit–qubit steering are introduced. The findings reveal that steering between the magnon and cavity is asymmetric, while steering between the two qubits is symmetric in our system. Increasing the atom–field coupling improves steering from magnon to field, while reducing steering between the two qubits. Moreover, increasing magnon–field coupling enhances and elevates the lower bounds of qubit–qubit steering. Further, adding the damping rates causes deterioration of the cavity–magnon steering and qubit–qubit steering. However, the steering persistence is slightly greater when damping originates from the cavity field rather than the magnon modes based on the coupling parameters.