Inequalities For Functions Research Articles

Inequalities of Ostrowski Type for Functions Whose Derivative Module Is Relatively Convex on Time Scales

In this article, we discuss several novel generalized Ostrowski-type inequalities for functions whose derivative module is relatively convex in time scales calculus. Our core findings are proved by using the integration by parts technique, Hölder’s inequality, and the chain rule on time scales. These derived inequalities expand the existing literature, enriching specific integral inequalities within this domain.

Switching-like event-triggered control of uncertain NCSs under delay distribution and DoS attacks

The paper addresses the switching-like event-triggered control for uncertain networked control systems with time-varying delay under DoS attacks. First of all, to reduce the communication burden, a switching-like event-triggered mechanism is designed to automatically select the trigger condition according to whether the system is under DoS attacks, which have the advantage of reducing the number of data packets transmitted. Secondly, unlike the traditional assumption of time-varying delay, here it satisfies the condition that the probability is known, and combines the networked control systems to propose a novel time-delay system model, which can obtain a larger upper bound on the delay. Then, by using both the Lyapunov functional method and linear matrix inequality technique, we obtain sufficient conditions of uncertain networked control systems to achieve exponentially stable in the mean square sense. Furthermore, under the common limitations of the maximum continuous packet losses caused by the DoS attacks and delay, the stability criterion is derived, which can be used to estimate the communication parameters and security controller gain. Finally, through two simulation examples, the larger upper bound of time delay, less trigger times, faster convergence rate are obtained, which verify the validity of our theoretical analysis.

Some Refinements and Reverses of Callebaut's Inequality for Isotonic Functionals via a Result Due to Cartwright and Field

In this paper we obtain some refinements and reverses of Callebaut's inequality for isotonic functionals via a result of Young's inequality due to Cartwright and Field.

An Explicit Form of Harnack’s Inequality for Poisson’s Dirichlet Problem

The explicit form of Harnack’s inequality for non-negative harmonic functions in the open ball plays an important role in harmonic analysis and elliptic partial differential equations. In this paper, we establish the explicit form of Harnack’s inequality for Poisson’s Dirichlet problem in the open ball with non-negative boundary data. We involve the Green’s function for Laplacian operator to find it out. Harnack’s inequality for positive harmonic functions can be followed from Harnack’s inequality for Poisson’s Dirichlet problem when the source function is set to be zero.

Generalized n-Polynomial p-Convexity and Related Inequalities

In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the newly defined class of functions. Additionally, we derive refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are generalized n-polynomial p-convex. When p=−1, our definition evolves into a new definition for the class of convex functions so-called generalized n-polynomial harmonically convex functions. The results obtained in this study generalize regarding those found in the existing literature. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields.

New LMI approach to [formula omitted] control of discrete-time singular systems

This article studies the problem of feedback H∞ control for discrete-time singular system. The objective is to design the controller to make the closed-loop system admissible and satisfy the specified H∞ performance. Firstly, the admissibility of the closed-loop system is analyzed via the Lyapunov function and linear matrix inequality (LMI) method, the admissibility condition of the singular system with H∞ performance is given. Then, applying a useful matrix inequality lemma, the controller design condition for discrete-time singular system is shown by a LMI. Specially, the benefit of the proposed design result is their less conservatism, the validity of the viewpoint is proposed by numeric illustrations.

A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities

We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math. Ann., 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed.

Interval-based KKT framework for support vector machines and beyond

Our article proves inequalities for interval optimization and shows that feasible and descent directions do not intersect in constrained cases. Mainly, we establish some new interval inequalities for interval-valued functions by defining LC-partial order. We use LC-partial order to study Karush–Kuhn–Tucker (KKT) conditions and expands Gordan's theorems for interval linear inequality systems. By applying Gordan's theorem, we can determine the best outcomes for interval optimization problems (IOPs) that have constraints, such as Fritz John and KKT conditions. The optimality conditions are observed with inclusion relations rather than equality. We can use the KKT condition for binary classification with interval data and support vector machines(SVMs). We present some examples to illustrate our results.

Data-driven control of singularly perturbed hybrid systems with multi-rate sampling

This paper addresses the exponential stability problem of singularly perturbed hybrid systems with multi-rate sampling. Firstly, a hybrid model is established to describe a class of nonlinear singularly perturbed systems by designing a data-driven mechanism and multi-rate sampling controller. Secondly, a novel error function and several inequalities are proposed to obtain the properties of the error subsystems. Thirdly, we estimate the bounds of fast and slow sampling intervals, and analyze the effect of the perturbation parameter and fast-slow sampling ratio on the sampling intervals. Finally, a DC motor model is illustrated to demonstrate the effectiveness of the proposed results.

Where I'm Livin’ and How I'm Feelin’: Associations among community stress, gender, and mental-emotional health among Black Americans

RationaleStructural racism is a primary avenue for the perpetuation of racial health disparities. For Black Americans, both historically and contemporarily, the neighborhood context serves as one of the most striking examples of structural racism, with stressful neighborhood contexts contributing to the well-documented inequalities in psychological functioning among this population. ObjectiveThus, in this study, we adapted an intersectional-ecological framework to investigate the links between community stress and multiple dimensions of mental-emotional health for Black men and women. MethodsDrawing on cross-sectional data from 842 Black Americans from the Milwaukee area, we tested both objective (Area Deprivation Index; ADI) and subjective (perceived neighborhood disadvantage; PND) indicators of community stress as simultaneous predictors of negative and positive affect and the odds of psychological disorder (depression, anxiety) in multilevel models, examining gender differences in these linkages. ResultsResults showed greater objective community stress was related to lower levels of negative affect for both men and women and lower odds of psychological disorder for women specifically. Greater subjective community stress was related to higher levels of negative affect and lower levels of positive affect for both men and women and to higher odds of psychological disorder for women specifically. ConclusionsFindings highlight the complex intersectional nature of the links between community stress and Black Americans' mental-emotional health. Specifically, findings demonstrate the pernicious psychological effects of perceived community stress and allude to Black Americans', particularly women's, active resistance and resilience to objective disadvantage, potentially through investing in social relationships in their neighborhoods.

Some Functional Inequalities and Their Applications on Finsler Measure Spaces

Some Functional Inequalities and Their Applications on Finsler Measure Spaces

A Functional Inequality and a New Class of Probabilities in the N-Person Red-and-Black Game

In this paper, we explore a model of an N-player, non-cooperative stochastic game, drawing inspiration from the discrete formulation of the red-and-black gambling problem, as initially introduced by Dubins and Savage in 1965. We extend upon the work of Pontiggia from 2007, presenting a main theorem that broadens the conditions under which bold strategies by all players can achieve a Nash equilibrium. This is obtained through the introduction of a novel functional inequality, which serves as a key analytical tool in our study. This inequality enables us to circumvent the restrictive conditions of super-multiplicativity and super-additivity prevalent in the works of Pontiggia and others. We conclude this paper with a series of illustrative examples that demonstrate the efficacy of our approach, notably highlighting its ability to accommodate a broader spectrum of probability functions than previously recognized in the existing literature.

Verification Methods for the Lyapunov–Krasovskii Functional Inequalities

Verification Methods for the Lyapunov–Krasovskii Functional Inequalities

Conditions for equality in Anderson’s theorem

The classical Anderson theorem is a well-known result relevant to multivariate statistics and probability. It establishes an integral inequality for symmetric quasiconcave functions over a shifted symmetric convex set. The aim of this note is twofold. First, we provide necessary and sufficient conditions for equality in Anderson’s theorem, extending the result of Soms (1991). Second, we propose a set of tractable requirements that guarantee strict inequality in Anderson’s theorem. Our results are used to characterize the independence of Gaussian-distributed random variables.

Time trends in limited lung function among German middle-aged and older adults

Limited lung function represents a serious health impairment. However, studies investigating changes in limited lung function over time are rare. Thus, the current study investigates time-related changes in limited lung function and potential social inequalities. Data from the 2008 and 2017 waves of the population-based German Aging Survey were used in a repeated cross-sectional study design (N = 8778), including participants aged 40 years and older. Lung function was assessed by the peak flow test. Socio-economic indicators included educational attainment, income and occupational group. Additionally, smoking history, occupational exposure to fumes and gases, and physical exercise were used as potentially explanatory variables for the observed changes. We found that the prevalence of limited lung function decreased strongly over time on a descriptive level from 9.0 to 5.4%. In line with these results, a decreasing trend emerged (OR = 0.48) when controlling for age and gender differences. When additionally controlling for changes in socio-economic indicators and explanatory variables there were still significant decreases over time, but the decline was slightly reduced (OR = 0.57). Moreover, similar significant relative decreases over time occurred for middle-aged and older participants, female and male participants, and those belonging to the different socio-economic groups. Thus, limited lung function generally decreased over time. This decrease could partially be explained by beneficial developments in socio-economic indicators, smoking, occupational exposures, and physical exercise. Future studies might investigate how changes in medicinal treatment and prevention efforts have contributed to the observed beneficial trends in lung health.

The Effectiveness of Early Rehabilitation in Limiting the Progression of Idiopathic Scoliosis.

Background: The purpose of this study was to assess specific rehabilitation methods' effectiveness in early idiopathic scoliosis (IS) development, focusing on lower limb functional inequality's role in scoliosis progression. Materials and Methods: This study comprised 812 patients aged 6-16 years at risk of developing idiopathic scoliosis (IS). The mean (SD) age was 10.66 (3.16) years. Patients were categorized into high- and medium-risk groups based on the angle of trunk rotation (ATR) size. Specific scoliosis physiotherapy was used, and the average follow-up period was 28.1 ± 14.5 months. Changes in ATR, Cobb angle, and functional length of the lower limbs pre- and post-treatment were statistically analyzed across three age groups (6-9, 10-12, and 13-16 years) and three scoliosis locations. Results: Significant effectiveness of early rehabilitation was observed in the high-risk group of children aged 6-9 years. In the medium-risk group, significant reductions in ATR were observed in both the 6-9 and 10-12 age groups across all three scoliosis locations. Additionally, there was a significant decrease in the Cobb angle in the thoracolumbar region and a significant reduction in lower limb inequality across all age groups and scoliosis locations. Conclusions: The early implementation of specific physiotherapy may enhance the efficacy of idiopathic scoliosis treatment by attenuating factors contributing to its progression.

IMPLICIT EXTENDED DISCONTINUOUS GALERKIN SCHEME FOR SOLVING SINGULARLY PERTURBED BURGERS' EQUATIONS

We present the implicit-modal discontinuous Galerkin scheme for solving the coupled viscous and singularly perturbed Burgers’ equations. This scheme overcomes overshoot and undershoots phenomena in the singularly perturbed Burgers’ equations. We present the stability analysis and obtain suitable ranges for penalty terms and time steps. Also, we gain the constant of trace inequality for the approximate function and its first derivatives based on Legendre basis functions. The numerical results have good agreement with the analytical and available approximate solutions.

Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains

Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains

Secure memory adaptive event‐triggered filter design of nonlinear network system under hybrid network attacks with application to tunnel diode circuit

AbstractThis article investigates the problem of secure memory adaptive event‐triggered filter design for nonlinear network systems under hybrid network attacks. First, the introduction of memory adaptive event‐triggered mechanism and numerical quantization is employed to enhance system performance, optimize network channel utilization, and prevent network congestion. Second, we consider the impact of deceptive attacks and DoS attacks occurring in the network channel on data transmission, which form the hybrid network attacks studied in this article. Subsequently, based on the established error model, we derive sufficient conditions for the mean‐square exponential stability of the system under a given performance index using Lyapunov–Krasovskii function and linear matrix inequality (LMI) techniques. Furthermore, the filter parameters are determined using the LMI method. Finally, we illustrate the feasibility and generality of the conclusions through numerical simulation examples and a tunnel diode circuit system.

New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions

Abstract In this study, based on two new local fractional integral operators involving generalized Mittag-Leffler kernel, Hermite-Hadamard inequality about these two integral operators for generalized h h -preinvex functions is obtained. Subsequently, an integral identity related to these two local fractional integral operators is constructed to obtain some new Ostrowski-type local fractional integral inequalities for generalized h h -preinvex functions. Finally, we propose three examples to illustrate the partial results and applications. Meanwhile, we also propose two midpoint-type inequalities involving generalized moments of continuous random variables to show the application of the results.