Inequalities For Functions Research Articles

Sharp coefficient inequalities of starlike functions connected with secant hyperbolic function

This article comprises the study of class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document} that represents the class of normalized analytic functions f satisfying ςf′(z)/f(ς)≺sech(ς)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\varsigma \\mathsf{f}}^{\\prime }(z)/\\mathsf{f}( {\\varsigma })\\prec \\sec h ( \\varsigma ) $\\end{document}. The geometry of functions of class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document} is star-shaped, which is confined in the symmetric domain of a secant hyperbolic function. We find sharp coefficient results and sharp Hankel determinants of order two and three for functions in the class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document}. We also investigate the same sharp results for inverse coefficients.

On the gradient rearrangement of functions

AbstractIn this paper, we introduce a symmetrization technique for the gradient of a $$\text {BV}$$ BV function, which separates its absolutely continuous part from its singular part (sum of jump and Cantorian part). In particular, we prove a $$\text {L }^{\text {1}}$$ L 1 comparison between the function and the symmetrization just mentioned. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.

The Stability of Multi-Coefficients Pexider Additive Functional Inequalities in Banach Spaces

The Hyers–Ulam stability of multi-coefficients Pexider additive functional inequalities in Banach spaces is investigated. In order to do this, the fixed point method and the direct method are used.

Sharp spectral gap estimates for higher-order operators on Cartan–Hadamard manifolds

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc. 139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. 367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

A recurrent neural network approach for nonconvex interval quadratic programming

In the paper, a new approach is presented to tackle an interval optimization problem with nonconvex interval objective function and affine interval equality and inequality constraints utilizing a recurrent neural network. The approach involves converting the nonconvex interval quadratic programming into a cubic optimization via convex combinations and considering its weight as a decision variable. The recurrent neural network is then established to solve the cubic optimization through subgradient techniques and an exterior penalty function method. Moreover, it is proven that there is a unique global solution to the recurrent neural network, and its trajectory can reach the feasible region in a finite time and remain there. Additionally, it is demonstrated that the trajectory of recurrent neural network converges exponentially or in finite time towards a singleton belonging to the set of critical points of the cubic optimization. This approach is beneficial for interval quadratic programming problems where the weight is unknown or challenging to determine in practical applications. Finally, the effectiveness of this approach is verified through two numerical examples.

Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\varepsilon }$$-range

Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\varepsilon }$$-range

Hecke-type series involving infinite products

In this paper, we study Hecke-type series involving infinite products. In particular, we establish some Hecke-type series involving infinite products and then obtain truncated versions of these series as well as truncated forms of some other known series of such types. Finally, as an application, we deduce six infinite families of inequalities for various partition functions. Our proofs of the main results heavily rely on a formula from the work of Liu (2013).

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