Jensen's Inequality Research Articles

What are the key stability challenges in high-bandwidth, non-minimum phase systems with time-varying, and non-smooth delays?

The analysis and control of stability in high-bandwidth systems characterized by non-minimum phase delays represent a formidable challenge within the realm of control theory and engineering. This research aims to address the pivotal question of whether it is feasible to enhance the stability of such intricate systems. These systems inherently possess uncertain and swiftly changing delay characteristics, rendering them exceptionally demanding to control effectively. In the course of this investigation, we embark on a comprehensive exploration of the theoretical underpinnings of the stability of high-bandwidth, non-minimum phase delay systems. This encompassing inquiry encompasses a meticulous consideration of both derivative-delay and piecewise continuous delay components. To underpin our analysis, we judiciously incorporate feedback mechanisms, drawing upon mathematical tools such as the Jensen inequality and Lyapunov-based methodologies to rigorously establish stability conditions. Furthermore, our exploration extends to encompass the concept of input-output stability and complements it with the notion of asymptotic stability, thereby ensuring that the systems in question exhibit uniform stability across diverse temporal domains. The outcomes of our investigation furnish compelling evidence that by harnessing the power of discrete-time Lyapunov-Krasovskii functionals, it becomes conceivable to circumscribe the maximum delay within predefined thresholds. This achievement holds the promise of enhancing stability in non-minimum phase delay systems characterized by high bandwidth. These findings have far-reaching implications, profoundly influencing the design and control paradigms across a spectrum of engineering applications. Notably, this impact extends to areas such as communication networks, real-time control systems, and robotics, where the mitigation of instability due to non-minimum phase delays has been an enduring challenge.

On further refinements of the Jensen inequality and applications

We aim to extend some recent refinements of the Jensen inequality. First, we establish even more precise estimates for the Jensen inequality. Then, we derive an improvement of the Jensen inequality with regard to ‐convexity. As an application, our main results are applied in deriving refinements of power mean inequalities and the Hölder inequality. Finally, we obtain more accurate estimates for ‐divergences, important in the information theory.

Further Results on the Input-to-State Stability of a Linear Disturbed System with Control Delay

In this paper, a theorem is obtained that gives sufficient input-to-state stability conditions for linear systems with control delay and additive disturbances. Stabilizing feedback is considered available in the absence of delay and disturbances. The mathematical tools are the Lyapunov–Krasovskii functional, the Jensen inequality and the double Hadamard inequality. The critical delay is highlighted.

Generalized integral Jensen inequality

In this paper we introduce necessary and sufficient conditions for a real-valued function to be preinvex. Some properties of preinvex functions and new versions of Jensen’s integral type inequality in this setting are given. Several examples are also involved.

Stability Analysis of Discrete-Time Neural Networks With a Time-Varying Delay: Extended Free-Weighting Matrices Zero Equation Approach.

This research investigates the stability of discrete-time neural networks (DNNs) with a time-varying delay by using the Lyapunov-Krasovskii functional (LKF) method. Recent researches acquired some less conservatism stability criteria for time-varying delayed systems via some augmented LKFs. However, the forward difference of such LKFs resulted in high-degree time-varying delay-dependent polynomials. This research aims to develop some augmented state-related vectors and the corresponding extended free-weighting matrices zero equations to avoid the appearance of such high-degree polynomials and help to provide more freedom for the estimation results. Besides, an augmented delay-product-type LKF is also established for ameliorating the stability conditions of the time-varying delayed DNNs. Then, based on the above methods and Jensen's summation inequality, the auxiliary-function-based summation inequality, and the reciprocally convex matrix inequality, some less conservatism stability criteria for time-varying delayed DNNs are formulated. The validity of the proposed time-varying delay-dependent stability criteria is illustrated by two numerical examples.

Inequalities for Convex Functions and Isotonic Sublinear Functionals

In this paper, versions of the famous Jensen inequality for sublinear isotonic functionals are proved. The obtained results generalize classic Jessen’s and McShane’s inequalities. Applications to generalized means and to Hölder’s and Minkowski’s inequalities are also given.

Time series regression models for zero-inflated proportions

Time series of proportions are often encountered in applications such as ecology, environmental science and public health. Strategies for such data include linear regression after logistic transformation. Though easy to fit, the transformation approach renders covariate effects uninterpretable on the scale on which they were observed owing to Jensen's inequality. An alternative to the transformation approach has been to directly model the response via the beta distribution. In this paper, we extend zero-inflated beta regression models for independent proportions to time series data that is bounded over the unit interval and that may take on zero values. Estimation is within the partial-likelihood framework and is computationally feasible to implement. We outline the asymptotic theory of our maximum partial likelihood estimators under mild regularity conditions and investigate their bias and variability using simulation studies. The utility of our method is illustrated using two real data examples.

Refinement of the classical Jensen inequality using finite sequences

This article is dedicated to a refinement of the classical Jensen inequality by virtue of some finite real sequences. Inequalities for various means are obtained from this refinement. Also, from the proposed refinement, the authors acquire some inequalities for Csiszâr $\Psi$- divergence and for Shannon and Zipf-Mandelbrot entropies. The refinement is further generalized through several finite real sequences.

Necessary and sufficient conditions for discrete inequalities of Jensen–Steffensen type with applications

AbstractIn this paper we give a necessary and sufficient condition for the discrete Jensen inequality to be satisfied for real (not necessarily nonnegative) weights. The result generalizes and completes the classical Jensen–Steffensen inequality. The validity of the strict inequality is studied. As applications, we first give the form of our result for strongly convex functions, then we study discrete quasi-arithmetic means with real (not necessarily nonnegative) weights, and finally we refine the inequality obtained.

Further improvements of the Jensen inequality in the integral sense by virtue of 6-convexity along with applications

<abstract><p>The Jensen inequality is of fundamental importance because of its influential and interesting consequences. In recent years, the Jensen inequality has been supposed to be the most engaging source for research. We present interesting improvements to the continuous version of Jensen's inequality through the application of the concept of 6-convexity. For real visualization and comparison to other results, some numerical experiments were provided. With the aid of the acquired results, improvements for the Hermite-Hadamard and Hölder inequalities were presented. Some relationships between the means were granted as applications of established improvements. In addition, some estimations of the Csiszár divergence and its associated cases were received as further applications of the obtained results. The major techniques employed in formulating the proposed improvements included the Jensen inequality and the concept of convexity.</p></abstract>

Multiple-term improvements of Jensen's inequality for (p,h)-convex and (p,h)-log convex functions

Multiple-term improvements of Jensen's inequality for (p,h)-convex and (p,h)-log convex functions

Prefixed-Time Local Intermittent Sampling Synchronization of Stochastic Multicoupling Delay Reaction-Diffusion Dynamic Networks.

This article focuses on the problem of prefixed-time synchronization for stochastic multicoupled delay dynamic networks with reaction-diffusion terms and discontinuous activation by means of local intermittent sampling control. Notably, unlike the existing common fixed-time synchronization, this article puts forward a new synchronization concept, prefixed-time synchronization, based on the fact that stochastic noise and discontinuous activation can be seen everywhere in practical engineering, which can effectively perfect and improve the existing works. Specifically, a local intermittent in the time domain and point sampling control strategy in the spatial domain is proposed instead of a simple single intermittent control approach, which greatly reduces the control cost. In addition, by some effective means, including the famous Young's inequality, Jensen's inequality, and Hölder's inequality, we obtain two different synchronization criteria of the networks without delay and with multicoupling delays and deeply reveal the quantitative relationship among control period, point sampling length, and network scale. Finally, a numerical example is given to verify the effectiveness of the developed method and the practicability by Chua's circuit model.

Inequalities for functions convex on the coordinates with applications to Jensen and Hermite-Hadamard type inequalities, and to new divergence functionals

Inequalities for functions convex on the coordinates with applications to Jensen and Hermite-Hadamard type inequalities, and to new divergence functionals

Strongly convex functions and extensions of related inequalities with applications to entropy

<abstract><p>We extended the Mercer inequlaity, Fejér-Hermite-Hadamard, and Jensen inequalities for strongly convex functions. Moreover, we obtained several results in information theory and mathematical analysis using obtained inequalities.</p></abstract>

Application of the fink identity to Jensen-type inequalities for higher order convex functions

The focus of this paper is the application of the Fink identity in obtaining Jensen-type inequalities for higher order convex functions. In addition to the basic form, we establish superadditivity and monotonicity relations that correspond to the Jensen inequality in this setting. We also obtain the corresponding Lah-Ribaric inequality. The obtained results are valid for functions of even degree of convexity. With this method, we derive some new bounds for the differences of power means, as well as some new H?lder-type inequalities.

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.

Existence results for the [formula omitted]-Hessian type system with the gradients via [formula omitted]-monotone matrices

In this work, we deal with the k-Hessian type system with the gradients SkσD2ui+α|∇ui|I=φi(|x|,−u1,−u2,…,−un), inΩ,ui=0, on∂Ω,i=1,2,…,n,where α≥0, n≥2, 1≤k≤N is a positive integer, I is the identity matrix and Ω stands for the open unit ball in RN(N≥2). Based on appropriate assumptions about φi(i=1,2,…,n), some results regarding existence of negative k-convex radial solution are established. More precisely, at least one solution and at least two solutions are obtained via the R+n-monotone matrices and the fixed point theory. Some basic inequality techniques such as Jensen inequality are applied, which allows us to overcome the difficulty that the k-Hessian type operator is related to the gradient terms. Finally, several examples are provided to show the validity of our main results.

New upper bounds on the Gaussian Q‐function via Jensen's inequality and integration by parts, and applications in symbol error probability analysis

AbstractUsing Jensen's inequality and integration by parts, some tight upper bounds are derived on the Gaussian Q‐function. The tightness of the bounds obtained by Jensen's inequality can be improved by increasing the number of exponential terms, and one of them is invertible. A piece‐wise upper bound is obtained and its application in the analysis of the symbol error probability of various modulation schemes in different channel models is shown.

Improvements of Jensen's inequality and its converse for strongly convex functions with applications to strongly f-divergences

In this paper, using the class of strongly convex functions, which is subclass of convex functions with stronger versions of analogous properties, we get improvements of Jessen's and Jensen's inequalities, their converses as well as related Jensen-type interpolating inequalities which present a starting point in many significant results in recent investigations. Obtained improvements we apply to so called strongly f-divergences, a concept of f-divergences for strongly convex functions. As outcome we derive stronger estimates for some well known divergences as the Kullback-Leibler divergence, χ2-divergence, Hellinger divergence, Bhattacharya distance and Jeffreys distance.

Refinements of discrete and integral Jensen inequalities with Jensen’s gap

Motivated by a paper of Dragomir, we give new refinements for both discrete and integral Jensen inequalities using the Jensen’s gap. As applications, we give refinements of various inequalities verifiable by Jensen’s inequality. Topics covered: norms, quasi-arithmetic means, Hölder’s inequality and f-divergences in information theory.