Jensen's Inequality Research Articles

Jensen's inequalities for pseudo-integrals

In this paper, we introduce a general$(oplus,otimes)$-convex function based on semirings $([a,b],oplus, otimes)$ with pseudo-addition $oplus$ andpseudo-multiplication $otimes.$ The generalization of the finiteJensen's inequality, as well as pseudo-integral with respect to$(oplus,otimes)$-convex functions, is obtained. This also generalizes Jensen's inequalities for Lebesgue integral and the results of Pap and v{S}trboja cite{12}. Meanwhile, we also prove Jensen's inequalities for pseudo-integrals on semirings $([a,b], sup, otimes)$with respect to nondecreasing functions and present correspondingresults for generalized fuzzy integrals.

Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.

Estimation of divergence measures via weighted Jensen inequality on time scales

The main purpose of the presented paper is to obtain some time scale inequalities for different divergences and distances by using weighted time scales Jensen’s inequality. These results offer new inequalities in h-discrete calculus and quantum calculus and extend some known results in the literature. The lower bounds of some divergence measures are also presented. Moreover, the obtained discrete results are given in the light of the Zipf–Mandelbrot law and the Zipf law.

Radical Convex Functions

In this article, we further explore convex functions by revealing new bounds, resulting from stronger convexity behavior. In particular, we define the so-called radical convex functions and study their properties. We will see that such convex functions are bounded above by new curves, rather than straight lines. Applications including discrete and continuous Jensen inequalities, subadditivity behavior, Hermite–Hadamard and integral inequalities will be presented.

New Hadamard-type integral inequalities via a general form of fractional integral operators

• Contain interesting results linking fractional analysis and inequality theory. • Hadamard type results for a new version of the Atangana-Baleanu (AB) integral operator. • Involve simulations for ABK-integraloperators which can be called a new and general version of the AB integral operators. The main motivation in this article is to prove a new and general integral identity and to obtain new integral inequalities of various Hadamard types with the help of this identity. Some basic inequalities such as Hölder, Young, power-mean and Jensen inequality have been used to obtain inequalities, and it has been determined that the main findings are generalizations and repetitions of many results that exist in the literature. Another impressive aspect of the study is that a new version of the Atangana–Baleanu integral operator is used, which is a very useful integral operator. We have given some simulations to demonsrate the consistency and harmony of this interesting operator for different values of the parameters.

The Hermite–Hadamard–Jensen–Mercer Type Inequalities for Riemann–Liouville Fractional Integral

In this paper, we give Hermite–Hadamard type inequalities of the Jensen–Mercer type for Riemann–Liouville fractional integrals. We prove integral identities, and with the help of these identities and some other eminent inequalities, such as Jensen, Hölder, and power mean inequalities, we obtain bounds for the difference of the newly obtained inequalities.

New Hermite\u2013Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense

In this paper, our aim is to consider the new class of log-convex fuzzy-interval-valued function known as log-s-convex fuzzy-interval-valued functions (log-s-convex fuzzy-IVFs). By this concept, we have introduced Hermite–Hadamard inequalities (HH-inequalities) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Moreover, some new Hermite–Hadamard–Fejér inequalities (HH–Fejér-inequalities) and Jensen’s inequalities via log-s-convex fuzzy-IVFs are also established and verified with the support of useful examples. Some special cases are also discussed which can be viewed as applications of fuzzy-interval HH-inequalities. The concepts and approaches of this paper may be the starting point for further research in this area.

New Hermite–Hadamard Inequalities in Fuzzy-Interval Fractional Calculus and Related Inequalities

It is a familiar fact that inequalities have become a very popular method using fractional integrals, and that this method has been the driving force behind many studies in recent years. Many forms of inequality have been studied, resulting in the introduction of new trend in inequality theory. The aim of this paper is to use a fuzzy order relation to introduce various types of inequalities. On the fuzzy interval space, this fuzzy order relation is defined level by level. With the help of this relation, firstly, we derive some discrete Jensen and Schur inequalities for convex fuzzy interval-valued functions (convex fuzzy-IVF), and then, we present Hermite–Hadamard inequalities (HH-inequalities) for convex fuzzy-IVF via fuzzy interval Riemann–Liouville fractional integrals. These outcomes are a generalization of a number of previously known results, and many new outcomes can be deduced as a result of appropriate parameter “γ” and real valued function “Ω” selections. We hope that our fuzzy order relations results can be used to evaluate a number of mathematical problems related to real-world applications.

Enhanced Results on Stability Criteria for Linear Time Delay Systems with Distributed Delay via Relaxed Double Integral Inequality

This paper investigates the matter of stability criteria for linear time delay systems with distributed delay. Firstly, a relaxed double integral inequality is established to estimate the double integral terms appearing within the derivative of Lyapunov-Krasovskii functionals (LKFs) with a triple integral term. Unlike the recently introduced Jensen's inequalities, Wirtinger based integral inequalities, refined Jensen's inequalities and therefore the auxiliary function based integral inequalities the proposed relaxed integral inequality provides large feasible solution region and fewer conservative results. Secondly, by constructing an augmented Lyapunov-Krasovskii functional with a triple integral term, the robust stability criteria for linear time delay systems with distributed delay are given in terms of linear matrix inequalities (LMIs), which may be easily computed by the LMI toolbox of MATLAB. Finally, two numerical examples are performed to indicate the effectiveness of the proposed criterion.

Elastic FemtoCaching: Scale, Cache, and Route

The advent of elastic Content Delivery Networks (CDNs) enable Content Providers (CPs) to lease cache capacity on demand and at different cloud and edge locations in order to enhance the quality of their services. This article addresses key challenges in this context, namely how to invest an available budget in cache space in order to match spatio-temporal fluctuations of demand, wireless environment and storage prices. Specifically, we jointly consider dynamic cache rental, content placement, and request-cache association in wireless scenarios in order to provide just-in-time CDN services. The goal is to maximize the an aggregate utility metric for the CP that captures both service benefits due to caching and fairness in servicing different end users. We leverage the Lyapunov drift-minus-benefit technique and Jensen's inequality to transform our infinite horizon problem into hour-by-hour subproblems which can be solved without knowledge of future file popularity and transmission rates. For the case of non-overlapping small cells, we provide an optimal subproblem solution. However, in the general overlapping case, the subproblem becomes a mixed integer non-linear program (MINLP). In this case, we employ a randomized cache lease method to derive a scalable solution. We show that the proposed algorithm guarantees the theoretical performance bound by exploiting the submodularity property of the objective function and pick-and-compare property of the randomized cache lease method. Finally, via real dataset driven simulations, we find that the proposed algorithm achieves 154% utility compared to similar static cache storage-based algorithms in a representative urban topology.

Event-triggered observer-based H ∞ control of switched linear systems with time-varying delay

This paper is devoted to study the issue of event-triggered observer-based control of switched linear systems subject to time-varying state delay and exogenous disturbance. Firstly, an observer-based event-triggered controller is provided under the proposed event-triggered mechanism. Then, based on the average dwell time scheme and Jensen's Inequality technique, delay-dependent less conservative sufficient conditions in terms of Linear Matrix Inequalities (LMIs) are obtained to guarantee the resulting augmented switched systems is exponential stabilisation and satisfies a prescribed control performance. Furthermore, observer-based event-triggered controller is designed, and it is shown that there exists a positive lower bound of inter-event intervals. Finally, numerical example and comparison results are provided to illustrate the effectiveness and advantages of the proposed method.

Robust passivity analysis of mixed delayed neural networks with interval nondifferentiable time-varying delay based on multiple integral approach

<abstract><p>New results on robust passivity analysis of neural networks with interval nondifferentiable and distributed time-varying delays are investigated. It is assumed that the parameter uncertainties are norm-bounded. By construction an appropriate Lyapunov-Krasovskii containing single, double, triple and quadruple integrals, which fully utilize information of the neuron activation function and use refined Jensen's inequality for checking the passivity of the addressed neural networks are established in linear matrix inequalities (LMIs). This result is less conservative than the existing results in literature. It can be checked numerically using the effective LMI toolbox in MATLAB. Three numerical examples are provided to demonstrate the effectiveness and the merits of the proposed methods.</p></abstract>

Generalized integral inequalities for ABK-fractional integral operators

<abstract> In this paper, we employ new version of the Atangana-Baleanu integral operator namely $ ABK $-fractional integrals to obtain two general integral identities complying second-order derivatives for a given function. Thus allowing us to derive new generalized Hermite-Hadamard type inequalities via $ ABK $- fractional integrals. Moreover we give several new versions of Mid-point and Trapezoid type inequalities by employing Hölder, Young and Jensen inequalities. </abstract>

Extended Dissipativity and Non-Fragile Synchronization for Recurrent Neural Networks With Multiple Time-Varying Delays via Sampled-Data Control

This paper deals with the extended dissipativity and non-fragile synchronization of delayed recurrent neural networks (RNNs) with multiple time-varying delays and sampled-data control. A suitable Lyapunov-Krasovskii Functional (LKF) is built up to prove the quadratically stable and extended dissipativity condition of delayed RNNs using Jensen inequality and limited Bessel-Legendre inequality approaches. A non-fragile sampled-data approach is applied to investigate the problem of neural networks with multiple time-varying delays, which ensures that the master system synchronizes with the slave system and is designed with respect to the solutions of Linear Matrix Inequalities (LMIs). The effectiveness of the suggested approach is established by providing suitable simulations using MATLAB LMI control toolbox. Finally, numerical examples and comparative results are provided to illustrate the adequacy of the planned control scheme.

Finite-Time MixedH∞/Passivity for Neural Networks With Mixed Interval Time-Varying Delays Using the Multiple Integral Lyapunov-Krasovskii Functional

In this article, we consider the finite-time mixed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> /passivity, finite-time stability, and finite-time boundedness for generalized neural networks with interval distributed and discrete time-varying delays. It is noted that this is the first time for studying in the combination of H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> , passivity, and finite-time boundedness. To obtain several sufficient criteria achieved in the form of linear matrix inequalities (LMIs), we introduce an appropriate Lyapunov-Krasovskii function (LKF) including single, double, triple, and quadruple integral terms, and estimating the bound of time derivative in LKF with the use of Jensen's integral inequality, an extended single and double Wirtinger's integral inequality, and a new triple integral inequality. These LMIs can be solved by using MATLAB's LMI toolbox. Finally, five numerical simulations are shown to illustrate the effectiveness of the obtained results. The received criteria and published literature are compared.

Exact converses to a reverse AM-GM inequality, with applications to sums of independent random variables and (super)martingales

For every given real value of the ratio $\mu:=A_X/G_X>1$ of the arithmetic and geometric means of a positive random variable $X$ and every real $v>0$, exact upper bounds on the right- and left-tail probabilities $\mathsf{P}(X/G_X\ge v)$ and $\mathsf{P}(X/G_X\le v)$ are obtained, in terms of $\mu$ and $v$. In particular, these bounds imply that $X/G_X\to1$ in probability as $A_X/G_X\downarrow1$. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function $f=\ln$, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a reverse Jensen inequality for the strictly concave quadratic function $f(x) \equiv -x^2$. As applications of the mentioned new results, improvements of the Markov, Bernstein--Chernoff, sub-Gaussian, and Bennett--Hoeffding probability inequalities are given.

Some Generalizations of Jensen's Inequality

In this article, we give some improvements and generalizations of the famous Jensen's and Jensen-Mercer inequalities for twice differentiable functions, where convexity property of the target function is not assumed in advance. They represent a refinement of these inequalities in the case of convex/concave functions with numerous applications in Theory of Means and Probability and Statistics.

New integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators

• Contain interesting results linking fractional analysis and inequality theory. • Hadamard type results in inequality theory for the Atangana-Baleanu integral operator. • New approaches and bounds for Hadamard-type inequalities. • New and motivated inequalities derived from the help of Jensen, Hólder and Young inequalities. Inequalities, including fractional integrals, have become a very popular method and have been the main motivation point for many studies in recent years. Studies have been carried out for many types of inequality, thereby introducing a new trend in inequality theory. In this study, new inequalities of Hermite-Hadamard type were obtained by using Atangana-Baleanu integral operators, which provide very useful and effective results with their use in fields such as fractional analysis, applied mathematics, mathematical biology and engineering. The fact that the main results were obtained for functions whose absolute value of the second derivative is convex. In the study, we have proved a new identity for twice differentiable convex functions and the modified version of the integral identity was given in the last section.

Robust processing of airborne laser scans to plant area density profiles

Abstract. We present a new algorithm for the estimation of the plant area density (PAD) profiles and plant area index (PAI) for forested areas based on data from airborne lidar. The new element in the algorithm is to scale and average returned lidar intensities for each lidar pulse, whereas other methods do not use the intensity information at all, use only average intensity values, or do not scale the intensity information, which can cause problems for heterogeneous vegetation. We compare the performance of the new algorithm to three previously published algorithms over two contrasting types of forest: a boreal coniferous forest with a relatively open structure and a dense beech forest. For the beech forest site, both summer (full-leaf) and winter (bare-tree) scans are analyzed, thereby testing the algorithm over a wide spectrum of PAIs. Whereas all tested algorithms give qualitatively similar results, absolute differences are large (up to 400 % for the average PAI at one site). A comparison with ground-based estimates shows that the new algorithm performs well for the tested sites. Specific weak points regarding the estimation of the PAD from airborne lidar data are addressed including the influence of ground reflections and the effect of small-scale heterogeneity, and we show how the effect of these points is reduced in the new algorithm, by combining benefits of earlier algorithms. We further show that low-resolution gridding of the PAD will lead to a negative bias in the resulting estimate according to Jensen's inequality for convex functions and that the severity of this bias is method dependent. As a result, the PAI magnitude as well as heterogeneity scales should be carefully considered when setting the resolution for the PAD gridding of airborne lidar scans.

Hankel Norm Model Reduction of Discrete-Time Interval Type-2 T–S Fuzzy Systems With State Delay

This article focuses on the model reduction problem of discrete-time time-delay interval type-2 Takagi-Sugeno (T–S) fuzzy systems. Compared with the type-1 T–S fuzzy system, the interval type-2 T–S fuzzy system has more advantages in expressing nonlinearity and capturing uncertainties. In addition, in order to simplify the analysis process, complex high-order systems can be approximated as low-order systems, which is called model reduction. In previous studies, there are few researches on model reduction of the interval type-2 T–S fuzzy system with time delay. Hankel norm is adopted to limit the error after model reduction. Based on Jensen's inequality, a linear matrix inequality (LMI) condition for the Hankel norm performance of the error system is obtained. A membership-function-dependent method based on piecewise linear membership functions is utilized to deal with mismatched membership functions where information of membership functions will be used for relaxing analysis results. Next, by a convex linearization design, the model reduction problem is formulated as a convex LMI feasibility/optimization condition. Numerical examples are given to verify the validity of the analysis.