This article explores a modern counterpart to the classical Jensen integral inequality for integrals, which provides an upper bound for convex functions evaluated at an integral. We extend this result to more general settings involving sums and products of integrals of multiple functions. Full details of the proofs are provided, and some examples illustrate the theory.

Abstract In this paper, the problem of exponential stability is studied for a class of integral systems with multiple delays. A novel integral inequality with multiple weighting matrices is first proposed. It is theoretically shown that the newly derived integral inequality provides a tighter lower bound than existing ones based on variants of the conventional Jensen integral inequality. By utilizing the newly derived integral inequality and by constructing an improved Lyapunov–Krasovskii functional, tractable delay‐dependent conditions are derived to ensure exponential stability of the system with a prescribed decay rate. An application to the design problem of a predictor‐based controller for input delayed systems is also presented. Numerical examples are given to illustrate the effectiveness and significant improvement of the obtained results.

ABSTRACTIn this paper, the problem of weighted L2‐gain analysis and anti‐windup (AW) fault tolerant control for a class of time varying delay uncertain saturated switched systems with actuator faults and external disturbances is studied by using the minimum dwell time method. First, the maximum allowable disturbance capability of the closed‐loop system is obtained by combining a sector nonlinear condition, an augmented Lyapunov–Krasovskii functional, the Jensen integral inequalities and the free weighting matrix method. Then the weighted L2‐gain of the closed‐loop system is analyzed based on this condition. Next by designing the AW fault tolerant controller, the maximum allowable disturbance capability of the closed‐loop system as well as the minimum weighted L2‐gain upper bound is obtained. Finally, a numerical simulation example is given to verify the effectiveness of the proposed method in this paper.

The aim of this paper is to integrate the concept of Ostrowski inequality with the Atangana–Baleanu fractional integral operator. First, we derive an equality that involves the Atangana–Baleanu fractional integral operator. Next, using this equality, we introduce novel generalizations of Ostrowski-type inequalities by applying several key mathematical tools like Hölder’s inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality all applied to the preinvexity of [Formula: see text]. Furthermore, several special cases are deduced from the main findings, contributing to the advancement of fractional calculus in the context of convexity and integral inequalities. These results offer potential applications in various fields where fractional calculus plays a significant role.

Fractional integral operators and convexity have a close link due to their fascinating properties in the mathematical sciences.In this paper, we first establish an integral identity involving the generalized Hattaf-fractional integral operators. By using the Jensen integral inequality, Young's inequality, power-mean inequality, and H\"{o}lder inequality, we then apply this identity to provide some new generalizations of Ostrowski type inequality for the convexity of $|\aleph|$. Furthermore, we deduce several special cases from the main results. The results of this novel investigation should lead to new discoveries in the area of fractional calculus and inequalities.

In studies of inequality theory, integral identities are developed to support numerous inequalities. Various fractional integral and derivative operators have been employed recently to accomplish these identities. In this article, we first establish an integral identity by employing Hattaf fractional integral operators. Then, we use this identity to give some novel generalizations of integral inequalities for the convexity of |ℵ| using the Jensen integral inequality, Young's inequality, power-mean inequality, and Hölder inequality. The main motivating goal of this study is to use Hattaf-fractional integral operators with strong kernel structure to derive new and general form of integral inequalities.

Advancements in integral inequalities of Ostrowski type via modified Atangana-Baleanu fractional integral operator

This study is dedicated to a comprehensive exploration aimed at advancing our understanding of stability within dynamic systems. The focus is particularly on the intricate domain of delayed systems characterized by gapped gamma distributions. The primary objective of this investigation revolves around evaluating the pragmatic application and efficacy of Jensen's integral inequality in combination with the powerful analytical tools provided by Linear Matrix Inequalities (LMIs). This evaluation is crucial for rigorously assessing exponential stability within these complex systems. Central to our investigative framework is the strategic deployment of augmented Lyapunov functions. These functions play a crucial role in unraveling the intricate stability properties of delayed systems featuring gapped gamma distributions, allowing for a nuanced examination of their inherent stability characteristics under various conditions. The mathematical formulation crafted in this exploration intricately captures the interplay between the distinctive attributes of the gapped gamma distribution and the complex dynamics of the loop traffic flow model within the overarching delayed system. This interconnection serves as the fundamental basis for the stability analysis, providing insights into the interdependence of these key elements. The noteworthy contribution of this study lies in the systematic construction of a robust analytical framework explicitly tailored for stability assessment. A comprehensive investigation is undertaken to elucidate critical aspects, including the convergence rate and the attainment of asymptotic stability within the considered delayed system. Additionally, a dedicated simulation section, focusing on Vehicle Active Suspension Control, has been incorporated to further validate and showcase the applicability of the proposed methodology.

On the generalization of Hermite-Hadamard type inequalities for [formula omitted]-convex function via fractional integrals

Abstract This paper is concerned with the problem of proportional-integral tracking control of a two-stage chemical reactor system subject to time delays, disturbances, uncertainties, and input quantization. In this work, an improved equivalent-input-disturbance estimator is incorporated into the proportional-integral tracking control system to compensate for the disturbances in the addressed model. Moreover, to minimize the communication congestion in the control networks, the quantized control input signals are considered while designing the controller. Further, a robust stability condition for the addressed system is established in the form of linear matrix inequalities by employing asymmetric Lyapunov–Krasovskii functional and Jensen's integral inequalities. Moreover, in accordance with the derived conditions, the control and observer gain matrices are determined. Finally, a numerical example is provided to demonstrate the validity of the proposed control scheme.

In this paper, the problem of robust stability analysis for a type of uncertain stochastic switched inertial neural networks (SSINNs) with time‐varying delay is investigated. First, the original second‐order system is converted into first‐order differential equations using the variable transformation method. Next, some sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained using Lyapunov‐Krasovskii functional (LKF), state‐dependent switching (SDS) method, and Jensen's integral inequality for estimating integral inequalities so that the augmented system is robust, global, and asymptotically stable in the mean square for the uncertain SSINNs with time‐varying delay. It is shown that the stability of the above considered system composed of all unstable subsystems can be achieved by using the SDS law. Finally, two numerical simulations are provided to demonstrate the effectiveness of the proposed SDS law.

<abstract><p>For a neutral system with mixed discrete, neutral and distributed interval time-varying delays and nonlinear uncertainties, the problem of exponential stability is investigated in this paper based on the $ H_\infty $ performance condition. The uncertainties are nonlinear time-varying parameter perturbations. By introducing a decomposition matrix technique, using Jensen's integral inequality, Peng-Park's integral inequality, Leibniz-Newton formula and Wirtinger-based integral inequality, utilization of a zero equation and the appropriate Lyapunov-Krasovskii functional, new delay-range-dependent sufficient conditions for the $ H_\infty $ performance with exponential stability of the system are presented in terms of linear matrix inequalities. Moreover, we present numerical examples that demonstrate exponential stability of the neutral system with mixed time-varying delays, and nonlinear uncertainties to show the advantages of our method.</p></abstract>

Some generalizations of reverse Hardy-type inequalities via Jensen integral inequality on time scales

In this article, first, we deduce an equality involving the Atangana–Baleanu (AB)-fractional integral operator. Next, employing this equality, we present some novel generalization of Ostrowski type inequality using the Hölder inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality for the convexity of |Υ|. We also deduced some new special cases from the main results. There exists a solid connection between fractional operators and convexity because of their fascinating properties in the mathematical sciences. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. It is assumed that the results presented in this article will show new directions in the field of fractional calculus.

We introduce new normalized fractional integral concepts on random variables. We also introduce the fractional coupled random variables. The Jensen integral inequality is generalized. Also, some fractional bounds estimating the expectation and variance are delivered. At the end, some other results related to the coupled random variables are proved. For this work, some results of the papers [On some statistical and probability inequalities. Journal of Inequalities and Special Functions, 2016] and [Some inequalities for the expectation and variance of a random variable whose PDFs are absolutely continuous using a pre-Chebychev inequality. Tamkang Journal of Mathematics, 2001] are deduced as some special cases.

Reverse Jensen Integral Inequalities for Operator Convex Functions in Terms of Fréchet Derivative

The integral inequalities concerning the inverse Hardy inequalities have been studied by a large number of authors during this century, of these articles have appeared, the work of Sulaiman in 2012, followed by Banyat Sroysang who gave an extension to these inequalities in 2013. In 2020 B. Benaissa presented a generalization of inverse Hardy inequalities. In this article, we establish a new generalization of these inequalities by introducing a weight function and a second parameter. The results will be proved using the Hölder inequality and the Jensen integral inequality. Several the reverses weighted Hardy’s type inequalities and the reverses Hardy’s type inequalities were derived from the main results.

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.

This paper investigates the stability problem of neural networks (NNs) with time-varying delay. Firstly, a new augmented vector and suitable Lyapunov–Krasovskii Functional (LKF) considering activation function are constructed by using more information of time delay. Secondly, a generalized free-weighting matrix integral inequality (GFMII) is chosen to estimate the derivative of single integral terms more accurately. Meanwhile, Jensen integral inequality and improved convex combination are combined to estimate integral terms with activation function; as a result, a novel stability criterion with less conservatism is established. Finally, two numerical examples are employed to illustrate the effectiveness of proposed methods.

We establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville, k-fractional, Hadamard fractional, Katugampola fractional, (k,s)-fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes.