Integral Inequalities Research Articles

New Class of Multivalent Functions Defined by Generalized (p,q)-Bernard Integral Operator

Making use of the generalized $(p, q)$-Bernardi integral operator, we introduce and study a new class $\mathcal{F J}_{p, q}^m(\alpha, \delta, \lambda, \gamma)$ of multivalent analytic functions with negative coefficients in the open unit disk $E$. Several geometric characteristics are obtained, like, coefficient estimate, radii of convexity, close-to-convexity and starlikeness, closure theorems, extreme points, integral means inequalities, neighborhood property and convolution properties for functions belonging to the class $\mathcal{F} \mathcal{J}_{p, q}^m(\alpha, \delta, \lambda, \gamma)$.

On fractional inequalities on metric measure spaces with polar decomposition

Abstract In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1<p\leq q<\infty is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.

A general form of Gronwall inequality with Stieltjes integrals

We present a new Gronwall inequality for Stieltjes integrals, which improves numerous existing results, and has a simple proof based on the quotient rule for Stieltjes integrals. As an application, we obtain uniqueness theorems for measure differential equations and nabla dynamic equations. Finally, we revisit the topic from the perspective of Stieltjes derivatives.

Quantum symmetric integral inequalities for convex functions

In the context of quantum symmetric calculus, this study proposed more refined version of Ostrowski and Hermite–Hadamard type inequalities. The function involved in these inequalities are convex functions. In order to reach the target, left and right quantum symmetric derivative and corresponding integral are used. Furthermore, the Hölder inequality is established in the frame work of left and right quantum symmetric integral. The new results refined the results about integral inequalities that exist in the literature.

Hermite-Hadamard Inequalities for a New Class of Generalized-(s,m) via Fractional Integral

This paper defines a new generalized (s,m)-σ convex function using the σ convex functions and provides some applications and exact results for this kind of functions. The new definition of the (s,m)-σ convex function class is used to obtain the Hermite Hadamard type integral inequalities existing in the literature, and new integral inequalities are obtained with the help of the σ-Riemann-Liouville fractional integral. Additionally, a new Hermite-Hadamard type fractional integral inequality is constructed using the σ-Riemann-Liouville fractional integral.

Takagi-Sugeno fuzzy sampled-data observer design for nonlinear permanent magnet synchronous generator-based wind turbine systems using auxiliary function-based integral inequality technique

This study focuses on designing the fuzzy sampled-data observer (FSDO) scheme for nonlinear permanent magnet synchronous generator (PMSG)-based wind turbine systems (WTS). The primary objective of this problem is to solve an unmeasurable state problem under the mismatched premise variables conditions among system, observer, and controller, respectively. To do this, firstly, the proposed wind turbine model is described in a Takagi-Sugeno fuzzy model due to the complex nonlinearities in PMSG-based WTS. Next, based on the measured output signals and sampled estimated states, the FSDO is designed to understand some unmeasurable state variables of the studied system. Then, a novel auxiliary function-based integral inequality (AFBII) is presented to approximate the integral quadratic terms related to sampling information. Thanks to the proposed AFBII technique and the introduction of slack variables, several new and less conservative stability requirements are derived in the formulations of linear matrix inequalities (LMIs) using the looped Lyapunov function. Finally, the simulation studies for the considered wind turbine model are validated numerically, and then some comparative results are provided to show the superiority and applicability of the theoretical observations.

New integral inequalities for synchronous functions via Atangana–Baleanu fractional integral operators

Chebyshev’s inequality is a well-known inequality in the field of inequality theory and yields remarkable results for functions that exhibit synchronous behavior. The study, designed to obtain Chebyshev type inequalities with the help of the Atangana–Baleanu fractional integral operator, which has become the most popular concept of fractional analysis in recent years, aims to obtain general forms for synchronous functions. In the process of obtaining results for synchronized functions within the scope of the study, functional tools such as Atangana–Baleanu fractional integral operator, various mathematical analysis operations and induction were used. The most striking aspect of the findings is that for some special values of the parameter of the fractional integral operator, some of the results can be reduced to the Chebyshev inequality. Compared to the studies in the literature, this study has a unique aspect in that the findings obtained with the help of the Atangana–Baleanu fractional integral operator contain general forms, as well as bringing together a popular concept of synchronous functions and fractional analysis.

Hermite-Hadamart Integral Inequality for Differentiable m- Convex Functions

Convexity in connection with integral inequalities is an interesting research domain in recent years. The convexity theory plays a fundamental role in the development of various branches of applied sciences since it includes the theory of convex functions that possesses two important attributes viz. a boundary point is where the maximum value is reached and any local minimum value is a global one. Convexities and inequalities are connected which has a basic character in many branches of pure and applied disciplines. The most important inequality related to convex function is the Hermite-Hadamard integral inequality. The extensions, enhancements and generalizations of this inequality has motivated the researchers in recent years. This paper is an extension of some inequalities connected with difference of the left-hand part as well as the right-hand part from the integral mean in Hermite- Hadamard’s inequality for the case of m- convex functions.

Some new (p,q)‐Hadamard‐type integral inequalities for the generalized m‐preinvex functions

This paper introduces and establishes new Hermite‐Hadamard type inequalities specifically tailored for ‐preinvex functions within the ‐calculus framework. These newly developed inequalities come with accompanying left‐right estimates, which enhance their practical utility. The primary objective of this research is to investigate the properties of ‐differentiable ‐preinvex functions and derive inequalities that extend and generalize existing results in the domain of integral inequalities. The techniques employed in this study hold broader implications, finding relevance in various fields where symmetry is paramount. The findings presented in this paper make a significant contribution to the field of analytic inequalities, offering valuable insights into the behavior and characteristics of ‐preinvex functions. Moreover, the established results demonstrate the wider applicability and generalization of analogous findings from prior literature. The techniques and inequalities introduced herein pave the way for further exploration and research in the realm of integral inequalities.

On existence of extremal integrable solutions and integral inequalities for nonlinear Volterra type integral equations

We prove the existence of maximal and minimal integrable solutions of nonlinear Volterra type integral equations. Two basic integral inequalities are obtained in the form of extremal integrable solutions which are further exploited for proving the boundedness and uniqueness of the integrable solutions of the considered integral equation.

Exponential State Estimation for Delayed Competitive Neural Network Via Stochastic Sampled-Data Control with Markov Jump Parameters Under Actuator Failure

Abstract This article examines the problem of estimating the states of Markovian jumping competitive neural networks, where the estimation is done using stochastic sampled-data control with time-varying delay. Instead of continuously measuring the states, the network relies on sampled measurements, and a sampled-data estimator is proposed. The estimator uses probabilistic sampling during two sampling periods, following a Bernoulli distribution. The article also takes into account the possibility of actuator failure in real systems. To ensure the exponentially mean-square stability of the delayed neural networks, the article constructs a Lyapunov-Krasovskii functional (LKF) that includes information about the bounds of the delay. The sufficient conditions for stability are derived in the form of linear matrix inequalities (LMIs) by employing modified free matrix-based integral inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.

Hermite-Hadamard inequalities for generalized (m-F)-convex function in the framework of local fractional integrals

This work presents new versions of the Hermite-Hadamard Inequality, for (m−F)-convex functions, defined on fractal sets Rς ( 0 ς ≤ 1). So, we show some new results for twice differentiable functions using local fractional calculus, as well as some new definitions. We will construct these new integral inequality using the generalized H¨older-integral inequality and the power mean integral inequality. Furthermore, we present some new inequalities for the midpoint and trapezoid formulas in a novel type of fractal calculus. The conclusions in this paper are substantial advancements and generalizations of prior research reported in the literature.

Applications of Differential Inequalities Employing A New ConvolutedOperator Constructed by The Supertrigonometric Function

In order to examine various geometric features, we expanded the supertrigonometric function (STF) and superhyperbolic function (SHF) into the open unit disk. A convolution differential operator of the STF provides the formulas. The suggested operator works with both integral and double differential inequalities. As a result, we present a collection of findings that includes recent works. The idea ofsubordination and superordination serves as a guide for our method, and we then developed the primary conclusion as a double side’s theorem.

A Simons-type Integral Inequality for Minimal Surfaces with Constant Kähler Angle in Complex Projective Spaces

A Simons-type Integral Inequality for Minimal Surfaces with Constant Kähler Angle in Complex Projective Spaces

Hybrid-driven based reliable control for uncertain semi-Markov jump cyber–physical systems via two-sided looped-functional

In this paper, the problem of input–output finite-time mean square stabilisation for semi-Markov jump cyber–physical systems is addressed under the frame of generally uncertain transition rates through the hybrid-triggering-based reliable control technique. And, the undertaken system is susceptible to several vulnerable elements such as linear fractional uncertainty, actuator faults, deception attacks and external disturbances. Moreover, a hybrid-triggered approach that fuses the both time-triggered and event-triggered strategies is implemented for the addressed system which significantly reduces the number of unimportant communication transmissions inside the network. Subsequently, a reliable control law is configured by utilising the established hybrid-trigger mechanism and taking into account the actuator fault. In addition, the deception attacks and hybrid-trigger mechanism are characterised by the stochastic variables that adhere to the Bernoulli distribution. Furthermore, by framing a mode-dependent two-side looped Lyapunov–Krasovskii functional and deploying the advanced integral inequalities, adequate criteria are developed within the context of linear matrix inequalities which ensures that the desired result. After that, the controller gain values is determined based on the inferred appropriate criteria. Ultimately, the simulation results, including the RLC circuit series, are shown in order to verify the suitability and potential of the demonstrated findings.

Design and Simulation of Torque Control for Hot Rolling Steel Mill

Lateral movement of a strip in the hot rolling process is an important unsolved technical problem. To minimize the strip thickness variations, the state space model formulation is usually used. It results in reducing productivity and, in the extreme case, strip tearing during tailing out at the finishing mill. This movement is caused by various asymmetric factors such as mechanical asymmetry of the mill and different temperatures along the strip width direction. Furthermore, the difference in the temperature and the thickness of steel strip resulting in deformation of the steel, which affects the bending torque of the steel strip. This study aims to understand the influences of these various factors on lateral movement. In order to reduce the lateral movement and the disturbance terms, that act on uncertain inter-connected systems, a new control approach, based on a combination of Integral Sliding Mode Control (ISMC) and Linear Matrix Inequalities (LMI) technique, is proposed. Simulation results of hot rolling system with three interconnected systems are presented to validate the new controller performance.

Exponential stability of non-instantaneous impulsive second-order fractional neutral stochastic differential equations with state-dependent delay

The aim of this manuscript mainly concerned with a new class of exponential stability for non-instantaneous impulsive second-order fractional neutral stochastic differential equations (NIIFNSDEs) with state-dependent delay driven by Poisson jump in a separable Hilbert spaces. Firstly, a more appropriate concept of mild solution is introduced. Secondly, sufficient conditions are derived for the existence of mild solution by means stochastic analysis, fractional calculus approach and Mönch fixed point theorem with appropriate hypotheses on non-linear continuous functions combined with solution operator. Finally, stability results are derived based on the pth moment exponential stable with the help of new impulsive integral inequality techniques. In addition, an example is provided to validate the theoretical results. This manuscript is the unique combination of the new theoretical investigation which is simulated numerically. Our work extends the works of Arthi et al. (2014), Das et al. (2016), Huang et al. (2018), Pandey et al. (2014), Wang et al. (2018), Yan and Jia (2016).

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

Hardy Type Inequalities in Classical and Grand Lebesgue Spaces $L_{p)}$, $0<p\le 1$, for Quasi-Monotone Functions

In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces $L_{p)} (0,1)$, $0<p\leq1$. In particular, they established similar results for the Hardy operator in classical weighted Lebesgue spaces. Moreover, it is proved that the grand Lebesgue space $L_{p) } (0,1)$ is a quasi-Banach function space. In this work, we are interested in Hardy inequalities applied to quasi-monotonic functions in classical Lebesgue spaces and grand Lebesgue spaces. We establish the boundedness of Hardy operator for quasi-monotone functions in grand Lebesgue spaces $L_{p)}$, $w(0,1),$ $0<p\leq1$. In addition some integral inequalities for the Hardy operator are proved in classical weighted Lebesgue spaces $L_{p,w}(0,1)$, $0<p<1,$ for quasi-monotone functions. All inequalities are proved with sharp constants. Some results of Rovshan A. Bandaliev et al. are deduced as particular cases. Also other estimates are obtained in classical Lebesgue spaces for Hardy's operator and its dual.

A new inequality and its application in descriptor Markovian jump systems with time-varying delays

This paper is focused on the dissipativity analysis of descriptor Markovian jump systems with time-varying delays. For achieving this aim, an augmented mode-dependent Lyapunov-Krasovskii functional (LKF) is constructed on the basis of the state decomposition method. Next, a novel integral inequality is proposed through the generalized free-weighting-matrix (GFWM) method as well as zero-valued equalities to estimate single integral terms more precisely. Then, an ameliorative stochastic admissibility and dissipative criterion is derived by utilizing the proposed integral inequality, the augmented LKF and other bounding techniques. Finally, the effectiveness and superiority of the presented results are confirmed with some numerical examples.