General Integral Inequalities Research Articles

A HILBERT-TYPE LOCAL FRACTIONAL INTEGRAL INEQUALITY WITH THE KERNEL OF A HYPERBOLIC COSECANT FUNCTION

By using Yang’s local fractional calculus theory, the method of weight function, and real-analysis techniques in the fractal set, a general Hilbert-type local fractional integral inequality with the kernel of a hyperbolic cosecant function is established. The necessary and sufficient condition for the constant factor of the general Hilbert-type local fractional integral inequality to be the best possible is discovered. Furthermore, two equivalent inequalities with the best constant factors were obtained.

Integral Inequalities Involving Strictly Monotone Functions

Functional inequalities involving special functions are very useful in mathematical analysis, and several interesting results have been obtained in this topic. Several methods have been used by many authors in order to derive upper or lower bounds of certain special functions. In this paper, we establish some general integral inequalities involving strictly monotone functions. Next, some special cases are discussed. In particular, several estimates of trigonometric and hyperbolic functions are deduced. For instance, we show that Mitrinović-Adamović inequality, Lazarevic inequality, and Cusa-Huygens inequality are special cases of our obtained results. Moreover, an application to integral equations is provided.

General Lebesgue integral inequalities of Jensen and Ostrowski type and applications

Jensen-Ostrowski-type inequalities for general Lebesgue integral are discussed. Here, we derive information inequalities for Lipschitzian derivatives by using new [Formula: see text]-divergence measure which will be useful in Information Theory. In this way, we establish some particular cases of these information inequalities. We also obtain information inequalities for general Lebesgue integral of differentiable functions whose derivatives in absolute value are [Formula: see text]-convex and its particular instances are established.

On new general inequalities for s-convex functions and their applications

In this work, we established some new general integral inequalities of Hermite–Hadamard type for s-convex functions. To obtain these inequalities, we used the Hölder inequality, power-mean integral inequality, and some generalizations associated with these inequalities. Also we compared some inequalities (e.g., Theorem 6 and Theorem 8). Finally, we gave some applications for special means.

New integral inequalities for Atangana-Baleanu fractional integral operators and various comparisons via simulations

Integral identities created in inequality theory studies help to prove many inequalities. Recently, different fractional integral and derivative operators have been used to achieve these identities. In this article, with the help of Atangana-Baleanu integral operators, an integral identity was first obtained and various integral inequalities for convex functions have been proved using this identity. In the last part of the article, various simulation graphs are given to reveal the consistency of Atangana-Baleanu fractional integral operators and Riemann-Liouville fractional integral operators for different ? values. The prominent motivating idea in this work is to obtain new and general form integral inequalities with the help of fractional integral operators with strong kernel structure.

Triangle inequality for quantum integral operator

We show that general integral triangle inequality does not hold for shifted q-integrals. Furthermore, we obtain a triangle inequality for shifted qintegrals. We also give an estimate for q-integrable product and use it to refine some recently obtained Ostrowski inequalities for quantum calculus.

Some General Fractional Integral Inequalities Involving LR–Bi-Convex Fuzzy Interval-Valued Functions

The main objective of this paper is to introduce a new class of convexity called left-right–bi-convex fuzzy interval-valued functions. We study this class from the perspective of fractional Hermite–Hadamard inequalities, involving a new fractional integral called the left-right–AB fractional integral. We discuss several special cases that demonstrate that our results are quite unifying. We provide non-trivial numerical examples regarding special means for positive real numbers in order to check the validity of our outcomes.

A generalized multiple integral inequality with application to time-varying delay systems

This paper aims to investigate a generalized multiple integral inequality that is applied to drop the conservativeness of the stability criterion for time-varying delayed system. First, a more general multiple integral inequality is introduced by utilizing Schmidt-orthogonal method with integral operation and inequality techniques. Then, the inequality obtained is adopted to study the time-delay system, which yields the delay-related stability criterion with fewer conservatism. Finally, numerical examples are given to certify that the acquired inequality plays a crucial role in stability analysis.

General Raina fractional integral inequalities on coordinates of convex functions

Integral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In this study, authors have established some generalized Raina fractional integral inequalities using an (l_{1},h_{1})-(l_{2},h_{2})-convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for the (l_{1},h_{1})-(l_{2},h_{2})-convex function on coordinates are given. Finally, we can say that our findings recapture some recent results as special cases.

On some generalized integral inequalities for functions whose second derivatives in absolute values are convex

AbstractIn this article, general integral inequalities are obtained for functions whose absolute value of the second derivative is convex. These inequalities are more general versions of some results in the literature and we recaptured these results with the selection of special parameters. In the study, graphs are also used to compare the inequalities that occur with the change of the µ parameter.

Some General Integral Operator Inequalities Associated with φ -Quasiconvex Functions

This paper deals with generalized integral operator inequalities which are established by using φ -quasiconvex functions. Bounds of an integral operator are established which have connections with different kinds of known fractional integral operators. All the results are deducible for quasiconvex functions. Some fractional integral inequalities are deduced.

Novel stability criterion for linear system with two additive time-varying delays using general integral inequalities

<abstract> The stability analysis strategy for continuous linear system with two additive time-varying delays is proposed in this paper. First, for the purpose of analysis, the novel Lyapunov-Krasovskii functional (LKF) consisting of integral terms based on the first-order derivative of the system state is constructed. Second, the derivative of LKF is estimated by utilizing the Wirtinger-based integral inequality and extended reciprocally convex matrix inequality. The delay-dependent stability criterions are established in terms of linear matrix inequalities (LMIs) framework. The results show that the system performances are improved based on both enlarging the maximum allowable upper bound of the time-delays and reducing the number of decision variables. Furthermore, the conservatism of obtained delay-dependent stability criterion is reduced. Finally, a numerical simulation is given to demonstrate the effectiveness of obtained theoretical results. </abstract>

Enhanced stability results for generalized neural networks with time-varying delay

This paper deals with the stability analysis problem for generalized neural network with time-varying delay. A proper Lyapunov-Krasovskii functional (LKF) with novel delay-product-dependent term is constructed, which fully considers the features of the improved Jensen integral inequality. Based on the new proposed LKF and a more general integral inequality issued from the improved Jensen one, an enhanced delay-dependent stability criterion is developed in terms of linear matrix inequality (LMI). Furthermore, a more refined stability criterion is established by relaxing the positive definiteness of the LKF. The effectiveness of the theoretical results are demonstrated by four numerical examples.

Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function

In the article, we establish some new general fractional integral inequalities for exponentially m-convex functions involving an extended Mittag-Leffler function, provide several kinds of fractional integral operator inequalities and give certain special cases for our obtained results.

SOME LOCAL FRACTIONAL INTEGRAL INEQUALITIES FOR GENERALIZED PREINVEX FUNCTIONS AND APPLICATIONS TO NUMERICAL QUADRATURE

In this paper, a new identity with parameters involving local fractional integrals is derived. Using this identity, some general local fractional integral inequalities for generalized preinvex functions are established. A parallel development is deduced for generalized preconcave functions. Taking special values for the parameters, some generalized midpoint inequalities, trapezoidal inequalities and Simpson inequalities are obtained. Finally, as some applications, error estimates of numerical integration for local fractional integrals are given.

Delayed impulsive control for exponential synchronization of stochastic reaction–diffusion neural networks with time-varying delays using general integral inequalities

The intention of this paper is to explore the exponential synchronization of delayed stochastic reaction–diffusion neural networks. By constructing appropriate Lyapunov–Krasovskii functional (LKF) with triple integral terms and by utilizing improved double integral inequalities, new synchronization criteria are developed in the form of linear matrix inequalities (LMIs) to assure the exponential synchronization of the neural networks under study. The resulting criteria use more data of the delay bounds, and by means of new inequalities, they are substantiated to be less conservative and also computationally appealing than certain existing works. At the end, numerical simulations are furnished to reveal the potency of the acquired theoretical results.

Multidimensional Hardy-type inequalities on time scales with variable exponents

A new Jensen inequality for multivariate superquadratic functions is derived and proved. The derived Jensen inequality is then employed to obtain the general Hardy-type integral inequality for sup ...

General fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function

In this paper some new general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function are presented. These results produce inequalities for several kinds of fractional integral operators. Some interesting special cases of our main results are also pointed out.

Stability analysis of nonlinear telerobotic systems with time-varying communication channel delays using general integral inequalities

Existing controller synthesis schemes for teleoperation of nonlinear master-slave systems with time-varying communication channel are subject to conservative stability criteria. This paper presents a new stability criteria for bilateral tele-operation systems with proportional-derivative, position-position and position-force control architectures, considering both passive/non-passive human operator. The communication channel pertains unknown asymmetric time-varying delays with given lower and upper-bounds. Using the Lyapunov Krasovskii approach, the stability analysis of the dynamic system is examined and the stability conditions are derived in the form of Linear Matrix Inequalities (LMIs). Advanced techniques have been employed to find less conservative integral inequalities in the process of stability analysis of the closed-loop teleoperator. Illustrative numerical examples and simulation results emphasize the effectiveness of the proposed stability criteria in enlarging the stability bounds and providing efficient control performances.

Event triggered reliable synchronization of semi-Markovian jumping complex dynamical networks via generalized integral inequalities

This paper focuses on the synchronization problem of semi-Markovian jumping complex dynamical networks with time-varying coupling delays against actuator failures. In an aim to shrink the treatment of network resources event triggered control strategy is proposed to achieve the synchronization criteria. By constructing Lyapunov–Krasovski functional, some delay dependent criteria that assures the synchronization of CDN are derived with the help of the general integral inequalities. It should be noted that the general integral inequality used here is general than that of Jensen inequality, the Wirtinger-based inequality, the Bessel-Legendre inequality, the Wirtinger-based double integral inequality, and the auxiliary function-based integral inequalities. The resulting LMIs can be easily verified with the help of the available softwares. Finally, simulation results are proposed to verify the effectiveness of the general integral inequality and designed control law.