Integral Inequalities Research Articles

Some Novel Integral Inequalities on the Co-ordinates for Geometrically Exponentially Convex Functions

The main purpose of this study is to define geometrically exponentially convex functions, which are a more general version, by expanding geometrically convex functions and to create the relevant lemmas. Some properties of geometrically exponentially convex functions are proven using definitions and lemmas. While obtaining the main findings, in addition to basic analysis information, Young and Hölder inequalities, well known in the literature, were also used for the powers of some functions. In the new theorems obtained, some special results were obtained for α=0.

Integral Inequalities Involving New Conformable Derivatives

Integral Inequalities Involving New Conformable Derivatives

Fractal-fractional estimations of Bullen-type inequalities with applications

The study of inequalities inside fractal domains has been stimulated by the growing interest in fractional calculus for the applied and mathematical sciences. This work uses extended fractional integrals in fractal domains to prove new Bullen-type inequalities for differentiable convex functions. By showing how these operators and inequalities can convert classical inequalities into fractal sets, it fills a vacuum in the literature on fractal-fractional integral inequalities. Using extended fractional integral operators, we derive new fractal-fractional estimates and Bullen-type inequalities, accompanied by thorough mathematical derivations and graphical validations. We show optimal results derived from new Bullen-type inequalities for fractal domains. We confirm our results with mathematical examples and graphs, improving models for complicated problems in fractal contexts. In particular, our findings have important ramifications for special means on fractal sets, probability density functions, and quadrature formulas, as well as for future study and applications.

Контактная задача для трансверсально изотропного слоя с трением

It is investigated the spatial contact problem of asymmetric interaction of two punches on a transversally isotropic layer with friction forces taken into account in an unknown contact domain. The planes of isotropy are parallel to the layer faces. The friction forces are taken into account along one coordinate axis. The lower layer face is subjected to sliding support. The layer material is characterized by five independent elastic parameters. The problem is reduced to an integral equation with respect to the contact pressure with the help of a double Fourier transformation together with the Coulomb law. The anisotropy degree is characterized be three dimensionless parameters arising in the kernel of the integral equation, two of which satisfy characteristic equation. In particular cases, the integral equation coincides with those well-known for the corresponding contact problems with friction for the isotropic layer and half-space. The B.A. Galanov method is used for numerical solutions. A system of the integral equation and integral inequality is considered. A rectangle is taken which a priori contains the unknown contact domain. By introducing special nonlinear operators, the system is reduced to only one nonlinear equation of the Hammerstein type which can be solved by the successive approximations method. The contact domain is determined by nodes at which the function required is positive. The punches are taken in the form of asymmetric elliptic paraboloids. The method allows us to investigate percolation, i.e. the process of junction of the discrete contact domains due to increasing the forces applied and settlements of the punches. Calculations are made for different friction coefficients, materials and relative thicknesses of the transversally isotropic layer.

Hermite–Hadamard-type inequalities for multiplicative harmonic s -convex functions

UDC 517.5 We study the concept of multiplicative harmonic s -convex functions and establish Hermite–Hadamard integral inequalities for this class of functions. Furthermore, we derive a set of Hermite–Hadamard-type inequalities applicable to the product and quotient of multiplicative harmonic s -convex functions. In addition, we deduce new inequalities involving multiplicative integrals for the product and quotient of harmonic convex and multiplicative harmonic s -convex functions. Some results for the class of multiplicative harmonic convex functions are obtained as special cases of our results. The obtained results are verified by providing examples with included graphs.

New Event-Triggered Synchronization Criteria for Fractional-Order Complex-Valued Neural Networks with Additive Time-Varying Delays

This paper explores the synchronization control issue for a class of fractional-order Complex-valued Neural Networks (FOCVNNs) with additive time-varying delays (TVDs) utilizing a sampled-data-based event-triggered mechanism (SDBETM). First, an innovative free-matrix-based fractional-order integral inequality (FMBFOII) and an improved fractional-order complex-valued integral inequality (FOCVII) are proposed, which are less conservative than the existing classical fractional-order integral inequality (FOII). Secondly, an SDBETM is inducted to conserve network resources. In addition, a novel Lyapunov–Krasovskii functional (LKF) enriched with additional information regarding the fractional-order derivative, additive TVDs, and triggering instants is constructed. Then, through the integration of the innovative FOCVII, LKF, SDBETM, and other analytical methodologies, we deduce two criteria in the form of linear matrix inequalities (LMIs) to ensure the synchronization of the master–slave FOCVNNs. Finally, numerical simulations are illustrated to confirm the validity of the proposed results.

Some Properties of the Potential Field of an Almost Ricci Soliton

In this article, we are interested in finding necessary and sufficient conditions for a compact almost Ricci soliton to be a trivial Ricci soliton. As a first result, we show that positive Ricci curvature and, for a nonzero constant c, the integral of Ric(cξ,cξ) satisfying a generic inequality on an n-dimensional compact and connected almost Ricci soliton (Mn,g,ξ,σ) are necessary and sufficient conditions for it to be isometric to the n-sphere Sn(c). As another result, we show that, if the affinity tensor of the soliton vector field ξ vanishes and the scalar curvature τ of an n-dimensional compact almost Ricci soliton (Mn,g,ξ,σ) satisfies τnσ−τ≥0, then (Mn,g,ξ,σ) is a trivial Ricci soliton. Finally, on an n-dimensional compact almost Ricci soliton (Mn,g,ξ,σ), we consider the Hodge decomposition ξ=ξ¯+∇h, where divξ¯=0, and we use the bound on the integral of Ricξ¯,ξ¯ and an integral inequality involving the scalar curvature to find another characterization of the n-sphere.

A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate.

Containment control for non-linear fractional-order multi-agent systems via refined sample data controller

Abstract This manuscript concentrates on the problem of designing a sampled data controller (SDC) for the consensus of a fractional-order multi-agent system (FOMAS) with Lipschitz non-linearity via an algebraic approach. The solution of the FOMAS is represented by using the Laplace transform approach. An upper bound of the sampling period is determined through various integral inequality techniques. Distinguished from the existing works, the estimate for an upper bound is more accurate which involves the Lipschitz constant of the non-linear function. Finally, numerical examples are given to validate the correctness of results. Furthermore, the comparison results are presented to show the proposed method determines a better upper bound of the sampling period.

The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications

Introducing adaptation parameters σ,σ1, formal parameters λi(i=1,2,3,4),κ,τ, and type parameters μ,ν, the integration operator is defined as T:Lpp(1−μσˆ)−1(R+)→Lppνσˆ−1(R+), Tf(y)=∫R+eλ1xμyν+κe−λ2xμyνeλ3xμyν+τe−λ4xμyνf(x)dx,y∈R+. Using the weight function method, a general Hilbert-type integral inequality is obtained, thereby proving the boundedness of the operator. The constant factor of the general Hilbert-type inequality is the best possible if and only if the adaptation parameters satisfy σ=σ1. From this, the formula for calculating the operator norm is obtained. In terms of application, some results from the references have been consolidated by discussing the combination of formal parameters and type parameters, and many new operator inequalities of different types and forms have been derived.

A New Differentiable Sphere Theorem and Its Applications

In this paper, we use the Lichnerowicz Laplacian to prove new results: the sphere theorem and the integral inequality for Einstein's infinitesimal deformations, which allow us to characterize spherical space forms. Our version of the sphere theorem states that a closed connected Riemannian manifold $(M, g)$ of even dimension $n>3$ is diffeomorphic to a Euclidean sphere or a real projective space if the inequality $Ric_{\rm max}(x) < n K_{\rm min}(x) g$ is true at each point $x\in M$, where $Ric_{\rm max}(x)$ is the maximum of the Ricci curvature, and $K_{\rm min}(x)$ is the minimum of the sectional curvature of $(M, g)$ at $x$. Since this inequality implies positive sectional curvature; therefore, our result partially answers Hopf's old open question.

Some new integral inequalities for F-convex functions via ABK-fractional operator

In this paper, we present Hermite-Hadamard type and midpoint-type integral inequalities for F-convex function via ABK-fractional integral operators. In the context of differentiable mapping Ψ, we provide multiple new identities and prove new results for F-convex functions using ABK-fractional integral operators whose derivatives in the absolute values are F-convex. We establish correlations between our results and a number of significant results in the literature. The results presented in this paper might encourage more study in this field.

Structural and Parametric Synthesis of a Hybrid Repetitive Control System for Uncertain Multi-Mode Plant

Article deals with the problem of constructing a hybrid adaptive-robust repetitive control system for one class of a MIMO plants which operating with structural and parametric uncertainty and constant action of an external disturbances. A procedure for structural and parametric synthesis of a control system is proposed. At the structural synthesis stage with the help of hyperstability criterion the authors are developing a decentralized combined controller. After that using the method of continuous models the authors are built a continuous-discrete control system. The main feature of a control loop structure synthesis stage is to ensure the validity of V. M. Popov’s integral inequality. For this purpose, the authors determining the special estimates that guarantee positivity for nonlinear non-stationary part of the system under consideration. At the system’s parametric synthesis stage the authors are use the environment of engineering and technical calculations "Matlab — Simulink" to perform optimization modeling of the continuous and hybrid repetitive control systems. This action is carried out using one of the functional optimization methods — a genetic algorithm. Authors use the criterion of generalized operation for automatic control systems to form the functionality for assessing the quality of the proposed system. At the simulation stage initially for continuous repetitive system it was searched its controller parameters which ensure the minimum value of the specified functional. Then, with a given discretization step of the control loop elements, the hybrid periodic system was optimized in order to improve the quality of its operation.

A family of quadrature formulas with their error bounds for convex functions and applications

In numerical analysis, the quadrature formulas serve as a pivotal tool for approximating definite integrals. In this paper, we introduce a family of quadrature formulas and analyze their associated error bounds for convex functions. The main advantage of these new error bounds is that from these error bounds, we can find the error bounds of different quadrature formulas. This work extends the traditional quadrature formulas such as the midpoint formula, trapezoidal formula, Simpson's formula, and Boole's formula. We also use the power mean and Hölder's integral inequalities to find more general and sharp bounds. Furthermore, we give numerical example and applications to quadrature formulas of the newly established inequalities.

Relaxed stability criteria of delayed neural networks using delay-parameters-dependent slack matrices

This note aims to reduce the conservatism of stability criteria for neural networks with time-varying delay. To this goal, on the one hand, we construct an augmented Lyapunov–Krasovskii functional (LKF), incorporating some delay-product terms that capture more information about neural states. On the other hand, when dealing with the derivative of the LKF, we introduce several parameter-dependent slack matrices into an affine integral inequality, zero equations, and the S-procedure. As a result, more relaxed stability criteria are obtained by employing the so-called Lyapunov–Krasovskii Theorem. Two numerical examples show that the proposed stability criteria are of less conservatism compared with some existing methods.

Blow-up and global existence for a new class of parabolic p(x,⋅)-Kirchhoff equation involving double phase operator

In this paper, we are concerned with study solutions for double phase problems involving a Kirchhoff function, and nonlinearity with subcritical growth:{ut+M([u]p,q,as)Lp,q,asu=|u|r−2u,inU×[0,T),u(x,t)=0in∂U×[0,T),u(x,0)=u0(x),inU, where, Lp,q,ss is double phase operator, and M is a Kirchhoff function. Combining the Faedo-Galerking approximation with Gronwall's inequality, the existence of a unique weak solution is established. More interestingly, we study the global existence and finite time blow of solution when the initial energy is sub-critical and supercritical. Finally, the stabilization of the solution with positive initial energy is established based on Komornik's integral inequality.

A matrix-separation-based integral inequality for aperiodic sampled-data synchronization of delayed neural networks considering communication delay

This paper achieves the synchronization of delayed neural networks (DNNs) considering aperiodic sampled-data control and communication delay. First of all, based on the master-slave DNNs with aperiodic sampling synchronization controller, a synchronization error system is constructed. Then, an augmented functional containing both the error state and its derivative is constructed. Compared with the existing researches, the augmented functional introduces more cross information of error states to the criterion. Next, an integral inequality based on the separation of internal integral variable and matrix is developed. Compared to the inequalities that treat the internal variable and the matrix as unified ones, the developed inequality provides a tighter estimate of the derivative of the augmented functional. On this basis, a criterion with less conservative is developed for the aperiodic sampled-data synchronization of DNNs considering communication delay. Finally, to indicate the superiority of the developed method on improving the acceptable sampling upper bound of synchronization, three numerical examples are provided.

Tensorial and Hadamard Product Integral Inequalities for Convex Functions of Continuous Fields of Operators in Hilbert Spaces

Let H be a Hilbert space and Ω a locally compact Hausdorff space endowed with a Radon measure μ with ∫_{Ω}1dμ(t)=1. In this paper we show among others that, if f is continuous differentiable convex on the open interval I, (A_{τ})_{τ∈Ω} is a continuous field of positive operators in B(H) such that Sp(A_{τ}) ⊂I for each τ∈Ω and B and operator such that Sp(B)⊂I, then we have ∫_{Ω}(f′(A_{τ})A_{τ})dμ(τ)⊗1-∫_{Ω}f′(A_{τ})dμ(τ)⊗B ≥∫_{Ω}f(A_{τ})dμ(τ)⊗1-1⊗f(B) ≥(∫_{Ω}A_{τ}dμ(τ)⊗1-(1⊗B))(1⊗f′(B)) and the Hadamard product inequality ∫_{Ω}(f′(A_{τ})A_{τ})dμ(τ)∘1-∫_{Ω}f′(A_{τ})dμ(τ)∘B ≥∫_{Ω}f(A_{τ})dμ(τ)∘1-1∘f(B) ≥∫_{Ω}A_{τ}dμ(τ)∘f′(B)-1∘(f′(B)B).

Integral inequalities of h‐superquadratic functions and their fractional perspective with applications

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of ‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of ‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research.

On Bernstein and Turán-type integral mean estimates for polar derivative of a polynomial

Let p(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p(z)$\\end{document} be a polynomial of degree n having no zero in |z|<k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|z|< k$\\end{document}, k≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k\\leq 1$\\end{document}, then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max|z|=1|p′(z)|≤n1+knmax|z|=1|p(z)|,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\max _{|z|=1}|p'(z)|\\leq \\frac{n}{1+k^{n}}\\max _{|z|=1}|p(z)|, $$\\end{document} provided |p′(z)|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|p'(z)|$\\end{document} and |q′(z)|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|q'(z)|$\\end{document} attain their maxima at the same point on the circle |z|=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|z|=1$\\end{document}, where q(z)=znp(1z‾)‾.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q(z)=z^{n}\\overline{p\\bigg(\\frac{1}{\\overline{z}}\\bigg)}. $$\\end{document} In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results.