Inequalities For Functions Research Articles

On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral

In this paper, we aim to construct $n$ dimensional Jensen, Hardy and Hermite-Hadamard type inequalities for multiple diamond-alpha integral on time scales. The cases of Hardy type inequality with a weighted function and Hermite-Hadamard type inequality with three variables are also considered minutely.

Bernstein–Remez inequality for algebraic functions: A topological approach

By taking full advantage of the structure of complex algebraic curves and by using compactness arguments, in this paper we give a self-contained proof that holomorphic algebraic functions verify a uniform Bernstein–Remez inequality. Namely, their growth over a bounded, open, complex set is uniformly controlled by their size on a compact complex subset of sufficiently high cardinality. Up to our knowledge, the first known demonstration on the existence of such an inequality for a specific subset of algebraic functions is contained in Nekhoroshev’s 1973 breakthrough on the genericity of close-to-integrable Hamiltonian systems that are stable over long time. Despite its pivotal rôle, this passage of Nekhoroshev’s proof has remained unnoticed so far. This work aims at extending and generalizing Nekhoroshev’s arguments to a modern framework. We stress the fact that our proof is different from the one contained in Roytwarf and Yomdin’s seminal work (1998), where Bernstein-type inequalities are proved for several classes of functions.

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained.

Infinite divisibility of the Whittaker distribution

In this paper, by using an integral representation of Ismail and Kelker for the quotient of Tricomi hypergeometric functions, we investigate the infinite divisibility and self-decomposability of the recently defined four-parameter lifetime Whittaker distribution, which is a natural extension of the classical gamma, exponential, chi-square, generalized Lindley, Lindley, beta prime, and Lomax distributions. We also show that the Whittaker distribution belongs to the class of hyperbolically completely monotone distributions and generalized gamma convolutions, and it is a super-Gaussian distribution. By using some results for the moments of the Whittaker distribution, we also deduce some Turán type inequalities for the Whittaker functions of the second kind and as an application we show that the effective variance of the Whittaker distribution is bounded from below.

Synchronization of Generally Uncertain Markovian Inertial Neural Networks With Random Connection Weight Strengths and Image Encryption Application.

This article focuses on the synchronization problem of delayed inertial neural networks (INNs) with generally uncertain Markovian jumping and their applications in image encryption. The random connection weight strengths and generally uncertain Markovian are discussed in the INNs model. Compared with most existing INNs models that have constant connection weight strengths, our model is more practical because connection weight strengths of INNs may randomly vary due to the external and internal environment and human factor. The delay-range-dependent synchronization conditions (DRDSCs) could be obtained by adopting the delay-product-term Lyapunov-Krasovskii functional (DPTLKF) and higher order polynomial-based relaxed inequality (HOPRII). In addition, the desired controllers are obtained by solving a set of linear matrix inequalities. Finally, two examples are shown to demonstrate the effectiveness of the proposed results.

Event-triggered H ∞ control of complex dynamical networks subject to stochastic cyber-attacks by new two-sided delay-dependent LKF functional

ABSTRACT This article deals with the problem of event-triggered control of complex dynamical networks (CDNs) subject to stochastic cyber-attacks and random coupling delays. Hackers may abuse the system by injecting malicious activity into communication networks. Two types of non-linear functions for cyber-attacks, time-varying coupling delays, respectively, and a set of random variables satisfying the Bernoulli distribution are proposed. Against complex dynamical networks, an attacker has the capacity to change the transmitted data. So the traditional control method can be a major challenge, and it is essential to respond to cyber-attacks of this kind. An event-triggered mechanism (ETM) was implemented to avoid losing limited bandwidth. The input delay method is then used to build a time-varying sample under stochastic cyber-attacks. The stability criterion of the developed system model and the display expression of control parameters are then established using a new two-sided delay-dependent Lyapunov–Krasovskii functional (LKF) and linear matrix inequality (LMI) techniques. Lastly, two numerical examples, including the inverted pendulum application, are presented to illustrate the effectiveness and feasibility of the derived theoretical results.

On the framework of Lp summations for functions

We develop the framework of Lp operations for functions: one based on an Lp analogue of the supremal convolution and another based on an Lp analogue of the infimal convolution. On one hand, we establish the general forms of the Lp-Borell-Brascamp-Lieb inequality for various functionals on the space of s-concave functions. On the other hand, we propose the j-th quermassintegral for convex functions and prove its associated variational formula in the rotationally invariant case.

Periodic Intermittent Adaptive Control with Saturation for Pinning Quasi-Consensus of Heterogeneous Multi-Agent Systems with External Disturbances.

A periodic intermittent adaptive control method with saturation is proposed to pin the quasi-consensus of nonlinear heterogeneous multi-agent systems with external disturbances in this paper. A new periodic intermittent adaptive control protocol with saturation is designed to control the internal coupling between the follower agents and the feedback gain between the leader and the follower. In particular, we use the saturation adaptive law: when the quasi-consensus error converges to a certain range, the adaptive coupling edge weight and the adaptive feedback gain will not be updated. Furthermore, we propose three saturated adaptive pinning control protocols. The quasi-consensus is achieved through its own pinning as long as the agents remain connected to each other. Using the Lyapunov function method and inequality technique, the convergence range of the quasi-consensus error of a heterogeneous multi-agent system is obtained. Finally, the rationality of the proposed control protocol is verified through numerical simulation. Theoretical derivation and simulation results show that the novel proposed periodic intermittent adaptive control method with saturation can successfully be used to achieve the pinning of quasi-consensus of nonlinear heterogeneous multi-agent systems.

Strichartz inequality for orthonormal functions associated with special Hermite operator

Abstract In this article, we derive the restriction theorem for the special Hermite transform and obtain the Strichartz estimate for the system of orthonormal functions associated with the special Hermite operator. Further, we discuss the optimal behavior of the constant as a limit of a large number of functions.

Linear formation of Cucker–Smale model with distributed time delays

In this paper, linear formation problem of Cucker–Smale model with distributed time delays is investigated. By using Lyapunov functional method and inequality techniques, combining with the connection between primitive system and projection system, the consistency of individual velocity is verified. Then, sufficient conditions for forming flocking and linear formations are obtained. In addition, the theoretical results are illustrated by some examples. The results obtained extend the application range of the classical Cucker–Smale model.

A Matrix Li–Yau–Hamilton Estimate for the Green Function on Kähler Manifolds

Abstract We prove a matrix Li–Yau–Hamilton inequality for the Green function on complete Kähler manifolds with nonnegative holomorphic bisectional curvature. This estimate is an elliptic analogue of the matrix estimate of Cao and Ni for the heat equation on Kähler manifolds. It is also the complex counterpart of the matrix estimate on Riemannian manifolds obtained previously by the author.

Functional Inequalities in Stratified Lie Groups with Sobolev, Besov, Lorentz and Morrey Spaces

Functional Inequalities in Stratified Lie Groups with Sobolev, Besov, Lorentz and Morrey Spaces

Rearrangement Inequalities of the One-Dimensional Maximal Functions Associated with General Measures

Rearrangement Inequalities of the One-Dimensional Maximal Functions Associated with General Measures

Refined Bohr inequality for functions in and in complex Banach spaces

In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and take values in the unit disk . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions defined on a balanced domain . All the results are proved to be sharp.

On a New Approach for Stability and Controllability Analysis of Functional Equations

We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the stability of functional equations. We also present a small list of aggregation functions on the classical, well-known special functions to investigate the best approximation error estimates using a different concept of perturbation stability.

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on N-dimensional RCD(K,N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{RCD}(K,N)$$\\end{document} spaces (X,d,HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,\ extsf {d},\\mathscr {H}^N)$$\\end{document}. Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements seem to be new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.

Lower N-weighted Ricci curvature bound with 𝜀-range and displacement convexity of entropies

In this paper, we provide a characterization of a lower [Formula: see text]-weighted Ricci curvature bound for [Formula: see text] with [Formula: see text]-range introduced by Lu–Minguzzi–Ohta [Comparison theorems on weighted Finsler manifolds and space-times with [Formula: see text]-range, Anal. Geom. Metr. Spaces 10(1) (2022) 1–30] in terms of a convexity of entropies over Wasserstein space. We further derive various interpolation inequalities and functional inequalities.

On Hermite–Hadamard–Fejér-Type Inequalities for η-Convex Functions via Quantum Calculus

In this paper, we use qa- and qb-integrals to establish some quantum Hermite–Hadamard–Fejér-type inequalities for η-convex functions. By taking q→1, our results reduce to classical results on Hermite–Hadamard–Fejér-type inequalities for η-convex functions. Moreover, we give some examples for quantum Hermite–Hadamard–Fejér-type inequalities for η-convex functions. Some results presented here for η-convex functions provide extensions of others given in earlier works for convex and η-convex functions.

Dissipativity-Based Consensus Tracking Control of Nonlinear Multiagent Systems With Generally Uncertain Markovian Switching Topologies and Event-Triggered Strategy.

This article focuses on the dissipativity-based consensus tracking control (DBCTC) problems of time-varying delayed leader-following nonlinear multiagent systems (LFNMASs) with the event-triggered transmission strategy. The switching topologies of the LFNMASs are subject to the uncertain and partially unknown generally Markovian jumping process. The control inputs of the following agents are updated according to the proposed event-triggered transmission strategy, which could reduce the communication burden. Based on the event-triggered transmission condition and distributed consensus protocol, some dissipativity-based criteria obtained by adopting the delay-product-term Lyapunov-Krasovskii functional (DPTLKF) and higher order polynomial-based relaxed inequality (HOPRII) are proposed to guarantee the LFNMAS consensus. The validity of the main results is verified by two simulation examples.

Certain Weighted Fractional Integral Inequalities for Convex Functions

In this study, by using the monotonicity properties of functions, several inequalities for convex functions are obtained with the help of a weighted fractional integral operator which provides a function f to be integrated in fractional order with respect to another function. It is also seen that the results obtained were generalizations of the previous results presented.