Jensen's Inequality Research Articles

Some novel estimates of Jensen and Hermite-Hadamard inequalities for h-Godunova-Levin stochastic processes

<abstract><p>It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings.</p></abstract>

Correction: Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $ (h_1, h_2) $-Godunova-Levin functions

Correction: Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $ (h_1, h_2) $-Godunova-Levin functions

Robust stability and passivity analysis for discrete-time neural networks with mixed time-varying delays via a new summation inequality

<abstract><p>The summation inequality is essential in creating delay-dependent criteria for discrete-time systems with time-varying delays and developing other delay-dependent standards. This paper uses our rebuilt summation inequality to investigate the robust stability analysis issue for discrete-time neural networks that incorporate interval time-varying leakage and discrete and distributed delays. It is a novelty of this study to consider a new inequality, which makes it less conservative than the well-known Jensen inequality, and use it in the context of discrete-time delay systems. Further stability and passivity criteria are obtained in terms of linear matrix inequalities (LMIs) using the Lyapunov-Krasovskii stability theory, coefficient matrix decomposition technique, mobilization of zero equation, mixed model transformation, and reciprocally convex combination. With the assistance of the LMI Control toolbox in Matlab, numerical examples are provided to demonstrate the validity and efficiency of the theoretical findings of this research.</p></abstract>

Jensen and Hermite-Hadamard type inclusions for harmonical $ h $-Godunova-Levin functions

<abstract><p>The role of integral inequalities can be seen in both applied and theoretical mathematics fields. According to the definition of convexity, it is possible to relate both concepts of convexity and integral inequality. Furthermore, convexity plays a key role in the topic of inclusions as a result of its definitional behavior. The importance and superior applications of convex functions are well known, particularly in the areas of integration, variational inequality, and optimization. In this paper, various types of inequalities are introduced using inclusion relations. The inclusion relation enables us firstly to derive some Hermite-Hadamard inequalities (H.H-inequalities) and then to present Jensen inequality for harmonical $ h $-Godunova-Levin interval-valued functions (GL-IVFS) via Riemann integral operator. Moreover, the findings presented in this study have been verified with the use of useful examples that are not trivial.</p></abstract>

Existence and controllability for conformable fractional stochastic differential equations with infinite delay via measures of noncompactness.

In this article, we consider conformable fractional stochastic differential equations (CFSDEs) driven by fBm with infinite delay via measures of noncompactness (MNC). As far as we know, there are few papers considering this issue. First, by virtue of a Mönch fixed point theorem and MNC, we explore the existence of solutions for CFSDEs. Subsequently, with the aid of Jensen inequality, Hölder inequality, stochastic analysis techniques, and semigroup theory, the controllability for this considered CFSDEs is investigated by employing a Mönch fixed point theorem. Thereafter, the controllability of CFSDEs with nonlocal conditions is discussed. Finally, the theoretical result is supported through an example.

A Refined Jensen Inequality Connected to an Arbitrary Positive Finite Sequence

The prime purpose of this paper is to provide a refinement of Jensen’s inequality in connection with a positive finite sequence. We deal with the refinement for particular cases and point out the relation between the new result with earlier results of Jensen’s inequality. As results, we obtain refinements of the quasi-arithmetic and power mean inequalities. Finally, several results are obtained in information theory with the help of the main results.

Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for (h1,h2)-Convex Functions Pertaining to Total Order Relation

There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω=⟨ωc,ωr⟩=⟨ω¯+ω̲2,ω¯−ω̲2⟩. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-(h1,h2)-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples.

Finite-time Lyapunov fluctuations and the upper bound of classical and quantum out-of-time-ordered expansion rate exponents.

This Letter demonstrates for chaotic maps [logistic, classical, and quantum standard maps (SMs)] that the exponential growth rate (Λ) of the out-of-time-ordered four-point correlator is equal to the classical Lyapunov exponent (λ) plus fluctuations (Δ^{(fluc)}) of the one-step finite-time Lyapunov exponents (FTLEs). Jensen's inequality provides the upper bound λ≤Λ for the considered systems. Equality is restored with Λ=λ+Δ^{(fluc)}, where Δ^{(fluc)} is quantified by k-higher-order cumulants of the (covariant) FTLEs. Exact expressions for Λ are derived and numerical results using k=20 furnish Δ^{(fluc)}∼ln(sqrt[2]) for all maps (large kicking intensities in the SMs).

The existence and averaging principle for stochastic fractional differential equations with impulses

In this paper, a class of fractional stochastic differential equations (SFDEs) with impulses is considered. By virtue of Mönch's fixed point theorem and Banach contraction principle, we explore the existence and uniqueness of solutions to the addressed system. Furthermore, with the aid of the Jensen inequality, Hölder inequality, Burkholder–Davis–Gundy inequality, Grönwall–Bellman inequality, and some novel assumptions, the averaging principle of our considered system is obtained. At the end of this paper, an example is provided to illustrate the theoretical results.

Some estimations of the Jensen difference and applications

The Jensen inequality is one of the most favorable inequalities during the last few decades due to its expressive characteristics and properties. This inequality has gained a very dominant position in the several fields of science as a result of its extensive applications. In the present note, we use an interesting approach for the determination of estimates for the Jensen difference. We acquire several estimates for the Jensen difference while utilizing the convex function definition, Jensen's inequality for concave functions, power mean, and Hölder inequalities. For the real visualizations of the acquired estimates, we provide some particular examples. We get different improvements for the Hölder inequality while taking specified functions in the received estimates. Moreover, we deduce some more results from the main work in the form of improvements for the Hermite‐Hadamard inequality. Furthermore, various relations for the famous quasi‐arithmetic and power means are acquired with the help of established estimates. At the end, we present various applications of the main work in information theory. The intended applications provide some new bounds for the Csiszár and Rényi divergences, Shannon entropy, and Bhattacharyya co‐efficient.

Fully nonlinear Hamilton–Jacobi equations of degenerate type

We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

Novel dynamic Hardy‐type inequalities on time scales

In the present article, we proved some new dynamic inequalities of Hardy‐type on an arbitrary time scale. We have used dynamic Minkowski inequality, dynamic Hölder's inequality, dynamic Jensen's inequality and Fubini theorem on time scales to prove our main results. As special case of our results, some Hardy‐type integral inequalities known in the literature have been recaptured, and some original discrete inequalities of Hardy‐type have been established.

Diamond-α Hardy-Type Inequalities on Time Scales

In the present article, we prove some new generalizations of dynamic inequalities of Hardy-type by utilizing diamond-α dynamic integrals on time scales. Furthermore, new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are proved. Moreover, we present Hardy inequalities for several functions by using the diamond-α dynamic integrals on time scales. The results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales. Our main results extend existing results of the integral and discrete Hardy-type inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Choquet integrals of set-valued functions with respect to set-valued fuzzy measures

As powerful tools for modeling nondeterministic problems, Choquet integrals have attracted much attention in recent years. This study generalizes the existing set-valued Choquet integrals and formulates the theory of Choquet integrals with respect to set-valued fuzzy measures. First, a set-valued Choquet integral of a real-valued function is defined using the set of Choquet integrals of real-valued functions with respect to fuzzy measure selections of a set-valued fuzzy measure. The integrand is then extended to set-valued functions, after which various properties, convergence theorems, and set-valued Jensen's inequalities are obtained. Second, following how single-valued Choquet integrals are defined, an alternative set-valued Choquet integral of real-valued function is formulated using Aumann integrals. Through these steps, the existing set-valued Choquet integrals are covered.

Generalized cyclic Jensen and information inequalities

We present Levinson’s type generalizations of cyclic refinements of Jensen’s inequality by employing recent class of functions that further characterize and extend the family of 3-convex functions. We get monotonic cyclic Jensen’s inequalities and particularly the renowned Jensen’s inequality for 3-convex functions at a point (f∈κ1c(I)). As an applications in information theory, we first introduce new Csiszár type cyclic divergence functional for 3-convex functions and establish cyclic-Kullback–Leibler and Hellinger distances. We give monotonicity of cyclic divergence functionals which enable us to construct monotonic Shannon, Relative and Zipf–Mandelbrot entropies.

Event-triggered control for uncertain stochastic systems under triple network attacks

This paper studies the control problem of uncertain stochastic systems, which takes into account the impact of network attacks. The types of network attacks considered are denial-of-service (DoS) attacks, deception attacks and replay attacks. In order to save network resources and improve communication utilization, the static event-triggered mechanism and adaptive event-triggered mechanism are cited respectively. Firstly, a new Lyapunov-Krasovskii functional is constructed, employing improved Wirtinger-based integral inequality and Jensens inequality, the criteria on stochastic stability in the mean square for uncertain stochastic systems are proposed. Secondly, the design methods of static event-triggered controller and adaptive event-triggered controller are given respectively. Finally, a practical example is given to manifest the effectiveness of the theoretical results.

A Study on Some New Generalizations of Reversed Dynamic Inequalities of Hilbert-Type via Supermultiplicative Functions

In this article, we establish some new generalizations of reversed dynamic inequalities of Hilbert-type via supermultiplicative functions by applying reverse Hölder inequalities with Specht’s ratio on time scales. We will generalize the inequalities by using a supermultiplicative function which the identity map represents a special case of it. Also, we use some algebraic inequalities such as the Jensen inequality and chain rule to prove the essential results in this paper. Our results (when T ≪ ℕ ) are essentially new.

MϕA-h-Convexity and Hermite-Hadamard Type Inequalities

We investigate a family of MϕA-h-convex functions, give some properties of it and several inequalities which are counterparts to the classical inequalities such as the Jensen inequality and the Schur inequality. We give the weighted Hermite-Hadamard inequalities for an MϕA-h-convex function and several estimations for the product of two functions.

Stochastic exciton-scattering theory of optical line shapes: Renormalized many-body contributions.

Spectral line shapes provide a window into the local environment coupled to a quantum transition in the condensed phase. In this paper, we build upon a stochastic model to account for non-stationary background processes produced by broad-band pulsed laser stimulation, as distinguished from those for stationary phonon bath. In particular, we consider the contribution of pair-fluctuations arising from the full bosonic many-body Hamiltonian within a mean-field approximation, treating the coupling to the system as a stochastic noise term. Using the Itô transformation, we consider two limiting cases for our model, which lead to a connection between the observed spectral fluctuations and the spectral density of the environment. In the first case, we consider a Brownian environment and show that this produces spectral dynamics that relax to form dressed excitonic states and recover an Anderson-Kubo-like form for the spectral correlations. In the second case, we assume that the spectrum is Anderson-Kubo like and invert to determine the corresponding background. Using the Jensen inequality, we obtain an upper limit for the spectral density for the background. The results presented here provide the technical tools for applying the stochastic model to a broad range of problems.

Net effect of environmental fluctuations in multiple global-change drivers across the tree of life

Jensen's inequality predicts that the response of any given system to average constant conditions is different from its average response to varying ones. Environmental fluctuations in abiotic conditions are pervasive on Earth; yet until recently, most ecological research has addressed the effects of multiple environmental drivers by assuming constant conditions. One could thus expect to find significant deviations in the magnitude of their effects on ecosystems when environmental fluctuations are considered. Drawing on experimental studies published during the last 30 years reporting more than 950 response ratios (n = 5,700), we present a comprehensive analysis of the role that environmental fluctuations play across the tree of life. In contrast to the predominance of interactive effects of global-change drivers reported in the literature, our results show that their cumulative effects were additive (58%), synergistic (26%), and antagonistic (16%) when environmental fluctuations were present. However, the dominant type of interaction varied by trophic level (autotrophs: interactive; heterotrophs: additive) and phylogenetic group (additive in Animalia; additive and positive antagonism in Chromista; negative antagonism and synergism in Plantae). In addition, we identify the need to tackle how complex communities respond to fluctuating environments, widening the phylogenetic and biogeographic ranges considered, and to consider other drivers beyond warming and acidification as well as longer timescales. Environmental fluctuations must be taken into account in experimental and modeling studies as well as conservation plans to better predict the nature, magnitude, and direction of the impacts of global change on organisms and ecosystems.