Jensen-type Inequalities Research Articles

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Strong F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {F}$\\end{document}-convexity and concavity and refinements of some classical inequalities

The concept of strong F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathcal {F}}$\\end{document}-convexity is a natural generalization of strong convexity. Although strongly concave functions are rarely mentioned and used, we show that in more effective and specific analysis this concept is very useful, and especially its generalization, namely strong F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathcal {F}}$\\end{document}-concavity. Using this concept, refinements of the Young inequality are given as a model case. A general form of the self-improving property for Jensen type inequalities is presented. We show that a careful choice of control functions for convex or concave functions can give a control over these refinements and produce refinements of the power mean inequalities.

Chebyshev–Jensen-Type Inequalities Involving χ-Products and Their Applications in Probability Theory

By means of the functional analysis theory, reorder method, mathematical induction and the dimension reduction method, the Chebyshev-Jensen-type inequalities involving the χ-products ⟨·⟩χ and [·]χ are established, and we proved that our main results are the generalizations of the classical Chebyshev inequalities. As applications in probability theory, the discrete with continuous probability inequalities are obtained.

Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes

<abstract><p>Mathematical inequalities supporting interval-valued stochastic processes are rarely addressed. Recently, Afzal et al. introduced the notion of $ \mathtt{h} $-Godunova-Levin stochastic processes and developed Hermite-Hadamard and Jensen type inequalities in the setting of interval-valued functions. This note introduces a more generalized class of Godunova-Levin stochastic process that unifies several previously published results through the use of Kulisch-Miranker type order relations that are rarely discussed in relation to stochastic processes. Further, it is the first time that fractional version of Hermite-Hadamard inequality has been developed by using interval-valued stochastic processes in conjunction with a classical operator. Moreover, we give new modified forms for Ostrowski type results and present a new way to treat Jensen type inclusions under interval stochastic processes by using a discrete sequential form. We end with an open problem regarding Milne type results and discuss the importance of different types of order relations related to inequality terms in interval-valued settings.</p></abstract>

Jensen-type inequalities in terms of Lipschitzianity

Jensen-type inequalities in terms of Lipschitzianity

Application of the fink identity to Jensen-type inequalities for higher order convex functions

The focus of this paper is the application of the Fink identity in obtaining Jensen-type inequalities for higher order convex functions. In addition to the basic form, we establish superadditivity and monotonicity relations that correspond to the Jensen inequality in this setting. We also obtain the corresponding Lah-Ribaric inequality. The obtained results are valid for functions of even degree of convexity. With this method, we derive some new bounds for the differences of power means, as well as some new H?lder-type inequalities.

Counter-examples to Breckner-convexity

In this paper, we examine convexity type inequalities. Let D be a nonempty convex subset of a linear space, c>0 and α:D-D→R be a given even function. The inequality f((x+y)/2) ≤ c f(x) + c f(y) + α(x-y) (x,y € D) is the focus of our examinations. We will construct an example to show that for c=1, this Jensen type inequality does not imply the convexity of the function. Then, we compare this inequality with Hermite–Hadamard type inequalities.

Generating probability distributions on intervals and spheres: Convex decomposition

This work studies the convexity of a class of positive definite probability measures with density functionsρ=cλ∑n=0∞g(n)Cnλwλ, that are generated through the interval characteristic sequences. We discuss when such measures supported in an interval are convex and demonstrate how convexity is preserved under multiplication by certain multipliers and under the intertwining product. A key message to be delivered from this work is: any integrable function f on the interval with a polynomial expansion that has relatively fast absolute convergence, can be decomposed into a non-unique pair of positive convex interval probabilities (f+,f−)∈Eλ′×Eλ′ up to normalization, in the sense that f=f+−f−, where Eλ′ is a class of convex functions. Consequently not only the study of an interval distribution is reduced to the study of the properties of class Eλ′, but also the discontinuous probabilistic Galerkin schemes can be simplified. As concrete examples the decomposition of polynomials and truncated normal distribution are computed and graphically illustrated. Such interval measures lead to localized spherical probability measures, and as a convex expression of probability measure is a Jensen type inequality for spherical random variables. Based on the spherical information entropy models, one dependent on the domain partition and the other not, we propose three minimization formulations for the probabilistic finite element method, including one that mixes the least-squares and entropy models.

More Accurate Jensen-Type Inequalities Based on the Lidstone Interpolation Formula

More Accurate Jensen-Type Inequalities Based on the Lidstone Interpolation Formula

Measure-Valued Low Mach Number Limits of Ideal Fluids

As a framework for handling low Mach number limits we consider a notion of measure-valued solution of the incompressible Euler system which explicitly formulates the usually suppressed dependence on the pressure variable. We clarify in which sense such a measure-valued solution is generated as a low Mach limit and state sufficient conditions for this convergence. For the special case of pressure-free solutions we are able to give such sufficient conditions only on the incompressible solution itself. As a necessary condition for low Mach limits we obtain a Jensen-type inequality. The same necessary condition actually holds true for limits of weak solutions or vanishing viscosity limits. We illustrate that this Jensen inequality is not trivially fulfilled. Since measure-valued solutions in the classical sense are always generated by weak solutions, this shows that our solution concept contains more information and that low Mach limits lead to a partial selection criterion for incompressible solutions.

Some New Estimates of Hermite–Hadamard, Ostrowski and Jensen-Type Inclusions for h-Convex Stochastic Process via Interval-Valued Functions

Mathematical programming and optimization problems related to fluid dynamics are heavily influenced by stochastic processes associated with integral and variational inequalities. Furthermore, symmetry and convexity are intrinsically related. Over the last few years, both have become increasingly interconnected so that we can learn from one and apply it to the other. The objective of this note is to convert ordinary stochastic processes into interval stochastic processes due to the wide range of applications in various disciplines. We have developed Hermite–Hadamard (H.H), Ostrowski-, and Jensen-type inequalities using interval h-convex stochastic processes. Our main results can be applied to a variety of new and well-known outcomes as specific situations. The results of this study are expected to stimulate future research on inequalities using fractional and fuzzy integral operators. Furthermore, we validate our main findings by providing some non-trivial examples. To demonstrate their general properties, we illustrate the connections between the examined results and those that have already been published. The results discussed in this article can be seen as improvements and refinements to results that have already been published. This is a fascinating subject that can be investigated in the future to identify equivalent inequalities for various convexity types.

Some well known inequalities for $ (h_1, h_2) $-convex stochastic process via interval set inclusion relation

<abstract><p>This note introduces the concept of $ (h_1, h_2) $-convex stochastic processes using interval-valued functions. First we develop Hermite-Hadmard $ (\mathbb{H.H}) $ type inequalities, then we check the results for the product of two convex stochastic process mappings, and finally we develop Ostrowski and Jensen type inequalities for $ (h_1, h_2) $-convex stochastic process. Also, we have shown that this is a more generalized and larger class of convex stochastic processes with some remark. Furthermore, we validate our main findings by providing some non-trivial examples.</p></abstract>

Improvements of some functional Jensen type inequalities via superquadracity

Improvements of some functional Jensen type inequalities via superquadracity

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>

Some new estimates of well known inequalities for $ (h_1, h_2) $-Godunova-Levin functions by means of center-radius order relation

<abstract><p>In this manuscript, we aim to establish a connection between the concept of inequalities and the novel Center-Radius order relation. The idea of a Center-Radius (CR)-order interval-valued Godunova-Levin (GL) function is introduced by referring to a total order relation between two intervals. Consequently, convexity and nonconvexity contribute to different kinds of inequalities. In spite of this, convex theory turns to Godunova-Levin functions because they are more efficient at determining inequality terms than other convexity classes. Our application of these new definitions has led to many classical and novel special cases that illustrate the key findings of the paper. Using total order relations between two intervals, this study introduces CR-$ (h_1, h_2) $-Goduova-Levin functions. It is clear from their properties and widespread usage that the Center-Radius order relation is an ideal tool for studying inequalities. This paper discusses various inequalities based on the Center-Radius order relation. With the CR-order relation, we can first derive Hermite-Hadamard ($ \mathcal{H.H} $) inequalities and then develop Jensen-type inequality for interval-valued functions ($ \mathcal{IVFS} $) of type $ (h_1, h_2) $-Godunova-Levin function. Furthermore, the study includes examples to support its conclusions.</p></abstract>

A new refinement of Jensen-type inequality with respect to uniformly convex functions with applications in information theory

A new refinement of Jensen-type inequality with respect to uniformly convex functions with applications in information theory

Some properties and inequalities for generalized class of harmonical Godunova-Levin function via center radius order relation

<abstract><p>There are many benefits derived from the speculation regarding convexity in the fields of applied and pure science. According to their definitions, convexity and integral inequality are linked concepts. The construction and refinement of classical inequalities for various classes of convex and nonconvex functions have been extensively studied. In convex theory, Godunova-Levin functions play an important role, because they make it easier to deduce inequalities when compared to convex functions. Based on Bhunia and Samanta's total order relation, harmonically cr-$ h $-Godunova-Levin function is defined in this paper. Utilizing center order (CR) relationship, various types of inequalities can be introduced. (CR)-order relation enables us to derive some Hermite-Hadamard ($ \mathcal{H.H} $) inequality along with a Jensen-type inequality for harmonically $ h $-Godunova-Levin interval-valued functions (GL-$ \mathcal{IVFS} $). Many well-known and new convex functions are unified by this kind of convexity. For further verification of the accuracy of our findings, we provide some numerical examples.</p></abstract>

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.

Some New Generalizations of Integral Inequalities for Harmonical cr-(h1,h2)-Godunova–Levin Functions and Applications

The interval analysis is famous for its ability to deal with uncertain data. This method is useful for addressing models with data that contain inaccuracies. Different concepts are used to handle data uncertainty in an interval analysis, including a pseudo-order relation, inclusion relation, and center–radius (cr)-order relation. This study aims to establish a connection between inequalities and a cr-order relation. In this article, we developed the Hermite–Hadamard (H.H) and Jensen-type inequalities using the notion of harmonical (h1,h2)-Godunova–Levin (GL) functions via a cr-order relation which is very novel in the literature. These new definitions have allowed us to identify many classical and novel special cases that illustrate our main findings. It is possible to unify a large number of well-known convex functions using the principle of this type of convexity. Furthermore, for the sake of checking the validity of our main findings, some nontrivial examples are given.

Operator Jensen’s Inequality for Operator Superquadratic Functions

In this work, an operator superquadratic function (in the operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. A general Bohr’s inequality for positive operators is thus deduced. A Jensen-type inequality is proved. Equivalent statements of a non-commutative version of Jensen’s inequality for operator superquadratic function are also established. Finally, several trace inequalities for superquadratic functions (in the ordinary sense) are provided as well.