scholarly journals Chebyshev-Type Integral Inequalities for Continuous Fields of Operators Concerning Khatri–Rao Products and Synchronous Properties

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 422
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Radon Measure ◽  
Geometric Mean ◽  
Integral Inequalities ◽  
Counting Measure ◽  
Integrable Functions ◽  
Discrete Inequalities ◽  
Finite Space ◽  
Continuous Fields ◽  
Adjoint Operators ◽  
Locally Compact Hausdorff Space

We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Ω equipped with a finite Radon measure μ . Under certain assumptions on synchronous Khatri–Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri–Rao products. We also establish Chebyshev-type inequalities involving Khatri–Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev–Grüss integral inequality via oscillations when μ is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Ω to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri–Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.

scholarly journals Integral Inequalities of Chebyshev Type for Continuous Fields of Hermitian Operators Involving Tracy–Singh Products and Weighted Pythagorean Means

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1256
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Hausdorff Space ◽  
Geometric Mean ◽  
Integral Inequalities ◽  
Kronecker Products ◽  
Counting Measure ◽  
Discrete Inequalities ◽  
Finite Space ◽  
Continuous Fields ◽  
Continuous Field ◽  
Locally Compact Hausdorff Space

In this paper, we establish several integral inequalities of Chebyshev type for bounded continuous fields of Hermitian operators concerning Tracy-Singh products and weighted Pythagorean means. The weighted Pythagorean means considered here are parametrization versions of three symmetric means: the arithmetic mean, the geometric mean, and the harmonic mean. Every continuous field considered here is parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. Tracy-Singh product versions of the Chebyshev-Grüss inequality via oscillations are also obtained. Such integral inequalities reduce to discrete inequalities when the space is a finite space equipped with the counting measure. Moreover, our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.

scholarly journals On certain new CAUCHY–TYPE fracitioanl integral inequalities and OPIAL–TYPE fractional derivative inequalities

2015 ◽  
Vol 46 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Amit Chouhan
Keyword(s):  
Fractional Derivative ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Future Research ◽  
Cauchy Type ◽  
Fractional Integral Operators ◽  
Integrable Functions ◽  
New Inequalities

The aim of this paper is to establish several new fractional integral and derivative inequalities for non-negative and integrable functions. These inequalities related to the extension of general Cauchy type inequalities and involving Saigo, Riemann-Louville type fractional integral operators together with multiple Erdelyi-Kober operator. Furthermore the Opial-type fractional derivative inequality involving H-function is also established. The generosity of H-function could leads to several new inequalities that are of great interest of future research.

scholarly journals Sharp integral inequalities based on general Euler two-point formulae

2005 ◽  
Vol 46 (4) ◽  
pp. 555-574 ◽  
Author(s):  
J. Pečarić ◽  
I. Perić ◽  
A. Vukelić
Keyword(s):  
Bounded Variation ◽  
Integral Inequalities ◽  
Functions Of Bounded Variation ◽  
Quadrature Formulae ◽  
Lipschitzian Functions ◽  
Integrable Functions

AbstractWe consider a family of two-point quadrature formulae, using some Euler-type identities. A number of inequalities, for functions whose derivatives are either functions of bounded variation, Lipschitzian functions or R-integrable functions, are proved.

scholarly journals Non-weak compactness of the integration map for vector measures

1993 ◽  
Vol 54 (3) ◽  
pp. 287-303 ◽  
Author(s):  
S. Okada ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Volterra Operator ◽  
Continuous Operator ◽  
Weakly Compact ◽  
Vector Measures ◽  
Mean Convergence ◽  
Integrable Functions ◽  
The Mean ◽  
The Fourier Transform ◽  
Adjoint Operators

AbstractLet m be a vector measure with values in a Banach space X. If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map Im which sends each f of L1(m) to the element ∫fdm (of X). Many properties of the (continuous) operator Im are closely related to the nature of the space L1(m). In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that Im is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1 ([−π, π]), the Volterra operator on L1 ([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps Im (or their restrictions) for suitable measures m.

Relaxation of quasi-convex integrals of arbitrary order

1994 ◽  
Vol 124 (5) ◽  
pp. 927-946 ◽  
Author(s):  
Micol Amar ◽  
Virginia De Cicco
Keyword(s):  
Integral Representation ◽  
Total Variation ◽  
Radon Measure ◽  
Arbitrary Order ◽  
Lower Semicontinuous ◽  
Integrable Functions ◽  
Lower Semicontinuous Envelope ◽  
Representation Result

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩf(∇ku)dxon the spaceBVk(Ω:ℝm) of the integrable functions, whose thef-th derivative in the sense of distributions is a Radon measure with bounded total variation.

scholarly journals DISCRETE POINCARE-TYPE INEQUALITIES

1998 ◽  
Vol 29 (2) ◽  
pp. 145-153
Author(s):  
WING-SUM CHEUNG
Keyword(s):  
Discrete Analogue ◽  
Integral Inequalities ◽  
Independent Variables ◽  
Type Integral ◽  
Discrete Inequalities ◽  
Poincaré Type Inequalities

In this paper some discrete analogue of Poincare-type integral inequalities involving many independent variables are established. These in turn can be used to serve as generators of other interesting discrete inequalities.

scholarly journals Certain Inequalities Involving Generalized Erdélyi-Kober Fractionalq-Integral Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Praveen Agarwal ◽  
Soheil Salahshour ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Integrable Functions ◽  
Special Cases

In recent years, a remarkably large number of inequalities involving the fractionalq-integral operators have been investigated in the literature by many authors. Here, we aim to present some new fractional integral inequalities involving generalized Erdélyi-Kober fractionalq-integral operator due to Gaulué, whose special cases are shown to yield corresponding inequalities associated with Kober type fractionalq-integral operators. The cases of synchronous functions as well as of functions bounded by integrable functions are considered.

A Converse of the Loewner–Heinz Inequality, Geometric Mean and Spectral Order

2014 ◽  
Vol 57 (2) ◽  
pp. 565-571 ◽  
Author(s):  
Mitsuru Uchiyama
Keyword(s):  
Real Number ◽  
Geometric Mean ◽  
Heinz Inequality ◽  
Spectral Order ◽  
Adjoint Operators

AbstractLet A, B be non-negative bounded self-adjoint operators, and let a be a real number such that 0 < a < 1. The Loewner–Heinz inequality means that A ≤ B implies that Aa ≦ Ba. We show that A ≤ B if and only if (A + λ)a ≦ (B + λ)a for every λ > 0. We then apply this to the geometric mean and spectral order.

Operator m-convex functions

2018 ◽  
Vol 25 (1) ◽  
pp. 93-107
Author(s):  
Jamal Rooin ◽  
Akram Alikhani ◽  
Mohammad Sal Moslehian
Keyword(s):  
Hilbert Space ◽  
Convex Functions ◽  
Weight Functions ◽  
Operator Inequality ◽  
Jensen Inequality ◽  
Hilbert Space Operators ◽  
Operator Version ◽  
Continuous Fields ◽  
Adjoint Operators ◽  
Comprehensive Study

AbstractThe aim of this paper is to present a comprehensive study of operatorm-convex functions. Let{m\in[0,1]}, and{J=[0,b]}for some{b\in\mathbb{R}}or{J=[0,\infty)}. A continuous function{\varphi\colon J\to\mathbb{R}}is called operatorm-convex if for any{t\in[0,1]}and any self-adjoint operators{A,B\in\mathbb{B}({\mathscr{H}})}, whose spectra are contained inJ, we have{\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)}. We first generalize the celebrated Jensen inequality for continuousm-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements. Introducing the notion of operatorm-convexity, we extend the Choi–Davis–Jensen inequality for operatorm-convex functions. We also present an operator version of the Jensen–Mercer inequality form-convex functions and generalize this inequality for operatorm-convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen–Mercer operator inequality for operatorm-convex functions, we construct them-Jensen operator functional and obtain an upper bound for it.

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