scholarly journals Operator inequalities of Jensen type

2013 ◽  
Vol 1 ◽  
pp. 9-21
Author(s):  
M. S. Moslehian ◽  
J. Mićić ◽  
M. Kian
Keyword(s):  
Convex Function ◽  
Type Operator ◽  
Operator Inequalities ◽  
Continuous Convex Function ◽  
Adjoint Operators

Abstract We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if f : [0;1) → ℝ is a continuous convex function with f(0) ≤ 0, thenfor all operators Ci such that (i=1 , ... , n) for some scalar M ≥ 0, where and

scholarly journals External Jensen-type operator inequalities via superquadraticity

2020 ◽  
pp. 31-31
Author(s):  
Mohsen Kian ◽  
Mario Krnic ◽  
Mohsen Delavar
Keyword(s):  
Type Operator ◽  
Jensen Inequality ◽  
Operator Inequalities ◽  
Superquadratic Functions ◽  
Adjoint Operators ◽  
External Form

In this paper we establish several Jensen-type operator inequalities for a class of superquadratic functions and self-adjoint operators. Our results are given in the so-called external form. As an application, we give improvements of the H?lder?McCarthy inequality and the classical discrete and integral Jensen inequality in the corresponding external forms. In addition, the established Jensen-type inequalities are compared with the previously known results and we show that our results provide more accurate estimates in some general settings.

scholarly journals Interpolating Jensen-type operator inequalities for log-convex and superquadratic functions

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4523-4535
Author(s):  
Mojtaba Bakherad ◽  
Mohsen Kian ◽  
Mario Krnic ◽  
Seyyed Ahmadi
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Type Operator ◽  
Operator Inequality ◽  
Operator Inequalities ◽  
Superquadratic Functions

Motivated by some recently established Jensen-type operator inequalities related to a convex function, in the present paper we derive several more accurate Jensen-type operator inequalities for certain subclasses of convex functions. More precisely, we obtain interpolating series of Jensen-type inequalities utilizing log-convex and non-negative superquadratic functions. In particular, we obtain the corresponding refinements of the Jensen-Mercer operator inequality for such classes of functions.

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (1) ◽  
pp. 155-158 ◽  
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

scholarly journals Another Aspect of Triangle Inequality

2011 ◽  
Vol 2011 ◽  
pp. 1-5
Author(s):  
Kichi-Suke Saito ◽  
Runling An ◽  
Hiroyasu Mizuguchi ◽  
Ken-Ichi Mitani
Keyword(s):  
Convex Function ◽  
Triangle Inequality ◽  
Unit Interval ◽  
Usual Sense ◽  
Continuous Convex Function

We introduce the notion of ψ-norm by considering the fact that an absolute normalized norm on C2 corresponds to a continuous convex function ψ on the unit interval [0,1] with some conditions. This is a generalization of the notion of q-norm introduced by Belbachir et al. (2006). Then we show that a ψ-norm is a norm in the usual sense.

scholarly journals Hausdorff dimension of the nondifferentiability set of a convex function

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5827-5831
Author(s):  
Reza Mirzaie
Keyword(s):  
Riemannian Manifold ◽  
Hausdorff Dimension ◽  
Convex Function ◽  
Open Subset ◽  
Upper Bound ◽  
Continuous Convex Function

We find an upper bound for the Hausdorff dimension of the nondifferentiability set of a continuous convex function defined on a Riemannian manifold. As an application, we show that the boundary of a convex open subset of Rn, n ? 2, has Hausdorff dimension at most n - 2.

scholarly journals Kantorovich type operator inequalities via the Specht ratio

2004 ◽  
Vol 377 ◽  
pp. 69-81 ◽  
Author(s):  
Jun Ichi Fujii ◽  
Yuki Seo ◽  
Masaru Tominaga
Keyword(s):  
Type Operator ◽  
Operator Inequalities

Reverse Jensen-Mercer Type Operator Inequalities

2016 ◽  
Vol 31 ◽  
pp. 87-99 ◽  
Author(s):  
Ehsan Anjidani ◽  
Mohammad Reza Changalvaiy
Keyword(s):  
Hilbert Space ◽  
Selfadjoint Operator ◽  
Type Operator ◽  
Operator Inequalities ◽  
Linear Map ◽  
Positive Linear Map

Let $A$ be a selfadjoint operator on a Hilbert space $\mathcal{H}$ with spectrum in an interval $[a,b]$ and $\phi:B(\mathcal{H})\rightarrow B(\mathcal{K})$ be a unital positive linear map, where $\mathcal{K}$ is also a Hilbert space. Let $m,M\in J$ with $m

scholarly journals A continuity property related to an index of non-separability and its applications

1992 ◽  
Vol 46 (1) ◽  
pp. 67-79 ◽  
Author(s):  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Convex Function ◽  
Compact Subset ◽  
Metric Space ◽  
Dense Subset ◽  
Countable Family ◽  
Set Valued Mapping ◽  
Continuous Convex Function ◽  
Fréchet Differentiable

For a set E in a metric space X the index of non-separability is β(E) = inf{r > 0: E is covered by a countable-family of balls of radius less than r}.Now, for a set-valued mapping Φ from a topological space A into subsets of a metric space X we say that Φ is β upper semi-continuous at t ∈ A if given ε > 0 there exists a neighbourhood U of t such that β(Φ(U)) < ε. In this paper we show that if the subdifferential mapping of a continuous convex function Φ is β upper semi-continuous on a dense subset of its domain then Φ is Fréchet differentiable on a dense Gδ subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small.

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