scholarly journals Inequalities of Convex Functions and Self-Adjoint Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Zlatko Pavić
Keyword(s):  
Convex Functions ◽  
Operator Mean ◽  
Operator Means ◽  
Adjoint Operators ◽  
Arithmetic Operator

The paper offers generalizations of the Jensen-Mercer inequality for self-adjoint operators and generally convex functions. The obtained results are applied to define the quasi-arithmetic operator means without using operator convexity. The version of the harmonic-geometric-arithmetic operator mean inequality is derived as an example.

scholarly journals Operator log-convex functions and operator means

2010 ◽  
Vol 350 (3) ◽  
pp. 611-630 ◽  
Author(s):  
Tsuyoshi Ando ◽  
Fumio Hiai
Keyword(s):  
Convex Functions ◽  
Operator Means

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.

scholarly journals Extension of Jensen's Inequality for Operators without Operator Convexity

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Jadranka Mićić ◽  
Zlatko Pavić ◽  
Josip Pečarić
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Quasiarithmetic Means ◽  
Adjoint Operators

We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.

scholarly journals Some operator inequalities for operator means and positive linear maps

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3751-3758
Author(s):  
Jianguo Zhao
Keyword(s):  
Type Inequality ◽  
Real Numbers ◽  
Positive Real ◽  
Operator Inequalities ◽  
Linear Map ◽  
Operator Mean ◽  
Linear Maps ◽  
Operator Means ◽  
Arbitrary Operator ◽  
Positive Linear Maps

In this note, some operator inequalities for operator means and positive linear maps are investigated. The conclusion based on operator means is presented as follows: Let ? : B(H) ? B(K) be a strictly positive unital linear map and h-1 IH ? A ? h1IH and h-12 IH ? B ? h2IH for positive real numbers h1, h2 ? 1. Then for p > 0 and an arbitrary operator mean ?, (?(A)??(B))p ? ?p?p(A?*B), where ?p = max {?2(h1,h2)/4)p, 1/16?2p(h1,h2)}, ?(h1h2) = (h1 + h-1 1)?(h2 + h-12). Likewise, a p-th (p ? 2) power of the Diaz-Metcalf type inequality is also established.

scholarly journals Some quantum $f$-divergence inequalities for convex functions of self-adjoint operators

2017 ◽  
Vol 37 (1) ◽  
pp. 125-139
Author(s):  
Ali Taghavi ◽  
Vahid Darvish ◽  
Haji Mohammad Nazari ◽  
Silvestry Server Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Convex Functions ◽  
Trace Class ◽  
Trace Class Operators ◽  
Adjoint Operators ◽  
New Inequalities

In this paper, some new inequalities for convex functions of self-adjoint operators are obtained. As applications, we present some inequalities for quantum $f$-divergence of trace class operators in Hilbert Spaces.

scholarly journals Additive inequalities for weighted harmonic and arithmetic operator means

2019 ◽  
Vol 73 (1) ◽  
pp. 1
Author(s):  
Sever Dragomir
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Weighted Arithmetic ◽  
Operator Means ◽  
The Difference ◽  
Arithmetic Operator

In this paper we establish some new upper and lower bounds for the difference between the weighted arithmetic and harmonic operator means under various assumptions for the positive invertible operators A, B. Some applications when A, B are bounded above and below by positive constants are given as well.

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