On the Ky Fan inequality and some of its applications

In this paper, we extend some significant Ky Fan type inequalities in a large setting to operators on Hilbert spaces and derive their equality conditions. Among other things, we prove that if f : [0, ∞) → [0, ∞) is an operator monotone function with f (1) = 1, f ′ (1) = µ, and associated mean σ, then for all operators A and B on a complex Hilbert space H such that 0 < A, B ≤ 1 2 I, we havewhere I is the identity operator on H , A ′ := I − A, B ′ := I − B, and ∇ µ is the µ-weighted arithmetic mean.

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“…where 4 = 0 for all t ∈ sp(T ). Since γ = 0, 1, we have sp(T ) = {1}, which concludes that T = I, and so A = B.…”

Section: Ky Fan Type Inequalitiesmentioning

confidence: 99%

“…Several mathematicians obtained extensions, refinements, and various related results. For more information on Ky Fan type inequalities, see [4,11,13,15,16], also [10,17], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Operator Ky Fan type inequalities

Habibzadeh

Rooin

Moslehian

2018

Linear Algebra and its Applications

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“…Utilizing the monotonicity of (1 − x n )/n as a function of n, Rooin [63] established some Ky Fan type inequalities. Applying the convexity of the function f (x) = (e x − α)/(1 + e x ) (α > 0) and the Jensen inequality, the authors of [17] generalized the Ky Fan inequality. Of course, Ky Fan [22,1951] proved that the function…”

Section: Ky Fan Mean Inequalitymentioning

confidence: 99%

Ky Fan inequalities

Moslehian

2011

Preprint

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“…New generalizations of Aczél's inequality and Popoviciu's inequality were given [8] , which has broad applications to mathematical analysis. In [9], a generalization of the Ky Fan discrete multivariate inequality was obtained and some of its applications were also given.…”

Section: Introductionmentioning

confidence: 99%

An Elegant Inequality

Liu

2021

Preprint

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A new inequality, (x) p +(1−x) 1 p ≤ 1 for p ≥ 1 and 1 2 ≥ x ≥ 0 is found and proved. The inequality looks elegant as it integrates two number pairs (x and 1 − x, p and 1 p ) whose summation and product are one. Its right hand side, 1, is the strict upper bound of the left hand side. The equality cannot be categorized into any known type of inequalities such as H ölder, Minkowski etc. In proving it, transcendental equations have been met with, so some novel techniques have been built to get over the difficulty.

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On the Ky Fan inequality and some of its applications

Abstract: a b s t r a c t In this paper, by applying Jensen's inequality for convex functions, a generalization of the Ky Fan discrete multivariate inequality is obtained and some of its applications are also given.

Cited by 4 publications

References 6 publications

Operator Ky Fan type inequalities

Operator Ky Fan type inequalities

Ky Fan inequalities

An Elegant Inequality

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