On Sherman Method to Deriving Inequalities for Some Classes of Functions Related to Convexity

Niezgoda, Marek

doi:10.1007/978-981-13-3013-1_11

Trends in Mathematics

2018

DOI: 10.1007/978-981-13-3013-1_11

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On Sherman Method to Deriving Inequalities for Some Classes of Functions Related to Convexity

Marek Niezgoda

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Cited by 3 publications

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“…See [1,2,3,7,9,10,14,15,16] for some applications and generalizations of Sherman's inequality (1.6). Statement (1.5) is called Sherman's condition.…”

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confidence: 99%

A note on majorization properties of the Lieb function

Niezgoda

2020

ELA

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In this note, the Lieb function $(A,B) \to \Phi (A,B) = \tr \exp ( A + \log B )$ for an Hermitian matrix $A$ and a positive definite matrix $B$ is studied. It is shown that $\Phi$ satisfies a majorization property of Sherman type induced by a doubly stochastic operator. The variant for commuting matrices is also considered. An interpretation is given for the case of the orthoprojection operator onto the space of block diagonal matrices.

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“…See [1,2,3,7,9,10,14,15,16] for some applications and generalizations of Sherman's inequality (1.6). Statement (1.5) is called Sherman's condition.…”

mentioning

confidence: 99%

A note on majorization properties of the Lieb function

Niezgoda

2020

ELA

Self Cite

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A Jensen–Sherman type inequality with a control map for G-invariant convex functions

Niezgoda

2020

Positivity

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Nonlinear Sherman-type inequalities

Niezgoda

2018

Advances in Nonlinear Analysis

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An important class of Schur-convex functions is generated by convex functions via the well-known Hardy–Littlewood–Pólya–Karamata inequality. Sherman’s inequality is a natural generalization of the HLPK inequality. It can be viewed as a comparison of two special inner product expressions induced by a convex function of one variable. In the present note, we extend the Sherman inequality from the (bilinear) inner product to a (nonlinear) map of two vectorial variables satisfying the Leon–Proschan condition. Some applications are shown for directional derivatives and gradients of Schur-convex functions.

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On Sherman Method to Deriving Inequalities for Some Classes of Functions Related to Convexity

Cited by 3 publications

References 19 publications

A note on majorization properties of the Lieb function

A note on majorization properties of the Lieb function

A Jensen–Sherman type inequality with a control map for G-invariant convex functions

Nonlinear Sherman-type inequalities

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