The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

Li, Yongtao; Gu, Xian-Ming; Zhao, Jianxing

doi:10.3390/sym10090380

Cited by 15 publications

(10 citation statements)

References 12 publications

(15 reference statements)

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“…Proof. The equivalence between (iv) and (vi) is given in Theorem 2.2, the equivalence among (i), (iii) and (vi), one can find in [11] as well as (ii), (iii) and (iv) in [15], the equivalence between (iii) and (v) is shown in [16].…”

Section: Resultsmentioning

confidence: 85%

A note on the proofs of generalized Radon inequality

Xiao

2018

Mathematica Moravica

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In this paper, we introduce and prove several generalizations of the Radon inequality. The proofs in the current paper unify and also are simpler than those in early published work. Meanwhile, we find and show the mathematical equivalences among the Bernoulli inequality, the weighted AM-GM inequality, the Hölder inequality, the weighted power mean inequality and the Minkowski inequality. Finally, some applications involving the results proposed in this work are shown.2010 Mathematics Subject Classification. Primary: 26D15. Key words and phrases. The Bergström inequality, the Radon inequality, the weighted power mean inequality, equivalence, the Hölder inequality.Full paper.

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Section: Resultsmentioning

confidence: 85%

A note on the proofs of generalized Radon inequality

Xiao

2018

Mathematica Moravica

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“…Next we apply the Arithmetic mean-geometric mean inequality [13] and Variant Young's inequality and obtain:…”

Section: The ϵ−Independent Prior Estimate (Energy Estimates)mentioning

confidence: 99%

A Rigorous Homogenization for a Two-Scale Convergence Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil

Sakyi

Amoako-Yirenkyi

Dontwi

2020

JAMCS

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We present a rigorous homogenization approach to modelling piping ow erosion in a spatially heterogeneous soil. The aim is to provide a justication to a formal homogenization approach to piping ow erosion with deposition in a spatially heterogeneous soil. Under the assumption that the soil domain is perforated periodically with cylindrical repeating microstructure, we begin by proving that a solution to the proposed set of microscopic equations exist. Two-scale convergence is then used to study the asymptotic behaviour of solutions to the microscopic problem as the microscopic length scale approaches zero(0). We thus derive rigorously a homogenized model or macro problem as well as explicit formula for the eective coecients. A strong observation from the numerical simulation was that, soil particle concentration in the water/soil mixture decreases but at a decreasing rate whereas soil particle deposition increases at regions with increasing amount of particle concentration in the ow causing a reduction in bare pore spaces across the soil domain.

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“…The refinements or extensions of inequalities take place widely in inequality theory. There are many approaches or techniques to obtain new versions of inequalities and studies on the equivalences of inequalities (See [3,8,10,12]). Some of these approaches are based on the convexity of functions as in Jensen type, Hermite-Hadamard type inequalities.…”

Section: Introductionmentioning

confidence: 99%