# On the sum of squared logarithms inequality and related inequalities

@article{Dannan2014OnTS, title={On the sum of squared logarithms inequality and related inequalities}, author={Fozi M. Dannan and Patrizio Neff and Christian Thiel}, journal={arXiv: Classical Analysis and ODEs}, year={2014} }

We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors $a,b\in\mathbb{R}_+^n$ so that $\sum_{i=1}^n(\log a_i)^2\ \leq\ \sum_{i=1}^n(\log b_i)^2\,.\notag $ Generalizations of some inequalities from information theory are obtained, including a generalized information inequality and a generalized log sum inequality, which states for $a,b\in\mathbb{R}_+^n$ and…

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## 19 References

### On the generalised sum of squared logarithms inequality

- Mathematics
- 2014

Assume n≥2$n\geq2$. Consider the elementary symmetric polynomials ek(y1,y2,…,yn)$e_{k}(y_{1},y_{2},\ldots, y_{n})$ and denote by E0,E1,…,En−1$E_{0},E_{1},\ldots,E_{n-1}$ the elementary symmetric…

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AbstractWe consider a family of isotropic volumetric–isochoric decoupled strain energies
$$F \mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll}\frac{\mu}{k}\,e^{k\,\|{\rm…

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We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$\begin{aligned} F\mapsto W_{\mathrm{eH}}(F):=\widehat{W}_{\mathrm{eH}}(U):=\left \{ \begin{array}{l@{\quad}l}…

### The exponentiated Hencky-logarithmic strain energy: part III—coupling with idealized multiplicative isotropic finite strain plasticity

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AbstractWe investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric–isochoric decoupled strain energies
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- 2013

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### Elements of Information Theory

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### Classical and New Inequalities in Analysis

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