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Shigeru Furuichi

Publications and source records attributed to Shigeru Furuichi.

At least 37 records · Page 2Linked to original sources
arXiv2021

Refined Young inequality and its application to divergences

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also study some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys-Tsallis divergence and the Jensen-Shannon-Tsallis divergence.

math.CA↗
arXiv2021

A sharper bound for the Jensen's operator inequality

The primary goal of this paper is to improve the operator version of Jensen inequality. As an application, we provide an improvement for the celebrated Ando's inequality. Additionally, we give a tight bound for the operator Hölder inequality.

math.FA↗
arXiv2020

The logarithmic mean of two convex functionals

The purpose of this paper is to introduce the logarithmic mean of two convex functionals that extends the logarithmic mean of two positive operators. Some inequalities involving this functional mean are discussed as well. The operator versions of the functional theoretical results obtained here are immediately deduced without referring to the theory of operator means.

math.FA↗
arXiv2020

Singular Values Inequalities for Matrix Means

In this article, we show multiple inequalities for the singular values of the difference of matrix means. The obtained results refine and complement some well established results in the literature. Although we target singular values inequalities, we will show several matrix means inequalities, as well.

math.FA↗
arXiv2020

On weighted means and their inequalities

In \cite{PSMA}, Pal et al. introduced some weighted means and gave some related inequalities by using an approach for operator monotone functions. This paper discusses the construction of these weighted means in a simple and nice setting that immediately leads to the inequalities established there. The related operator version is here immediately deduced as well. According to our constructions of the means, we study all cases of the weighted means from three weighted arithmetic/geometric/harmonic means, by the use of the concept such as stable and stabilizable means. Finally, the power symmetric means are studied and new weighted power means are given.

math.FA↗
arXiv2020

Refined inequalities on the weighted logarithmic mean

Inspired by the recent work by R.Pal et al., we give further refined inequalities for a convex Riemann integrable function, applying the standard Hermite-Hadamard inequality. Our approach is different from their one in \cite{PSMA2016}. As corollaries, we give the refined inequalities on the weighted logarithmic mean and weighted identric mean. Some further extensions are also given.

math.CA↗
arXiv2020

On the operator Aczél inequality and its reverse

In this paper, we present some operator and eigenvalue inequalities involving operator monotone, doubly concave and doubly convex functions. These inequalities provide some variants of operator Aczél inequality and its reverse via generalized Kantorovich constant.

math.FA↗
arXiv2020

Some remarks on Tsallis relative operator entropy

This paper intends to give some new estimates for Tsallis relative operator entropy ${{T}_{v}}\left( A|B \right)=\frac{A{{\natural}_{v}}B-A}{v}$. Let $A$ and $B$ be two positive invertible operators with the spectra contained in the interval $J \subset (0,\infty)$. We prove for any $v\in \left[ -1,0 \right)\cup \left( 0,1 \right]$, $$ (\ln_v t)A+\left( A{{\natural}_{v}}B+tA{{\natural}_{v-1}}B \right)\le {{T}_{v}}\left( A|B \right) \le (\ln_v s)A+{{s}^{v-1}}\left( B-sA \right) $$ where $s,t\in J$. Especially, the upper bound for Tsallis relative operator entropy is a non-trivial new result. Meanwhile, some related and new results are also established. In particular, the monotonicity for Tsallis relative operator entropy is improved. Furthermore, we introduce the exponential type relative operator entropies which are special cases of the perspective and we give inequalities among them and usual relative operator entropies.

math.FA↗
arXiv2019

On The Operator Hermite--Hadamard Inequality

The main target of this paper is to discuss operator Hermite--Hadamard inequality for convex functions, without appealing to operator convexity. Several forms of this inequality will be presented and some applications including norm and mean inequalities will be shown too.

math.FA↗
arXiv2019

Inequalities related to some types of entropies and divergences

The aim of this paper is to discuss new results concerning some kinds of parametric extended entropies and divergences. As a result of our studies for mathematical properties on entropy and divergence, we give new bounds for the Tsallis quasilinear entropy and divergence by applying the Hermite-Hadamard inequality. We also give bounds for biparametrical extended entropies and divergences which have been given in \cite{7}. In addition, we study $(r,q)$-quasilinear entropies and divergences as alternative biparametrical extended entropy and divergence, and then we give bounds for them. Finally we obtain inequalities for an extended Lin's divergence and some characterizations of Fermi-Dirac entropy and Bose-Einstein entropy.

cs.IT↗
arXiv2019

Note on constants appearing in refined Young inequalities

In this short note, we give the refined Young inequality with Specht's ratio by only elementary and direct calculations. The obtained inequality is better than one previously shown by the author in 2012. In addition, we give a new property of Specht's ratio. These imply an alternative proof of the refined Young ineqaulity shown by author in 2012. We also give a remark on the relation to Kantorovich constant. Finally, we give a proposition for a corresponding general function.

math.CA↗
arXiv2019

Some new Karamata type inequalities and their applications to some entropies

Some new inequalities of Karamata type are established with a convex function in this paper. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by Pe{\v} carić and Mićić. Applying the obtained results, we give reverses for information inequality (Shannon inequality) in different types, namely ratio type and difference type, under some conditions. Also, we provide interesting inequalities for von Neumann entropy and quantum Tsallis entropy which is a parametric extension of von Neumann entropy. The inequality for von Neumann entropy recovers the non-negativity and gives a refinement for the weaker version of Fannes's inequality for only special cases. Finally, we estimate bounds for the Tsallis relative operator entropy.

math-ph↗
arXiv2018

Some Inequalities involving Heron and Heinz Means of two Convex Functionals

In this paper we first introduce the Heron and Heinz means of two convex functionals. Afterwards, some inequalities involving these functional means are investigated. The operator versions of our theoretical functional results are immediately deduced. We also obtain new refinements of some known operator inequalities via our functional approach in a fast and nice way.

math.FA↗
arXiv2018

Some inequalities for interpolational operator means

Using the properties of geometric mean, we shall show for any $0\le α,β\le 1$, \[f\left( A{{\nabla }_{α}}B \right)\le f\left( \left( A{{\nabla }_{α}}B \right){{\nabla }_{β}}A \right){{\sharp}_{α}}f\left( \left( A{{\nabla }_{α}}B \right){{\nabla }_{β}}B \right)\le f\left( A \right){{\sharp}_{α}}f\left( B \right)\] whenever $f$ is a non-negative operator log-convex function, $A,B\in \mathcal{B}\left( \mathcal{H} \right)$ are positive operators, and $0\le α,β\le 1$. As an application of this operator mean inequality, we present several refinements of the Aujla subadditive inequality for operator monotone decreasing functions. Also, in a similar way, we consider some inequalities of Ando's type. Among other things, it is shown that if $Φ$ is a positive linear map, then \[Φ\left( A{{\sharp}_{α}}B \right)\le Φ\left( \left( A{{\sharp}_{α}}B \right){{\sharp}_{β}}A \right){{\sharp}_{α}}Φ\left( \left( A{{\sharp}_{α}}B \right){{\sharp}_{β}}B \right)\le Φ\left( A \right){{\sharp}_{α}}Φ\left( B \right).\]

math.FA↗
arXiv2018

On further refinements for Young inequalities

In this paper, sharp results on operator Young's inequality are obtained. We first obtain sharp multiplicative refinements and reverses for the operator Young's inequality. Secondly, we give an additive result, which improves a well-known inequality due to Tominaga. We also provide some estimates for $A{{\sharp}_{v}}B-A{{\nabla }_{v}}B$ in which $v\notin \left[ 0,1 \right]$.

math.FA↗
arXiv2018

Further improvements of Young inequality

We focus on the improvements for Young inequality. We give elementary proof for known results by Dragomir, and we give remarkable notes and some comparisons. Finally, we give new inequalities which are extensions and improvements for the inequalities shown by Dragomir.

math.CA↗