# Shigeru FuruichiNihon University | Nichidai · Information science

Shigeru Furuichi

Professor

## About

193

Publications

12,992

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1,902

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Citations since 2017

Introduction

Please find all my manuscripts in arXiv.
“Advances in mathematical inequalities” is a book so that I can not share due to copyright with publisher.
Thank you very much for your understanding.

**Skills and Expertise**

## Publications

Publications (193)

Entropy is an important concept in many fields related to communications [...]

The main goal of this article is to present new inequalities for the spectral geometric mean A♮ t B of two positive definite operators A, B on a Hilbert space. The obtained results complement many known inequalities for the geometric mean A♯ t B. In particular, explicit comparisons between A♮ t B and A♯ t B are given, with applications towards Ando...

In information theory, the well-known log-sum inequality is a fundamental tool which indicates the non-negativity for the relative entropy. In this article, we establish a set of inequalities which are similar to the log-sum inequality involving two functions defined on scalars. The parametric extended log-sum inequalities are shown. We extend thes...

The ordering between Wigner--Yanase--Dyson function and logarithmic mean is known. Also bounds for logarithmic mean are known. In this paper, we give two reverse inequalities for Wigner--Yanase--Dyson function and logarithmic mean. We also compare the obtained results with the known bounds of the logarithmic mean. Finally, we give operator inequali...

This paper aims to characterize the function appearing in the weighted Hermite-Hadamard inequality. We provide improved inequalities for the weighted means as applications of the obtained results. Modifications of the weighted Hermite-Hadamard inequality are also presented. Our results contain some exciting inequalities and extensions of the known...

To better understand the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results known for positive definite matrices. More precisely, we have several results that allow a better understanding of...

The main goal of this exposition is to present further analysis of the Kantorovich and Ando operator inequalities. In particular, a new proof of Ando's inequality is given, a new non-trivial refinement of Kantorovich inequality is shown, and some equivalent forms of Kantorovich inequality are presented with a Minkowski-type application.

This article improves the triangle inequality for complex numbers, using the Hermite-Hadamard inequality for convex functions. Then, applications of the obtained refinement are presented to include some operator inequalities. The operator applications include numerical radius inequalities and operator mean inequalities.

In this paper, we introduce operator geodesically convex and operator convex-log functions and characterize some properties of them. Then we apply these classes of functions to present several operator Azcél and Minkowski-type inequalities extending some known results. The concavity counterparts are also considered.

In this paper, we give new singular value inequalities and determinant inequalities including the inverse of A, B, and A+B for sector matrices. We also give the matrix inequalities for sector matrices with a positive multilinear map. Our obtained results give generalizations for the known results.

An upper bound of the logarithmic mean is given by a convex conbination of the arithmetic mean and the geometric mean. In addition, a lower bound of the logarithmic mean is given by a geometric bridge of the arithmetic mean and the geometric mean. In this paper, we study the bounds of the logarithmic mean. We give operator inequalities and norm ine...

A considerable amount of literature in the theory of inequality is devoted to the study of Jensen's and Young's inequality. This article presents a number of new inequalities involving the log-convex functions and the geometrically convex functions. As their consequences, we derive the refinements for Young's inequality and Jensen's inequality. In...

Refining and reversing weighted arithmetic-geometric mean inequalities have been studied in many papers. In this paper, we provide some bounds for the differences between the weighted arithmetic and geometric means , using known inequalities. We improve the results given by Furuichi-Ghaemi-Gharakhanlu and Sababheh-Choi. We also give some bounds on...

In this article, we employ a standard convexity argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.

The main goal of this article is to present new inequalities for the spectral geometric mean $A\natural_t B$ of two positive definite operators $A,B$ on a Hilbert space. The obtained results complement many known inequalities for the geometric mean $A\sharp_t B$. In particular, explicit comparisons between $A\natural_t B$ and $A\sharp_t B$ are give...

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.

We give a refined Young inequality which generalizes the inequality by Zou--Jiang. We also show the upper bound for the logarithmic mean by the use of the weighted geometric mean and the weighted arithmetic mean. Furthermore, we show some inequalities among the weighted means.

Some mathematical inequalities among various weighted means are studied. Inequalities on weighted logarithmic mean are given. Besides, the gap in Jensen's inequality is studied as a convex function approach. Consequently, some non-trivial inequalities on means are given. Some operator inequalities are also shown.

In this article, we employ a standard convex argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.

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 sho...

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 studied some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we s...

In (Pal et al. in Linear Multilinear Algebra 64(12):2463–2473, 2016), 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 the...

Mercer’s inequality for convex functions is a variant of Jensen’s inequality, with an operator version that is still valid without operator convexity. This paper is two folded. First, we present a Mercer-type inequality for operators without assuming convexity nor operator convexity. Yet, this form refines the known inequalities in the literature....

This article proposes a new two-parameter generalized entropy, which can be reduced to the Tsallis and Shannon entropies for specific values of its parameters. We develop a number of information-theoretic properties of this generalized entropy and divergence, for instance, the sub-additive property, strong sub-additive property, joint convexity, an...

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 theor...

In this article, we present a new treatment of the arithmetic-geometric mean inequality and its siblings, the Heinz and the Young inequalities. New refinements via calculus computations and convex analysis are presented and a new Heinz-type inequality is presented for any symmetric operator mean.

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 theor...

In this article, we explore the celebrated Grüss inequality, where we present a new approach using the Grüss inequality to obtain new refinements of operator means inequalities. We also present several operator Grüss-type inequalities with applications to the numerical radius and entropies.

In this article, we explore the celebrated Gr\"{u}ss inequality, where we present a new approach using the Gr\"{u}ss inequality to obtain new refinements of operator means inequalities. We also present several operator Gr\"{u}ss-type inequalities with applications to the numerical radius and entropies.

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.

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.

The main goal of this article is to present several refinements and reverses of well known operator inequalities. These inequalities include operator means, operator monotone functions, operator log-convex functions and positive linear maps. Among many other results, we show that for any 0 ≤ α, β ≤ 1, f (A∇ α B) ≤ f ((A∇ α B) ∇ β A) α f ((A∇ α B) ∇...

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 d...

In information theory, the so-called log-sum inequality is fundamental and a kind of generalization of the non-nagativity for the relative entropy. In this paper, we show the generalized log-sum inequality for two functions defined for scalars. We also give a new result for commutative matrices. In addition, we demonstrate further results for gener...

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...

In this paper, we introduce operator geodesically convex and operator convex-log functions and characterize some properties of them. Then apply these classes of functions to present several operator Azc\'{e}l and Minkowski type inequalities extending some known results. The concavity counterparts are also considered.

Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called $g-$convexity as a generalization of $\log-$convexity. Then we prove that $g...

Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called g−convexity as a generalization of log −convexity. Then we prove that g−conv...

In this work, we present a new technique for the oscillatory properties of solutions of higher-order differential equations. We set new sufficient criteria for oscillation via comparison with higher-order differential inequalities. Moreover, we use the comparison with first-order differential equations. Finally, we provide an example to illustrate...

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 Acz\'{e}l inequality and its reverse via generalized Kantorovich constant.

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+\l...

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 appoarch 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...

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 \(v\in \left[ -1,0 \right) \cup \left( 0,1 \right] \), $$\begin{...

In this short paper, we establish a reverse of the derived inequalities for sector matrices by Tan and Xie, with Kantorovich constant. Then, as application of our main theorem, some inequalities for determinant and unitarily invariant norm are presented.

In this paper, we give new singular value inequalities and determinant inequalities including the inverse of $A$, $B$ and $A+B$ for sectorial matrices. We also give a square of $\Re A \sharp_v \Re B \leq \Re A\sharp_v B $ for $v\in [0,1]$ using Kantorovich constant for accretive matrices. In addition, we give the matrix inequalities for sectorial m...

Based on the data collected during a full-scale experiment, the order/disorder characteristics of a compartment fire are researched. We discuss methods, algorithms and the novelty of our entropic approach. From our analysis, we claim that the permutation type hypoentropies can be successfully used to detect unusual data and to perform relevant anal...

Some new inequalities of Karamata type are established with a convex function. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by PečnaričA and MičAičA.
Applying the obtained results, we give reverses for information inequality (Shannon inequality) in different types, namely ratio type and d...

Based on the data collected during a full-scale experiment, the order/disorder characteristics of a compartment fire are researched. We discuss methods, algorithms and the novelty of our entropic approach. From our analysis, we claim that the permutation type hypoentropies can be successfully used to detect unusual data and to perform relevant anal...

The main target of this paper is to discuss operator Hermite--Hadamard inequality for convex functions, without appealing to operator convexity. However, this will be at the cost of additional conditions or weaker estimates. Several forms of this inequality will be presented and some applications including norm and mean inequalities will be shown t...

In this article, we present some new inequalities for numerical radius of Hilbert space operators via convex functions. Our results generalize and improve earlier results by El-Haddad and Kittaneh. Among several results, we show that if $A\in \mathbb{B}\left( \mathcal{H} \right)$ and $r\ge 2$, then \[{{w}^{r}}\left( A \right)\le {{\left\| A \right\...

In this article, we present some new inequalities for numerical radius of Hilbert space operators via convex functions. Our results generalize and improve earlier results by El-Haddad and Kittaneh. Among several results, we show that if A ∈ B (H) and r ≥ 2, then w r (A) ≤ A r − inf x=1 ||A| − w (A)| r 2 x 2 where w (·) and ·· denote the numerical r...

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 biparamet...

The present paper is devoted to the study of Jensen-Mercer-type inequalities. Our results generalize and improve some earlier results in the literature.

I. Sason obtained the tight bounds for symmetric divergence measures are derived by applying the results established by G. L. Gilardoni. In this article, we are going to report two kinds of extensions for the above results, namely classical q-extension and non-commutative extension.

The aim of this paper is to discuss new results concerning some kinds of parametric extended entropies and divergences. As a sereis of our studies for mathematical properties on entropy and divergence, we give new bounds for Tsallis quasilinear entropy and divergence by applying Hermite-Hadamard inequality. We also give bounds for two parameter-ext...

The main goal of this article is to present a new approach, made up of integrals, to refining convex functions inequalities, with applications to several known inequalities like Jensen, Hermite-Hadamard and means inequalities. The first main result will be a Hilbert space operator version, which will be utilized to obtain the aforementioned refinem...

We focus on the improvement of operator Kantorovich type inequalities. Among the consequences, we improve the main result of the paper [H.R. Moradi, I.H. Gümüş, Z. Heydarbeygi, A glimpse at the operator Kantorovich inequality, Linear Multilinear Algebra, doi:10.1080/03081087.2018.1441799].

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. W...

We give some new refinements and reverses Young inequalities in both additive-type and multiplicative-type for two positive numbers/operators. We show our advantages by comparing with known results. A few applications are also given. Some results relevant to the Heron mean are also considered.

Some new inequalities of Karamata type are established with a convex function in this article. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by Pe\v{c}ari\'c and Mi\'ci\'c. Applying the obtained results, we give reverses for information inequality (Shannon inequality) in different types, n...

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 the difference A 1/2 A −1/2 BA −1/2 v A 1/2 −{(1...

In this article, we present exponential-type inequalities for positive linear mappings and Hilbert space operators, by means of convexity and the Mond-Pečarić method. The obtained results refine and generalize some known results. As an application, we present extensions for operator-like geometric and harmonic means inequalities.

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...

Using the properties of geometric mean, we shall show for any $0\le \alpha ,\beta \le 1$, \[f\left( A{{\nabla }_{\alpha }}B \right)\le f\left( \left( A{{\nabla }_{\alpha }}B \right){{\nabla }_{\beta }}A \right){{\sharp}_{\alpha }}f\left( \left( A{{\nabla }_{\alpha }}B \right){{\nabla }_{\beta }}B \right)\le f\left( A \right){{\sharp}_{\alpha }}f\le...

Functional version for the so-called Furuta parametric relative operator entropy is here investigated. Some related functional inequalities are also discussed. The theoretical results obtained by our functional approach immediately imply those of operator versions in a simple, fast and nice way.

In this article, we present exponential-type inequalities for positive linear mappings and Hilbert space operators, by means of convexity and the Mond-Pe\v cari\'c method. The obtained results refine and generalize some known results. As an application, we present extensions for operator-like geometric and harmonic means.

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...

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 H\"older inequality.

We establish a reverse inequality for Tsallis relative operator entropy involving a positive linear map. In addition, we present converse of Ando's inequality, for each parameter. We give examples to compare our results with the known results by Furuta and Seo. In particular, we establish an extension and a reverse of the L\"owner-Heinz inequality...

The main purpose of this article is to study estimates for the Tsallis relative operator entropy, by using the Hermite-Hadamard inequality. We obtain alternative bounds for the Tsallis relative operator entropy and in the process to derive these bounds, we established the significant relation between the Tsallis relative operator entropy and the ge...

We discuss the necessity of a restriction suggested by other authors in their attempt to solve the problem.

In this paper, we study some complementary inequalities to Jensen’s inequality for self-adjoint operators, unital positive linear mappings, and real valued twice differentiable functions. New improved complementary inequalities are presented by using an improvement of the Mond-Pečarić method. These results are applied to obtain some inequalities wi...

New sharp multiplicative reverses of the operator means inequalities are presented,
with a simple discussion of squaring an operator inequality. As a direct consequence, we extend
the operator P´olya-Szeg¨o inequality to arbitrary operator means. Furthermore, we obtain
some new lower and upper bounds for the Tsallis relative operator entropy, opera...

Jensen’s operator inequality for convexifiable functions is obtained. This result contains classical Jensen’s operator inequality as a particular case. As a consequence, a new refinement and a reverse of Young’s inequality are given.

We show the following result: Let $A,B\in \mathbb{B}\left( \mathcal{H} \right)$ be two strictly positive operators such that $A\le B$ and $m\mathbf{1}_\mathcal{H} \le B\le M{{\mathbf{1}}_{\mathcal{H}}}$ for some scalars $0<m<M$. Then \[{{B}^{p}}\le \exp \left( \frac{M{{\mathbf{1}}_{\mathcal{H}}}-B} {M-m} \ln m^p + \frac{B-m{{\mathbf{1}}_{\mathcal{H...

We give the tight bounds of Tsallis relative operator entropy by using Hermite-Hadamard's inequality. Some reverse inequalities related to Young's inequality are also given. In addition, operator inequalities for normalized positive linear map with Tsallis relative operator entropy are given.

In this short paper, we give two refinements of Jensen-Mercer's operator inequality. We then use these results to refine some inequalities related to quasi-arithmetic means of Mercer's type for operators.

In this paper we present some reverses of the Golden-Thompson type inequalities: Let $H$ and $K$ be Hermitian matrices such that $ e^s e^H \preceq_{ols} e^K \preceq_{ols} e^t e^H$ for some scalars $s \leq t$, and $\alpha \in [0 , 1]$. Then for all $p>0$ and $k =1,2,\ldots, n$ \begin{align*} \label{} \lambda_k (e^{(1-\alpha)H + \alpha K} ) \leq (\ma...

In this paper we present some inequalities involving operator decreasing functions and operator means. These inequalities provide some reverses of operator Acz\'el inequality dealing with the weighted geometric mean.

In this paper, we study further improvements of the reverse Young and Heinz inequalities for positive real numbers. We use these modified inequalities to obtain corresponding operator inequalities and matrix inequalities on the Hilbert–Schmidt norm.

By the use of Hermite-Hadamard inequality, we obtained the estimates for the Tsallis relative operator entropy in our previous paper \cite{RFM2017}. In the present paper, we give alternative tight bounds for the Tsallis relative operator entropy.

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.

In the paper [1], tight bounds for symmetric divergence measures applying the results established by G.L.Gilardoni. In this article, we report on two kinds of extensions for the Sason’s results, namely a classical q-extension and a non-commutative(quantum) extension. Especially, we improve Sason’s bound of the summation of the absolute value for th...

We give the Choi-Davis-Jensen type inequality without using convexity. Applying our main results, we also give new inequalities improving previous known results. In particular, we show some inequalities for relative operator entropies and quantum mechanical entropies.

Let $A$ and $B$ be two accretive operators. We first introduce the weighted geometric mean of $A$ and $B$ together with some related properties. Afterwards, we define the relative entropy as well as the Tsallis entropy of $A$ and $B$. The present definitions and their related results extend those already introduced in the literature for positive in...

For positive operator $A$ with $0<m\le A\le M$, the inequalities \[\left\langle {{A}^{r}}x,x \right\rangle \le \left\langle {{A}^{r}}x,x \right\rangle +\frac{r\left( 1-r \right)}{2}{{M}^{r-2}}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right)\le {{\left\langle Ax,x \right\rangle }^{r}},\qquad 0< r< 1\]...

We give the tight bounds of Tsallis relative operator entropy by using Hermite-Hadamard's inequality. Some reverse inequalities related to Young inequalities are also given. In addition, operator inequalities for normalized positive linear map with Tsallis relative operator entropy are given.

In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of v, namely \(v\in \mathbb {R}\). These modified inequalities are used to establish corresponding operator inequalities on a Hilbert space.

In this paper, we study further improvements of the reverse Young and Heinz inequalities for positive real numbers. We use these modified inequalities to obtain corresponding operator inequalities and matrix inequalities on the Hilbert-Schmidt norm.