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Wigner--Yanase--Dyson function and logarithmic mean

Shigeru Furuichi

Abstract

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 inequalities based on the obtained results.

Wigner--Yanase--Dyson function and logarithmic mean

Abstract

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 inequalities based on the obtained results.
Paper Structure (4 sections, 10 theorems, 54 equations)

This paper contains 4 sections, 10 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Reverse inequalities
  3. Comparisons of the bounds for logarithmic mean
  4. Conclusion

Key Result

Proposition 1.1

(HKPR2013) For $S,T\in M_+(n,\mathbb{C})$ and any $X\in M(n,\mathbb{C})$, if $1/2\le p \le 1 \le q \le 2$ or $-1\le q \le 0 \le p \le 1/2$, then we have In particular, $p \in[0,1] \Longrightarrow {\left\vert\left\vert\left\vert L(S,T)X \right\vert\right\vert\right\vert} \le {\left\vert\left\vert\left\vert W_p(S,T)X \right\vert\right\vert\right\vert}$.

Theorems & Definitions (18)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • ...and 8 more