Complementary and refined inequalities of Callebaut inequality for operators
Abstract
The Callebaut inequality says that \begin{align*} \sum_{ j=1}^n \left(A_j\sharp B_j\right)\leq \left(\sum_{ j=1}^n A_j \sigma B_j\right)\sharp\left(\sum_{ j=1}^n A_j \sigma^{\bot} B_j\right)\leq\left(\sum_{ j=1}^n A_j\right)\sharp \left(\sum_{ j=1}^nB_j\right)\,, \end{align*} where $A_j, B_j\,\,(1\leq j\leq n)$ are positive invertible operators and $\sigma$ and $\sigma^\perp$ are an operator mean and its dual in the sense of Kabo and Ando, respectively. In this paper we employ the Mond--Pečarić method as well as some operator techniques to establish a complementary inequality to the above one under mild conditions. We also present some refinements of a Callebaut type inequality involving the weighted geometric mean and Hadamard products of Hilbert space operators.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- 10.48550/arXiv.1410.1114
- arXiv:
- arXiv:1410.1114
- Bibcode:
- 2014arXiv1410.1114B
- Keywords:
-
- Mathematics - Functional Analysis;
- Mathematics - Operator Algebras;
- Primary 47A63;
- Secondary 15A60;
- 47A60
- E-Print:
- 15 pages, to appear in Linear Multilinear Algebra (LAMA)