Complementary and refined inequalities of Callebaut inequality for operators

TL;DR

This paper introduces a new complementary inequality to the Callebaut inequality for operators, along with refinements involving weighted geometric means and Hadamard products, using operator techniques and the Mond–Pecaric method.

Contribution

It provides a novel complementary inequality to the Callebaut inequality for operators and offers refinements involving weighted geometric means and Hadamard products.

Findings

Established a complementary inequality to the Callebaut inequality.

Presented refinements involving weighted geometric means.

Applied the Mond–Pecaric method and operator techniques.

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--Pe\v{c}ari\'c 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.