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Complementary and refined inequalities of Callebaut inequality for operators

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 σB_j\right)\sharp\left(\sum_{ j=1}^n A_j σ^{\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 $σ$ and $σ^\perp$ 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.