  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.
