The numerical inequality \begin{align*} \left(\sum_{j=1}^n x_jy_j\right)^2\leq \sum_{j=1}^n x_j^{1+s}y_j^{1-s}\sum_{j=1}^n x_j^{1-s}y_j^{1+s} \leq \left(\sum_{j=1}^n x_j^2\right)\left(\sum_{j=1}^ny_j^2\right), \end{align*} where $ x_j, y_j \,\,(1\leq j\leq n)$ are positive real numbers and $s\in[0,1]$ is called the Callebaut inequality, which is a refinement of the Cauchy--Schwarz inequality. An operator version of it 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, respectively. In this talk we employ some operator techniques to establish some Callebaut inequalities as well as some refinements of a Callebaut type inequality involving Hadamard products of Hilbert space operators.

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