Some reversed and refined Callebaut inequalities via Kontorovich constant

In this paper we employ some operator techniques to establish some refinements and reverses of the Callebaut inequality involving the geometric mean and Hadamard product under some mild conditions. In particular, we show \begin{align*} K&\left(\frac{M^{2t-1}}{m^{2t-1}},2\right)^{r'} \sum_{j=1}^n(A_j\sharp_{s}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-s}B_j) \nonumber\\&\,\,+\left(\frac{t-s}{t-1/2}\right)\left(\sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t}B_j) -\sum_{j=1}^n(A_j\sharp B_j)\circ \sum_{j=1}^n(A_j\sharp B_j)\right)\nonumber \\&\leq \sum_{j=1}^n(A_j\sharp_{t}B_j)\circ \sum_{j=1}^n(A_j\sharp_{1-t} B_j)\,, \end{align*} where $A_j, B_j\in{\mathbb B}({\mathscr H})\,\,(1\leq j\leq n)$ are positive operators such that $0<m' \leq B_j\leq m <M \leq A_j\leq M'\,\,(1\leq j\leq n)$, either $1\geq t\geq s>{\frac{1}{2}}$ or $0\leq t\leq s<\frac{1}{2}$, $r'=\min\left\{\frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$ and $K(t,2)=\frac{(t+1)^2}{4t}\,\,(t>0)$.

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