# Some reversed and refined Callebaut inequalities via Kontorovich
constant

Research paper by **Mojtaba Bakherad**

Indexed on: **04 Apr '16**Published on: **04 Apr '16**Published in: **Mathematics - Functional Analysis**

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#### Abstract

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 $K(t,2)=\frac{(t+1)^2}{4t}\,\,(t>0)$.