  Let $A_i$ and $B_i$ be positive definite matrices for all $i=1,\cdots,m.$ It
is shown that $$\left|\left|\sum_{i=1}^m(A_i^2\sharp
B_i^2)^r\right|\right|_1\leq\left|\left|\left(\left(\sum_{i=1}^mA_i\right)^{\frac{pr}{_2}}\left(\sum_{i=1}^mB_i\right)^{pr}\left(\sum_{i=1}^mA_i\right)^{\frac{rp}{_2}}\right)^{\frac{1}{p}}\right|\right|_1,$$for
all $p>0$ and for all $r\geq1.$
  We conjecture this inequality is also true for all unitarily invariant norms.
We give an affirmative answer to the case of $m=2,$ $p\geq1$, $r\geq1$ and for
all unitarily invariant norms. In other words, it is shown that
$$\left|\left|\left|\left(A^{^2}\sharp B^{^2}\right)^{r}+\left(C^{^2}\sharp
D^{^2}\right)^{r}\right|\right|\right|\leq
\left|\left|\left|\left(\left(A+C\right)^{^\frac{rp}{_2}}\left(B+D\right)^{{rp}}\left(A+C\right)^{^\frac{rp}{_2}}\right)^{\frac{1}{_p}}\right|\right|\right|,$$for
all unitarly invariant norms, for all $p\geq1$ and for all $r\geq1$, where
$A,B,C,D$ are positive definite matrices. This gives an affirmative answer to
the conjecture posed by Dinh, Ahsani and Tam in the case of $m=2$. The
preceding inequalities directly lead to a recent result of Audenaert
\cite{ANIFP}.
