  There are various generalizations of the geometric mean $(a,b)\mapsto
a^{1/2}b^{1/2}$ for $a,b\in \mathbb{R}^+$ to positive matrices, and we consider
the standard positive geometric mean $(X,Y)\mapsto
X^{1/2}(X^{-1/2}YX^{-1/2})^{1/2}X^{1/2}$. Much research in recent years has
been devoted to relating the weighted version of this mean
$X\#_{t}Y:=X^{1/2}(X^{-1/2}YX^{-1/2})^{t}X^{1/2}$ for $t\in [0, 1]$ with
operators $e^{(1-t)X+tY}$ and $e^{(1-t)X/2}e^{tY}e^{(1-t)X/2}$ in
Golden-Thompson-like inequalities. These inequalities are of interest to
mathematical physicists for their relationship to quantum entropy, relative
quantum entropy, and R\'{e}nyi divergences. However, the weighted mean is
well-defined for the full range of $t\in\mathbb{R}$. In this paper we examine
the value of $|||e^H\#_te^K|||$ and variations thereof in comparison to
$|||e^{(1-t)H+tK}|||$ and $|||e^{(1-t)H}e^{tK}|||$ for any unitarily invariant
norm $|||\cdot|||$ and in particular the trace norm, creating for the first
time the full picture of interpolation of the weighted geometric mean with the
Golden-Thompson Inequality. We expand inequalities known for
$|||(e^{rH}\#_te^{rK})^{1/r}|||$ with $r>0$, $t\in [0,1]$ to the entire real
line, and comment on how the exterior inequalities can be used to provide
elegant proofs of the known inequalities for $t\in [0,1]$. We also characterize
the equality cases for strictly increasing unitarily invariant norms.
