The Schatten norm complement follows by convexity

For positive definite $A, B \in \mathbb{C}^{n \times n}$, let
$G = A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$ be their geometric mean and
$L = A^{1/2}(B^{1/2}A^{-1}B^{1/2})^{1/2}A^{1/2}$. The note proves
$\|A + B + G + L\|_p \le \|A + B + 2L\|_p$ for every Schatten norm,
$1 \le p \le \infty$. Write $H = A + B$ and
$K = A^{1/2}B^{1/2} + B^{1/2}A^{1/2}$. The Heron norm inequality of Dinh,
Dumitru and Franco gives $\|H + 2G\|_p \le \|H + K\|_p$. A polar-decomposition
argument shows $K \preceq G + L$, so that $0 \preceq H + K \preceq H + G + L$
and $\|H + K\|_p \le \|H + G + L\|_p$. Since
$(H + 2G) + (H + 2L) = 2(H + G + L)$, the triangle inequality then gives the
result.