  We show that Lieb's concavity theorem holds more generally for any unitary
invariant matrix function $\phi:\mathbf{H}_+^n\rightarrow \mathbb{R}_+^n$ that
is concave and satisfies H\"older's inequality. Concretely, we prove the joint
concavity of the function $(A,B)
\mapsto\phi\big[(B^\frac{qs}{2}K^*A^{ps}KB^\frac{qs}{2})^{\frac{1}{s}}\big] $
on $\mathbf{H}_+^n\times\mathbf{H}_+^m$, for any $K\in \mathbb{C}^{n\times m}$
and any $s,p,q\in(0,1], p+q\leq 1$. This result improves a recent work by Huang
for a more specific class of $\phi$.
