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We show that Lieb's concavity theorem holds more generally for any unitarilyinvariant matrix function $\phi:\mathbf{H}^n_+\rightarrow \mathbb{R}$ that ismonotone and concave. 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 $\mathbf{H}_+^m\times\mathbf{H}_+^n$, for any $K\in \mathbb{C}^{m\timesn},s\in(0,1],p,q\in[0,1], p+q\leq 1$.

Generality of Lieb's Concavity Theorem