Improvement on a Generalized Lieb's Concavity Theorem

We show that Lieb’s concavity theorem holds more generally for any unitary invariant matrix function $\varphi :{\mathbf{H}}_{+}^n\to {\mathbb{R}}_{+}^n$ 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}$+^m$,forany$K\in $\mathbb{C}$^{n\times m}$andany$s,p,q\in(0,1], p+q\leq 1$. This result improves a recent work by Huang for a more specific class of $\varphi$.