Generality of Liebs Concavity Theorem

We show that Liebs concavity theorem holds more generally for any unitarily invariant matrix function $phi:mathbf{H}^n_+rightarrow mathbb{R}$ that is monotone and concave. Concretely, we prove the joint concavity of the function $(A,B) mapstophibig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big] $ on $mathbf{H}_+^mtimesmathbf{H}_+^n$, for any $Kin mathbb{C}^{mtimes n},sin(0,1],p,qin[0,1], p+qleq 1$.