Summary: The operator function \((A,B) \rightarrow \operatorname{Tr} f(A,B)(K^*)K\), defined in pairs of bounded selfadjoint operators in the domain of a function \(f\) of two real variables, is convex for every Hilbert-Schmidt operator \(K\) if and only if \(f\) is operator convex. We obtain, as a special case, a new proof of Lieb’s concavity theorem for the function \((A,B)\rightarrow \operatorname{Tr} A^{p}K^{*}B^{q}K\), where \(p\) and \(q\) are non-negative numbers with sum \(p+q \leq 1\). In addition, we prove concavity of the operator function
\[ (A,B)\rightarrow \operatorname{Tr} \left[\frac{A}{A+\mu_1}K^*\frac{B}{B+\mu_2}K\right] \]
in its natural domain \(D_{2}(\mu_{1},\mu_{2})\).