Using a variational method, we prove a conjecture of Carlen, Frank and Lieb,
which concerns the joint convexity of the the trace function $$
\Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s, $$
where $-1\leq q< 0,~1\leq p\leq 2,~(p,q)\ne (1,-1),~s\geq\frac{1}{p+q}$, $A$
and $B$ are $N\times N$ positive semi-definite matrices and $K$ is a fixed
$N\times N$ matrix. This admits the Audenaert-Datta conjecture with
$s=\frac{1}{p+q}$ as a special case. Together with other known results, we will
give full range of $(p,q,s)$ for $\Psi_{p,q,s}$ to be joint convex/concave. As
a consequence, we obtain the full range of $(\alpha,z)$ for $\alpha-z$ R\'enyi
relative entropies to be monotone under the completely positive trace
preserving maps. We will also use this method to give simple proofs for some
known results on joint convexity/concavity of $\Psi_{p,q,s}$.