In a celebrated paper in 1973, Lieb proved what we now call Lieb's Concavity Theorem and resolved a conjecture of Wigner, Yanase and Dyson in 1963. This result, together with its many extensions, has found plenty of applications in mathematical physics and quantum information theory. One notable application of Lieb's Concavity Theorem is the data processing inequality for the quantum relative entropy, which is now one of the cornerstones in quantum information. In recent years, when studying the data processing inequalities for alpha-z Rényi quantum relative entropies, Audenaert and Datta conjectured that certain trace functionals are jointly convex. Later on, a stronger conjecture was made by Carlen, Frank and Lieb when reviewing the related problems. In this talk, I will present a simple variational method to study the concavity/convexity of trace functionals. This allows us to resolve these two conjectures and recover many known results easily. The talk is based on arXiv:1811.01205.