In this talk I will give a brief introduction to the convexity and concavity of trace functionals involving trace and matrices. They play an important role in quantum information theory. Since Lieb`s celebrated work in 1973 resolving the conjecture of Wigner-Yanase-Dyson, this topic has seen great progress. As an example, I will introduce a conjecture in quantum information theory by Audenaert-Datta and a stronger conjecture by Carlen-Frank-Lieb in recent years, explain the connection with convexity/concavity of trace functionals, and show how to solve them in a simple way. The key ingredients are some variational formulas coming from basic properties of Schatten norms. This is based on arXiv:1811.01205.