In this talk I will briefly review the history of convexity/concavity of certain trace functions from Wigner-Yanase-Dyson Conjecture (1963) and a recent conjecture of E. A. Carlen, R. L. Frank and E. H. Lieb in 2018. The WYD Conjecture is important in quantum physics and the CFL Conjecture is closely related to quantum information. The main aim of this talk is to fully study convexity/concavity of a family of trace functions. This confirms the CFL Conjecture and simplifies the proofs of many known results.