[{"article_processing_charge":"No","day":"13","article_type":"original","citation":{"chicago":"Zhang, Haonan. “From Wigner-Yanase-Dyson Conjecture to Carlen-Frank-Lieb Conjecture.” Advances in Mathematics. Elsevier, 2020. https://doi.org/10.1016/j.aim.2020.107053.","short":"H. Zhang, Advances in Mathematics 365 (2020).","mla":"Zhang, Haonan. “From Wigner-Yanase-Dyson Conjecture to Carlen-Frank-Lieb Conjecture.” Advances in Mathematics, vol. 365, 107053, Elsevier, 2020, doi:10.1016/j.aim.2020.107053.","apa":"Zhang, H. (2020). From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2020.107053","ieee":"H. Zhang, “From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture,” Advances in Mathematics, vol. 365. Elsevier, 2020.","ista":"Zhang H. 2020. From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture. Advances in Mathematics. 365, 107053.","ama":"Zhang H. From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture. Advances in Mathematics. 2020;365. doi:10.1016/j.aim.2020.107053"},"publication":"Advances in Mathematics","date_published":"2020-05-13T00:00:00Z","type":"journal_article","abstract":[{"text":"In this paper we study the joint convexity/concavity of the trace functions Ψp,q,s(A,B)=Tr(Bq2K∗ApKBq2)s, p,q,s∈R,\r\nwhere A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of (p,q,s)∈R3 for Ψp,q,s to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of (α,z) for α-z Rényi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψp,0,1/p for 0