A new proof of Lee's conjecture on the Frobenius norm via the matrix Cauchy-Schwarz inequality

In 2010, Eun-Young Lee conjectured that if $A,B$ are two n× n complex matrices and |A|, |B| are the absolute values of $A, B$, respectively, then \[ \|A+B\|_F≤ √{{1+√2}{2}}\||A|+|B|\|_F, \] where \|·\|F is the Frobenius norm of matrices. This conjecture has been proven by Lin and Zhang [J. Math. Anal. Appl. 516 (2022) 126542] by studying inequalities for the angle between two matrices induced by the Frobenius inner product. In this paper, we present a new proof of the same result, relying solely on the Cauchy-Schwarz inequality.