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 $n\times n$ complex matrices and $\mid A\mid ,\mid B\mid$ are the absolute values of $A,B$, respectively, then $$\mathrm{\parallel }A+B{\mathrm{\parallel }}_F\le \sqrt{{\displaystyle \frac{1+\sqrt{2}}{2}}}\mathrm{\parallel }\mid A\mid +\mid B\mid {\mathrm{\parallel }}_F,$$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_F$ 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.