In this short article, we mainly give some examples to prove that the main results of Section 4 of the paper [W. S. Liao and J. Wu, Ann. Func. Anal. 6 (2015), no. 3, 191–202] are not true. Then we give the corrected matrix inequalities.
"A note on the paper “Matrix inequalities for the difference between arithmetic mean and harmonic mean”." Ann. Funct. Anal. 10 (4) 509 - 514, November 2019. https://doi.org/10.1215/20088752-2019-0009