  Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020)
45--49), we present more results (bounds) on Harnack type inequalities for
matrices in terms of majorization (i.e., in partial products) of eigenvalues
and singular values. We discuss and compare the bounds derived through
different ways. Jiang and Lin's results imply Tung's version of Harnack's
inequality (Proc. Amer. Math. Soc. 15 (1964) 375--381); our results %with
simpler proofs are stronger and more general than Jiang and Lin's. We also show
some majorization inequalities concerning Cayley transforms. Some open problems
on spectral norm and eigenvalues are proposed.
