  The purpose of this paper is to establish new log-majorization results
concerning eigenvalues and singular values which generalize some previous work
related to a conjecture and an open question which were presented by R. Lemos
and G. Soares in \cite{lemos}. In addition, we present a complement of a
unitarily invariant norm inequality which was conjectured by R. Bhatia, Y. Lim
and T. Yamazaki in \cite{Bhatia2}, and recently proved by T.H. Dinh, R. Dumitru
and J.A. Franco in \cite{Dinh} for the Schatten p-norm with $1\leq p\leq
\infty$.
