  The aim of this paper is to study the logarithmic submajorisations
inequalities for operators in a finite von Neumann algebra. Firstly, some
logarithmic submajorisations inequalities due to Garg and Aulja are extended to
the case of operators in a finite von Neumann algebra. As an application, we
get some new Fuglede-Kadison determinant inequalities of operators in that
circumstance. Secondly, we improve and generalize to the setting of finite von
Neumann algebras, a generalized H\"{o}lder type generalized singular numbers
inequality.
