Let (M, τ) be a semi-finite von Neumann algebra, L0(M) be the set of all τ-measurable operators, µt(x) be the generalized singular number of x ∈ L0(M). We proved that if 1 : [0,∞) → [0,∞) is an increasing continuous function, then for any x, y in L0(M), µt(1(|x + y|)) ≤ µt(1( 12 ( |x| + |y| x∗ + y∗ x + y |x∗| + |y∗| ) )), 0 < t < τ(1). We also obtained that if f : [0,∞) → [0,∞) is a concave function, then µ( f ( 12 ( |x| + |y| x∗ + y∗ x + y |x∗| + |y∗| ) )) is submajorized by µ( f (|x|)) + µ( f (|y|)).