Further generalized refinement of Young’s inequalities for τ -mesurable operators

In this paper, we prove that if , > 0 and 0 ≤ ≤ 1. Then for all positive integer (1) - For ∈ , we have (2) - For ∈ , we have we also prove two similar inequalities for the cases ∈ and ∈ . These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and -norms of positive τ-measurable operators.