Improvements of Young inequality using the Kantorovich constant

Some improvements of Young inequality and its reverse for 2 positive numbers with Kantorovich constant K(t, 2) = (1+t) are given. Using these inequalities some operator inequalities and Hilbert-Schmidt norm versions for matrices are proved. In particular, it is shown that if a, b are positive numbers and 0 ⩽ ν ⩽ 1, then for all integers k ⩾ 1 : where mk = [2k ν] is the largest integer not greater than 2k ν, r0 = min{ν, 1 − ν}, rk = min{2rk−1, 1 − 2rk−1} and Rk = 1 − rk .