Abstract
In our previous paper, we obtained a reverse Hölder’s type inequality which gives an upper bound of the difference:
(∑akp)1/p(∑bkq)1/q−λ∑akbk
with a parameter λ>0, for n-tuples a=(a1,…,an) and b=(b1,…,bn) of positive numbers and for p>1, q>1 satisfying 1/p+1/q=1. In this paper for commutative positive operators A and B on a Hilbert space H and a unit vector x∈H, we give an upper bound of the difference
〈Apx,x〉1/p〈Bqx,x〉1/q−λ〈ABx,x〉.
As applications, considering special cases, we induce some difference and ratio operator
inequalities. Finally, using the geometric mean in the Kubo-Ando theory we shall give a
reverse Hölder’s type operator inequality for noncommutative operators.