Journal of Inequalities and Applications
Volume 7 (2002), Issue 5, Pages 633-645
doi:10.1155/S1025583402000322
The upper bound of a reserve Hölder’s type operator inequality and its applications
Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
Received 12 January 2000; Revised 15 March 2000
Copyright © 2002 Masaru Tominaga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.