Journal of Inequalities and Applications
Volume 7 (2002), Issue 5, Pages 633-645

The upper bound of a reserve Hölder’s type operator inequality and its applications

Masaru Tominaga

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.


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/qλakbk 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 xH, we give an upper bound of the difference Apx,x1/pBqx,x1/qλABx,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.