Generalized operator version of Bernoulli’s inequality

Abstract

In this paper, we present a generalized operator version of second-order Bernoulli’s inequality which includes and unifies the recent results by Hirzallah, Kittaneh, Krnić, Lovričević and Pečarić. Related eigenvalue inequalities and equalities for the difference of means of two positive invertible operators and on a separable Hilbert space under the condition that is compact, are obtained.

Keywords:

• positive operator
• operator mean
• Bernoulli’s inequality

AMSSubject Classifications:

• Primary 47A63
• Secondary 47A30
• 47A64
• 47A10
• 47B07
• 47B15

Acknowledgments

The authors thank the referee for valuable comments, simplification of the proof of Lemma 2.1 and pointing out Lemma 2.7 and Theorem 2.8. The first author also thanks the ‘Council of Scientific and Industrial Research, Human Resource Development Group, India’ for grant of Junior Research Fellowship vide letter no. 09/797(0008)/2011-EMR-I.