Some New Hilbert's Type Inequalities
© C.-J. Zhao and W.-S. Cheung. 2009
Received: 25 December 2008
Accepted: 24 April 2009
Published: 5 May 2009
Some new inequalities similar to Hilbert's type inequality involving series of nonnegative terms are established.
In recent years, several authors [1–10] have given considerable attention to Hilbert's type inequalities and their various generalizations. In particular, in , Pachpatte proved somenew inequalities similar to Hilbert's inequality [11, page 226] involving series of nonnegative terms. The main purpose of this paper is to establish their general forms.
2. Main Results
In , Pachpatte established the following inequality involving series of nonnegative terms.
Theorem 2 A.
We first establish the following general form of inequality (2.1).
This completes the proof.
Taking and changing , , and into , , and respectively, and with suitable changes, (2.11) reduces to Pachpatte [1, inequality (1)].
In , Pachpatte also established the following inequality involving series of nonnegative terms.
Theorem 2 B.
Inequality (2.12) can also be generalized to the following general form.
The proof is complete.
Taking and changing , , and into , , and respectively, and with suitable changes, (2.21) reduces to Pachpatte [1, Inequality ( 7)].
Proceeding now much as in the proof of Theorems 2.1 and 2.3, and with suitable modifications, it is not hard to arrive at the desired inequality. The details are omitted here.
In the special case where and , Theorem 2.5 reduces to the following result.
Theorem 2 C.
This is the new inequality of Pachpatte in [1,Theorem 4].
Taking and in Theorem 2.5, and in view of , we obtain the following theorem.
Theorem 2 D.
This is the new inequality of Pachpatte in [1,Theorem 3].
Research is supported by Zhejiang Provincial Natural Science Foundation of China(Y605065), Foundation of the Education Department of Zhejiang Province of China (20050392). Research is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and a HKU Seed Grant forBasic Research.
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