The Best Lower Bound Depended on Two Fixed Variables for Jensen's Inequality with Ordered Variables
© Vasile Cirtoaje. 2010
Received: 16 June 2010
Accepted: 4 November 2010
Published: 24 November 2010
We give the best lower bound for the weighted Jensen's discrete inequality with ordered variables applied to a convex function , in the case when the lower bound depends on , weights, and two given variables. Furthermore, under the same conditions, we give some sharp lower bounds for the weighted AM-GM inequality and AM-HM inequality.
2. Main Results
For proving Theorem 2.1, we will need the following three lemmas.
Using Corollary 2.5, we can prove the propositions below.
Using Corollary 2.12, we can prove the following proposition.
3. Proof of Lemmas
Proof of Lemma 2.2.
Proof of Lemma 2.3.
by Lemma 2.2, the conclusion follows.
Proof of Lemma 2.4.
4. Proof of Theorem
5. Proof of Propositions
Proof of Proposition 2.6.
Proof of Proposition 2.7.
Proof of Proposition 2.13.
The proposition is proved.
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