On Refinements of Aczél, Popoviciu, Bellman's Inequalities and Related Results
© G. Farid et al. 2010
Received: 11 July 2010
Accepted: 27 November 2010
Published: 6 December 2010
We give some refinements of the inequalities of Aczél, Popoviciu, and Bellman. Also, we give some results related to power sums.
A related result due to Bjelica  is stated in the following theorem.
Note that quotation of the above result in [4, page 58] is mistakenly stated for all . In 1990, Bjelica  proved that the above result is true for . Mascioni , in 2002, gave the proof for and gave the counter example to show that the above result is not true for . Díaz-Barreo et al.  mistakenly stated it for positive integer and gave a refinement of the inequality (1.4) as follows.
Moreover, Díaz-Barreo et al.  stated the above result as Popoviciu's generalization of Aczél's inequality given in . In fact, generalization of inequality (1.2) attributed to Popoviciu  is stated in the following theorem (see also [2, page 118]).
Díaz-Barreo et al.  gave a refinement of the above inequality for positive integer . They proved the following result.
In this paper, first we give a simple extension of a Theorem 1.2 with Aczél's inequality. Further, we give refinements of Theorems 1.2, 1.4, and 1.5. Also, we give some results related to power sums.
2. Main Results
To give extension of Theorem 1.2, we will use the result proved by Pečarić and Vasić in 1979 [9, page 165].
Now, applying Azcél's inequality on right-hand side of the above inequality gives us the required result.
We use the inequality (2.7) and the Hölder's inequality to prove the further refinements of the Theorems 1.2 and 1.4.
In , Hu and Xu gave the generalized results related to Theorems 2.4 and 2.5.
3. Some Further Remarks on Power Sums
In , we introduced Cauchy means related to power sums; here, we restate the means without weights.
In this section, we give exponential convexity of a positive difference of the inequality (3.1) by using parameterized class of functions. We define new means and discuss their relation to the means defined in . Also, we prove mean value theorem of Cauchy type.
It is worthwhile to recall the following.
3.1. Exponential Convexity
is a positive semidefinite matrix.
Let us introduce the following.
In , we have the following lemma.
Theorem 3.12 ..
3.2. Mean Value Theorems
We will use the following lemma  to prove the related mean value theorems of Cauchy type.
provided that the denominators are nonzero.
Putting in (3.30), we get (3.28).
This research was partially funded by Higher Education Commission, Pakistan. The research of the second author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant no. 117-1170889-0888.
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