- Research Article
- Open Access
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.
- Positive Integer
- Related Result
- Positive Real Number
- Positive Semidefinite
- Compact Interval
with equality if and only if the sequences and are proportional.
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]).
If , then reverse of the inequality (1.8) holds.
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.
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.
where are positive real numbers.
where are positive real numbers (c.f [9, page 165]).
We use the inequality (2.7) and the Hölder's inequality to prove the further refinements of the Theorems 1.2 and 1.4.
(i)If , then
(ii)If , then
First of all, we observe that and also , therefore by Theorem 1.2, we have
and denoting , ,
and denote , .
and denote , .
In , Hu and Xu gave the generalized results related to Theorems 2.4 and 2.5.
If is strictly increasing on , then strict inequality holds in (3.1).
This implies Lemma 2.1 by substitution, .
In , we introduced Cauchy means related to power sums; here, we restate the means without weights.
We proved that is monotonically increasing with respect to and .
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.
for all and all choices , and , such that .
Let . The following propositions are equivalent:
(i) is exponentially convex,
for every and for every , .
If is exponentially convex function, then is a log-convex function.
3.1. Exponential Convexity
then is strictly increasing function on for each .
therefore is strictly increasing function on for each .
is a positive semidefinite matrix.
(b)The function is exponentially convex.
(c)The function is log convex.
Since after some computation we have that so is continuous on , then by Proposition 3.4, we have that is exponentially convex.
it follows that . Now, by Corollary 3.5, we have that is log convex.
Let us introduce the following.
Let us note that , , and .
If in we substitute by , then we get , and if we substitute by in , we get .
In , we have the following lemma.
Theorem 3.12 ..
By raising power , we get (3.17) for , and .
From Remark 3.9, we get that (3.17) is also valid for or or .
If we substitute by , then monotonicity of implies the monotonicity of , and if we substitute by , then monotonicity of implies monotonicity of .
3.2. Mean Value Theorems
We will use the following lemma  to prove the related mean value theorems of Cauchy type.
then for are monotonically increasing functions.
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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