- Research Article
- Open Access

# On Refinements of Aczél, Popoviciu, Bellman's Inequalities and Related Results

- G Farid
^{1}Email author, - J Pečarić
^{1, 2}and - AtiqUr Rehman
^{1}

**2010**:579567

https://doi.org/10.1155/2010/579567

© G. Farid et al. 2010

**Received:**11 July 2010**Accepted:**27 November 2010**Published:**6 December 2010

## Abstract

We give some refinements of the inequalities of Aczél, Popoviciu, and Bellman. Also, we give some results related to power sums.

## Keywords

- Positive Integer
- Related Result
- Positive Real Number
- Positive Semidefinite
- Compact Interval

## 1. Introduction

The well-known Aczél's inequality [1] (see also [2, page 117]) is given in the following result.

Theorem 1.1.

with equality if and only if the sequences and are proportional.

A related result due to Bjelica [3] is stated in the following theorem.

Theorem 1.2.

Note that quotation of the above result in [4, page 58] is mistakenly stated for all . In 1990, Bjelica [3] proved that the above result is true for . Mascioni [5], 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. [6] mistakenly stated it for positive integer and gave a refinement of the inequality (1.4) as follows.

Theorem 1.3.

Moreover, Díaz-Barreo et al. [6] stated the above result as Popoviciu's generalization of Aczél's inequality given in [7]. In fact, generalization of inequality (1.2) attributed to Popoviciu [7] is stated in the following theorem (see also [2, page 118]).

Theorem 1.4.

If , then reverse of the inequality (1.8) holds.

The well-known Bellman's inequality is stated in the following theorem [8] (see also [2, pages 118-119]).

Theorem 1.5.

Díaz-Barreo et al. [6] gave a refinement of the above inequality for positive integer . They proved the following result.

Theorem 1.6.

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].

Lemma 2.1.

Theorem 2.2.

Proof.

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.

Theorem 2.3.

(i)If , then

(ii)If , then

Proof.

- (i)
First of all, we observe that and also , therefore by Theorem 1.2, we have

- (ii)
Since

and denoting , ,

Theorem 2.4.

Proof.

and denote , .

Theorem 2.5.

Proof.

and denote , .

Remark 2.6.

In [10], Hu and Xu gave the generalized results related to Theorems 2.4 and 2.5.

## 3. Some Further Remarks on Power Sums

The following theorem [9, page 152] is very useful to give results related to power sums in connection with results given in [11, 12].

Theorem 3.1.

Remark 3.2.

If is strictly increasing on , then strict inequality holds in (3.1).

This implies Lemma 2.1 by substitution, .

In this section, we use Theorem 3.1 to give some results related to power sums as given in [11–13], but here we will discuss only the nonweighted case.

In [11], 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 [11]. Also, we prove mean value theorem of Cauchy type.

It is worthwhile to recall the following.

Definition 3.3.

for all and all choices , and , such that .

Proposition 3.4.

Let . The following propositions are equivalent:

(i) is exponentially convex,

for every and for every , .

Corollary 3.5.

If is exponentially convex function, then is a log-convex function.

### 3.1. Exponential Convexity

Lemma 3.6.

then is strictly increasing function on for each .

Proof.

therefore is strictly increasing function on for each .

Theorem 3.7.

is a positive semidefinite matrix.

(b)The function is exponentially convex.

(c)The function is log convex.

- (b)
Since after some computation we have that so is continuous on , then by Proposition 3.4, we have that is exponentially convex.

- (c)

it follows that . Now, by Corollary 3.5, we have that is log convex.

Let us introduce the following.

Definition 3.8.

Remark 3.9.

Let us note that , , and .

Remark 3.10.

If in we substitute by , then we get , and if we substitute by in , we get .

In [11], we have the following lemma.

Lemma 3.11.

Theorem 3.12 ..

Proof.

By raising power , we get (3.17) for , and .

From Remark 3.9, we get that (3.17) is also valid for or or .

Remark 3.13.

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 [11] to prove the related mean value theorems of Cauchy type.

Lemma 3.14.

then for are monotonically increasing functions.

Theorem 3.15.

Proof.

as required.

Theorem 3.16.

provided that the denominators are nonzero.

Proof.

Putting in (3.30), we get (3.28).

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

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