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

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## 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 Open image in new window and Open image in new window 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 Open image in new window . In 1990, Bjelica [3] proved that the above result is true for Open image in new window . Mascioni [5], in 2002, gave the proof for Open image in new window and gave the counter example to show that the above result is not true for Open image in new window . Díaz-Barreo et al. [6] mistakenly stated it for positive integer Open image in new window 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 Open image in new window , 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 Open image in new window . 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 Open image in new window are positive real numbers.

where Open image in new window 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 Open image in new window , then

(ii)If Open image in new window , then

Proof.

- (i)
First of all, we observe that Open image in new window and also Open image in new window , therefore by Theorem 1.2, we have

- (ii)
Since

and denoting Open image in new window , Open image in new window ,

Theorem 2.4.

Proof.

and denote Open image in new window , Open image in new window .

Theorem 2.5.

Proof.

and denote Open image in new window , Open image in new window .

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 Open image in new window is strictly increasing on Open image in new window , then strict inequality holds in (3.1).

This implies Lemma 2.1 by substitution, Open image in new window .

In this section, we use Theorem 3.1 to give some results related to power sums as given in [11, 12, 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 Open image in new window is monotonically increasing with respect to Open image in new window and Open image in new window .

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 Open image in new window and all choices Open image in new window , and Open image in new window , such that Open image in new window .

Proposition 3.4.

Let Open image in new window . The following propositions are equivalent:

(i) Open image in new window is exponentially convex,

for every Open image in new window and for every Open image in new window , Open image in new window .

Corollary 3.5.

If Open image in new window is exponentially convex function, then Open image in new window is a log-convex function.

### 3.1. Exponential Convexity

Lemma 3.6.

then Open image in new window is strictly increasing function on Open image in new window for each Open image in new window .

Proof.

therefore Open image in new window is strictly increasing function on Open image in new window for each Open image in new window .

Theorem 3.7.

is a positive semidefinite matrix.

(b)The function Open image in new window is exponentially convex.

(c)The function Open image in new window is log convex.

- (a)Define a function(3.11)

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

- (c)Since Open image in new window is strictly increasing function on Open image in new window , so by Remark 3.2, we have(3.14)

it follows that Open image in new window . Now, by Corollary 3.5, we have that Open image in new window is log convex.

Let us introduce the following.

Definition 3.8.

Remark 3.9.

Let us note that Open image in new window , Open image in new window , and Open image in new window .

Remark 3.10.

If in Open image in new window we substitute Open image in new window by Open image in new window , then we get Open image in new window , and if we substitute Open image in new window by Open image in new window in Open image in new window , we get Open image in new window .

In [11], we have the following lemma.

Lemma 3.11.

Theorem 3.12 ..

Proof.

By raising power Open image in new window , we get (3.17) for Open image in new window , Open image in new window and Open image in new window .

From Remark 3.9, we get that (3.17) is also valid for Open image in new window or Open image in new window or Open image in new window .

Remark 3.13.

If we substitute Open image in new window by Open image in new window , then monotonicity of Open image in new window implies the monotonicity of Open image in new window , and if we substitute Open image in new window by Open image in new window , then monotonicity of Open image in new window implies monotonicity of Open image in new window .

### 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 Open image in new window for Open image in new window 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).

## Notes

### 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.

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