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

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Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

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

Theorem 1.1.

Let Open image in new window be a fixed positive integer, and let Open image in new window be real numbers such that

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.

Let Open image in new window be a fixed positive integer, and let Open image in new window be nonnegative real numbers such that

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.

Let Open image in new window be positive integers, and let Open image in new window be nonnegative real numbers such that (1.3) is satisfied, then for Open image in new window , one has

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.

Let Open image in new window be a fixed positive integer, and let Open image in new window be nonnegative real numbers such that

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.

Let Open image in new window be a fixed positive integer, and let Open image in new window be nonnegative real numbers such that (1.3) is satisfied. If Open image in new window , then

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.

Let Open image in new window be positive integers, and let Open image in new window ,   Open image in new window be nonnegative real numbers such that (1.3) is satisfied, then for Open image in new window , one has

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.

Let Open image in new window   be nonnegative real numbers such that Open image in new window , then for Open image in new window , one has

Theorem 2.2.

Let Open image in new window be a fixed positive integer, and let Open image in new window be nonnegative real numbers such that (1.3) is satisfied, then, for Open image in new window , one has

Proof.

By using condition (1.3) in Lemma 2.1 for Open image in new window , we have
These imply

Now, applying Azcél's inequality on right-hand side of the above inequality gives us the required result.

Let Open image in new window and Open image in new window be positive real numbers such that Open image in new window , then the well-known Hölder's inequality states that

where Open image in new window are positive real numbers.

If Open image in new window , then the well-known inequality of power sums of order Open image in new window and Open image in new window states that

where Open image in new window are positive real numbers (c.f [9, page 165]).

Now, if Open image in new window , then Open image in new window and using inequality (2.6) in (2.5), we get

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.

Let Open image in new window and Open image in new window be fixed positive integers such that Open image in new window , and let Open image in new window be nonnegative real numbers such that (1.3) is satisfied. Let one denote

(i)If Open image in new window , then

(ii)If Open image in new window , then

Proof.

  1. (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

     
We can write
By applying Theorem 1.2 for Open image in new window on right-hand side of the above equation, we get
By using inequality (2.11) on right-hand side of the above expression follows the required result.
  1. (ii)

    Since

     

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

It is given that Open image in new window and Open image in new window , therefore by using Theorem 1.2, for Open image in new window , on right-hand side of the above equation, we get
since Open image in new window , so by using (2.7)

Theorem 2.4.

Let Open image in new window and Open image in new window be fixed positive integers such that Open image in new window , and let Open image in new window be nonnegative real numbers such that (1.7) is satisfied. Also let Open image in new window , Open image in new window be defined in (2.8) and

Proof.

First of all, note that Open image in new window , therefore by generalized Aczél's inequality, we have

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

It is given that Open image in new window and Open image in new window , therefore by using Theorem 1.4, for Open image in new window , on right-hand side of the above equation, we get
by applying Hölder's inequality
by using inequality (2.20)
In [6], a refinement of Bellman's inequality is given for positive integer Open image in new window ; here, we give further refinements of Bellman's inequality for real Open image in new window . We will use Minkowski's inequality in the proof and recall that, for real Open image in new window and for positive reals Open image in new window , the Minkowski's inequality states that

Theorem 2.5.

Let Open image in new window and Open image in new window be fixed positive integers such that Open image in new window , and let Open image in new window be nonnegative real numbers such that (1.3) is satisfied. Also let Open image in new window and Open image in new window be defined in (2.8). If Open image in new window , then

Proof.

First of all, note that Open image in new window and Open image in new window , therefore by using Bellman's inequality, we have

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

It is given that Open image in new window and Open image in new window , therefore by using Bellman's inequality, for Open image in new window , on right-hand side of the above equation, we get
by applying Minkowski's inequality
and by using inequality (2.28)

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).

Here, it is important to note that if we consider
then Open image in new window is increasing on Open image in new window for Open image in new window . By using it in Theorem 3.1, we get

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.

A function Open image in new window is exponentially convex if it is continuous and

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.

(a)For Open image in new window , let Open image in new window be arbitrary real numbers, then the matrix

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.

Proof.
  1. (a)
    Define a function
     
This implies that Open image in new window is increasing function on Open image in new window . So using Open image in new window in the place of Open image in new window in (3.1), we have
Hence, the given matrix is positive semidefinite.
  1. (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.

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

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.

Let Open image in new window be a log-convex function and assume that if Open image in new window ,   Open image in new window ,   Open image in new window ,   Open image in new window , then the following inequality is valid:

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.

In Theorem 3.1, setting Open image in new window and Open image in new window ,  respectively, as defined in Lemma 3.14, we get the following inequalities:
If Open image in new window , then Open image in new window is strictly increasing function on Open image in new window , therefore by Theorem 3.1, we have
Now, by combining inequalities (3.24), we get
Finally, by condition (3.20), there exists Open image in new window , such that

as required.

Theorem 3.16.

Let Open image in new window , where Open image in new window is a compact interval such that Open image in new window and Open image in new window . If Open image in new window , then there exists Open image in new window such that the following equality is true:

provided that the denominators are nonzero.

Proof.

Let a function Open image in new window be defined as
Then, using Theorem 3.15, with Open image in new window , we have
Since Open image in new window , therefore (3.31) gives

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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Copyright information

© G. Farid et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan
  2. 2.Faculty of Textile TechnologyUniversity of ZagrebZagrebCroatia

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