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On Refinements of Aczél, Popoviciu, Bellman's Inequalities and Related Results
Journal of Inequalities and Applications volume 2010, Article number: 579567 (2010)
Abstract
We give some refinements of the inequalities of Aczél, Popoviciu, and Bellman. Also, we give some results related to power sums.
1. Introduction
The well-known Aczél's inequality [1] (see also [2, page 117]) is given in the following result.
Theorem 1.1.
Let be a fixed positive integer, and let
be real numbers such that

then

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.
Let be a fixed positive integer, and let
be nonnegative real numbers such that

then, for , one has

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.
Let be positive integers, and let
be nonnegative real numbers such that (1.3) is satisfied, then for
, one has

where

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 be a fixed positive integer, and let
be nonnegative real numbers such that

Also, let , then, for
, one has

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

Díaz-Barreo et al. [6] gave a refinement of the above inequality for positive integer . They proved the following result.
Theorem 1.6.
Let be positive integers, and let
,
be nonnegative real numbers such that (1.3) is satisfied, then for
, one has

where

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 be nonnegative real numbers such that
, then for
, one has

Theorem 2.2.
Let be a fixed positive integer, and let
be nonnegative real numbers such that (1.3) is satisfied, then, for
, one has

Proof.
By using condition (1.3) in Lemma 2.1 for , we have

These imply

Now, applying Azcél's inequality on right-hand side of the above inequality gives us the required result.
Let and
be positive real numbers such that
, then the well-known Hölder's inequality states that

where are positive real numbers.
If , then the well-known inequality of power sums of order
and
states that

where are positive real numbers (c.f [9, page 165]).
Now, if , then
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 and
be fixed positive integers such that
, and let
be nonnegative real numbers such that (1.3) is satisfied. Let one denote

(i)If , then

(ii)If , then

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

We can write

By applying Theorem 1.2 for 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.
-
(ii)
Since

and denoting ,
,
then

It is given that and
, therefore by using Theorem 1.2, for
, on right-hand side of the above equation, we get

since , so by using (2.7)

Theorem 2.4.
Let and
be fixed positive integers such that
, and let
be nonnegative real numbers such that (1.7) is satisfied. Also let
,
be defined in (2.8) and

then, for , one has

Proof.
First of all, note that , therefore by generalized Aczél's inequality, we have

Now,

and denote ,
.
Then

It is given that and
, therefore by using Theorem 1.4, for
, 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 ; here, we give further refinements of Bellman's inequality for real
. We will use Minkowski's inequality in the proof and recall that, for real
and for positive reals
, the Minkowski's inequality states that

Theorem 2.5.
Let and
be fixed positive integers such that
, and let
be nonnegative real numbers such that (1.3) is satisfied. Also let
and
be defined in (2.8). If
, then

Proof.
First of all, note that and
, therefore by using Bellman's inequality, we have

Now,

and denote ,
.
Then

It is given that and
, therefore by using Bellman's inequality, for
, 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.
Let , where
is interval in
and
. Also let
be a function such that
is increasing on
, then

Remark 3.2.
If is strictly increasing on
, then strict inequality holds in (3.1).
Here, it is important to note that if we consider

then is increasing on
for
. By using it in Theorem 3.1, we get

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.
Let be a positive
-tuple, then for
we defined

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.
A function is exponentially convex if it is continuous and

for all and all choices
, and
, such that
.
Proposition 3.4.
Let . The following propositions are equivalent:
(i) is exponentially convex,
(ii) is continuous and

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.
Let and
be the function defined as

then is strictly increasing function on
for each
.
Proof.
Since

therefore is strictly increasing function on
for each
.
Theorem 3.7.
Let be a positive
-tuple
such that
, and let

(a)For , let
be arbitrary real numbers, then the matrix

is a positive semidefinite matrix.
(b)The function is exponentially convex.
(c)The function is log convex.
Proof.
-
(a)
Define a function
(3.11)
then

This implies that is increasing function on
. So using
in the place of
in (3.1), we have

Hence, the given matrix is positive semidefinite.
-
(b)
Since after some computation we have that
so
is continuous on
, then by Proposition 3.4, we have that
is exponentially convex.
-
(c)
Since
is strictly increasing function on
, so by Remark 3.2, we have
(3.14)
it follows that . Now, by Corollary 3.5, we have that
is log convex.
Let us introduce the following.
Definition 3.8.
Let be a positive
-tuple
such that
for
, then for
, we define

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.
Let be a log-convex function and assume that if
,
,
,
, then the following inequality is valid:

Theorem 3.12 ..
Let be positive
-tuple
such that
for
, then for
such that
,
, one has

Proof.
Let be defined by (3.9). Now taking
,
,
,
, where
,
,
, and
in Lemma 3.11, we have

Since , by substituting
,
,
,
, and
, where
, in above inequality, we get

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.
Let , where
such that

Consider the functions ,
defined as

then for
are monotonically increasing functions.
Theorem 3.15.
Let , where
is a compact interval such that
and
. If
, then there exists
such that

Proof.
Since is compact and
, therefore let

In Theorem 3.1, setting and
, respectively, as defined in Lemma 3.14, we get the following inequalities:

If , then
is strictly increasing function on
, therefore by Theorem 3.1, we have

Now, by combining inequalities (3.24), we get

Finally, by condition (3.20), there exists , such that

as required.
Theorem 3.16.
Let , where
is a compact interval such that
and
. If
, then there exists
such that the following equality is true:

provided that the denominators are nonzero.
Proof.
Let a function be defined as

where and
are defined as

Then, using Theorem 3.15, with , we have

Since , therefore (3.31) gives

Putting in (3.30), we get (3.28).
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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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Farid, G., Pečarić, J. & Rehman, A. On Refinements of Aczél, Popoviciu, Bellman's Inequalities and Related Results. J Inequal Appl 2010, 579567 (2010). https://doi.org/10.1155/2010/579567
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DOI: https://doi.org/10.1155/2010/579567
Keywords
- Positive Integer
- Related Result
- Positive Real Number
- Positive Semidefinite
- Compact Interval