We give some refinements of the inequalities of Aczél, Popoviciu, and Bellman. Also, we give some results related to power sums.
1. Introduction
The wellknown 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 . DíazBarreo 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, DíazBarreo 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 wellknown Bellman's inequality is stated in the following theorem [8] (see also [2, pages 118119]).
Theorem 1.5.
Let be a fixed positive integer, and let be nonnegative real numbers such that (1.3) is satisfied. If , then
DíazBarreo 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 righthand side of the above inequality gives us the required result.
Let and be positive real numbers such that , then the wellknown Hölder's inequality states that
where are positive real numbers.
If , then the wellknown 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 righthand side of the above equation, we get
By using inequality (2.11) on righthand 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 righthand 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 righthand 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 righthand 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 logconvex 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
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
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 logconvex 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).
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. 11711708890888.
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