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

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

(1.1)

then

(1.2)

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

(1.3)

then, for , one has

(1.4)

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

(1.5)

where

(1.6)

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

(1.7)

Also, let , then, for , one has

(1.8)

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

(1.9)

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

(1.10)

where

(1.11)

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.

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

(2.1)

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

(2.2)

Proof.

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

(2.3)

These imply

(2.4)

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

(2.5)

where are positive real numbers.

If , then the well-known inequality of power sums of order and states that

(2.6)

where are positive real numbers (c.f [9, page 165]).

Now, if , then and using inequality (2.6) in (2.5), we get

(2.7)

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

(2.8)

(i)If , then

(2.9)

(ii)If , then

(2.10)

Proof.

(2.11)

We can write

(2.12)

By applying Theorem 1.2 for on right-hand side of the above equation, we get

(2.13)

(2.14)

and denoting , ,

then

(2.15)

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

(2.16)

since , so by using (2.7)

(2.17)

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

(2.18)

then, for , one has

(2.19)

Proof.

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

(2.20)

Now,

(2.21)

and denote , .

Then

(2.22)

It is given that and , therefore by using Theorem 1.4, for , on right-hand side of the above equation, we get

(2.23)

by applying Hölder's inequality

(2.24)

by using inequality (2.20)

(2.25)

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

(2.26)

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

(2.27)

Proof.

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

(2.28)

Now,

(2.29)

and denote , .

Then

(2.30)

It is given that and , therefore by using Bellman's inequality, for , on right-hand side of the above equation, we get

(2.31)

by applying Minkowski's inequality

(2.32)

and by using inequality (2.28)

(2.33)

Remark 2.6.

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

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

(3.1)

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

(3.2)

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

(3.3)

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

(3.4)

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

(3.5)

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

(3.6)

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

(3.7)

then is strictly increasing function on for each .

Proof.

Since

(3.8)

therefore is strictly increasing function on for each .

Theorem 3.7.

Let be a positive -tuple such that , and let

(3.9)

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

(3.10)

is a positive semidefinite matrix.

(b)The function is exponentially convex.

(c)The function is log convex.

Proof.

1. (a)
   Define a function

   

   (3.11)
    

then

(3.12)

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

(3.13)

Hence, the given matrix is positive semidefinite.

1. (b)

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

    
2. (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

(3.15)

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:

(3.16)

Theorem 3.12 ..

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

(3.17)

Proof.

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

(3.18)

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

(3.19)

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

(3.20)

Consider the functions , defined as

(3.21)

then for are monotonically increasing functions.

Theorem 3.15.

Let , where is a compact interval such that and . If , then there exists such that

(3.22)

Proof.

Since is compact and , therefore let

(3.23)

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

(3.24)

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

(3.25)

Now, by combining inequalities (3.24), we get

(3.26)

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

(3.27)

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:

(3.28)

provided that the denominators are nonzero.

Proof.

Let a function be defined as

(3.29)

where and are defined as

(3.30)

Then, using Theorem 3.15, with , we have

(3.31)

Since , therefore (3.31) gives

(3.32)

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