2 New refinements of generalized Aczél inequality
In order to prove the main results in this section, we need the following lemmas.
Let (
,
),
let be a real number,
(
),
and let .
Then (7)
Let (
,
),
let (
),
and let .
Then (8)
If ,
or ,
then (9)
The inequality is reversed for .
Let be real numbers,
let m be a natural number,
and let .
Then (10)
Lemma 2.5 Let ,
let (
),
and let .
Then (11)
Proof From the assumptions in Lemma 2.5, we find
and
(12)
Thus, by using inequality (7) we have
(13)
Noting the fact that there are
product terms in the expression
, and using the arithmetic-geometric mean’s inequality, we obtain
(14)
Therefore, we have
(15)
On the other hand, from Lemma 2.4 we have
(16)
Consequently, from (13), (15), and (16), we obtain the desired inequality (11). □
Lemma 2.6 Let ,
,
let ,
(
),
and let .
If ,
then (17)
If ,
then (18)
Proof Case I. When
. Let us consider the following product:
(19)
From the hypotheses of Lemma 2.6, it is easy to see that
and
(20)
Then, applying inequality (7), we have
(21)
There are
product terms in the expression
, and then we derive from the arithmetic-geometric mean’s inequality that
(22)
Therefore, we have
(23)
On the other hand, from Lemma 2.4 we find
(24)
Combining inequalities (21), (23), and (24) yields the desired inequality (17).
Case II. When . By the same method as in Lemma 2.5, it is easy to obtain the desired inequality (18). So we omit the proof. The proof of Lemma 2.6 is completed. □
Lemma 2.7 Let ,
let (
),
and let ,
.
Then (25)
Proof By the same method as in Lemma 2.5, applying Lemma 2.2, it is easy to obtain the desired inequality (25). So we omit the proof. □
Lemma 2.8 Let ,
let (
),
and let .
Then (26)
Proof After simply rearranging, we write by the component of in increasing order, where is a permutation of .
Then from Lemma 2.5 and Lemma 2.4 we get
(27)
The proof of Lemma 2.8 is completed. □
By the same method as in Lemma 2.8, we obtain the following two lemmas.
Lemma 2.9 Let ,
,
let ,
(
),
and let .
If ,
then (28)
If ,
then (29)
Lemma 2.10 Let ,
let (
),
and let ,
.
Then (30)
Now, we give the refinement and generalization of inequality (5).
Theorem 2.11 Let ,
,
,
,
,
and let .
Then (31)
Proof From the assumptions in Theorem 2.11, it is easy to verify that
(32)
It thus follows from Lemma 2.8 with the substitution
in (26) that
(33)
which implies
(34)
On the other hand, it follows from Lemma 2.1 that
(35)
Combining inequalities (34) and (35) yields inequality (31).
The proof of Theorem 2.11 is completed. □
Theorem 2.12 Let ,
(
),
let ,
,
,
,
and let .
If ,
then (36)
If ,
then (37)
Proof From the hypotheses of Theorem 2.12, we find that
and
Consequently, by the same method as in Theorem 2.11, and using Lemma 2.9 with a substitution () in (28) and (29), respectively, we obtain the desired inequalities (36) and (37). □
By the same method as in Theorem 2.11, and using Lemma 2.10, we obtain the following sharpened and generalized version of inequality (4).
Theorem 2.13 Let ,
,
,
,
,
let ,
and let .
Then (38)
Therefore, from Lemma 2.3 and Theorem 2.13 we get a new refinement and generalization of inequality (4).
Corollary 2.14 Let ,
,
,
,
,
let ,
and let .
If ,
then (39)
If ,
then (40)
Remark 2.15 If we set
in Corollary 2.14, then inequalities (39) and (40) reduce to Wu’s inequality ([[
11], Theorem 1]).
In particular, putting , , , , () in Theorem 2.13, we obtain a new refinement and generalization of inequality (2).
Corollary 2.16 Let ,
(
),
let ,
,
and let ,
.
Then (41)
Similarly, putting , , , , () in Theorem 2.12 and Theorem 2.11, respectively, we obtain a new refinement and generalization of inequality (3).
Corollary 2.17 Let ,
(
),
let ,
,
,
and let ,
.
Then (42)
From Lemma 2.3 and Theorem 2.11 we obtain the following refinement of inequality (5).
Corollary 2.18 Let ,
,
,
,
,
and let .
Then (43)
Similarly, from Lemma 2.3 and Theorem 2.12 we obtain the following refinement and generalization of inequality (5).
Corollary 2.19 Let ,
(
),
let ,
,
,
,
and let ,
.
Then (44)
If we set , then from Corollary 2.18 and Corollary 2.19 we obtain the following refinement of inequality (5).
Corollary 2.20 Let ,
(
),
,
let ,
,
,
,
and let .
Then (45)
3 Application
In this section, we show an application of the inequality newly obtained in Section 2.
Theorem 3.1 Let (
),
let ,
(
),
,
,
and let (
)
be positive integrable functions defined on with .
Then (46)
Proof For any positive integers
n, we choose an equidistant partition of
as
Noting that
(
), we have
Consequently, there exists a positive integer
N, such that
for all and .
By using Theorem 2.12, for any
, the following inequality holds:
(47)
we have
(48)
Noting that () are positive Riemann integrable functions on , we know that and are also integrable on . Letting on both sides of inequality (48), we get the desired inequality (46). The proof of Theorem 3.1 is completed. □
Remark 3.2 Obviously, inequality (46) is sharper than inequality (6).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.