1. Introduction
The famous Hardy–Hilbert inequality states the following [
1]:
where
,
is the best possible constant factor.
Under the same assumption condition as above, an analogous form of inequality (1), called the Hardy–Littlewood–Polya inequality, was transcribed in [
1], i.e.,
where the constant factor
is the best possible.
In [
2], Yang, Wu and Chen established an extended Hardy–Littlewood–Polya inequality, as follows:
where
,
.
For the continuous case, the following integral version of the Hardy–Hilbert inequality is well known (see [
1]).
If
and
, then
where the constant factor
is the best possible.
Recently, You [
3] considered the integral version of Hardy–Littlewood–Polya inequality; he extended the integral interval to the whole plane and obtained the following result:
Some relevant results about Hardy–Hilbert-type inequalities in the whole plane were provided in [
4,
5,
6,
7].
As is known to us, Hardy–Littlewood–Polya inequality and Hardy–Hilbert-type inequalities play important roles in mathematical analysis. These inequalities provide a lot of practical application cases in the theories of double series, double integrals and special functions.
Motivated by the above-mentioned works [
1,
2], we devote to establish a new generalization of Hardy–Littlewood–Polya inequality (2). To achieve this goal, we construct the following weighted coefficients for the series of the right-hand-side inequality (2):
Also, we extend the range of parameters by to in the newly established inequality.
Based on the above idea, in this article, by introducing multiple parameters and constructing weight coefficients, with the help of the techniques of real analysis, we acquire a discrete version of the Hardy–Littlewood–Polya inequality in the whole plane. Furthermore, we give the equivalent forms of the obtained inequality and characterize the equivalent conditions for the best possible constant factor linked to the parameters. Finally, we illustrate how the main results obtained can generate some new Hardy–Littlewood–Polya type inequalities.
2. Preliminaries and Lemmas
Definition 1. We define the homogeneous function of degree as follows:
which satisfies .
It follows that is a positive and continuous function with respect to . It is easy to observe that Claim 1. (i) For by (5), we obtain(ii) For , which implies that hence is strictly decreasing. We deduce that is decreasing with respect to and strictly decreasing in .
For convenience, let us state the conditions (C1) below, which will be used in what follows.
(C1)
, such that for
,
where
Claim 2. For by (6), we have for we have
Lemma 1. For we have the following inequalities: Proof. By using the decreasing property of this series, we find
Hence, the two-sided inequalities in (7) hold true. This proves Lemma 1.
Definition 2. We define the following weight functions: Lemma 2. The following inequalities are valid:where Proof. For
we define the following functions:
where for
It is easy to verify that
For fixed
by the aid of Claim 1 (ii), we conclude that both
and
are decreasing with respect to
and strictly decreasing in
. Using the decreasing property of the series, it follows that
Setting
in the above first integral, and setting
in the above second integral, respectively, we obtain
where
which satisfies, for
,
Hence, we derive the inequality (10). Moreover, inequality (11) can be deduced in the same way as above. The proof of Lemma 2 is complete.
3. Main Results
Theorem 1. Under the assumption condition (C1), we have the following parameterized Hardy–Littlewood–Polya inequality: Proof. By utilizing Hölder’s inequality [
8], we obtain
By virtue of inequalities (10) and (11), we deduce inequality (12). This completes the proof of Theorem 1.
Remark 1. In (12), for , we derive from Claim 1 that ,and we have the following inequality: Theorem 2. Theconstant factor in inequality (13) is the best possible.
Proof. For any
, we set
By the way of contradiction, if there exists a constant
such that inequality (13) is valid, when we replace
by
, it follows that
In view of (8),
and
Using inequality (11) and Lemma 1, we have
Combining the above results, we have
Taking a limit as , we acquire that , namely , which means that the constant factor is the best possible in (13). The proof of Theorem 2 is complete.
Remark 2. Letting , we can rewrite (12) as follows: Note that From , it follows that Moreover, from we conclude that
We can still find that Therefore, inequality (13) can be rewritten as Theorem 3. Suppose that ; if the constant factor in (12) (or (14)) is the best possible, then .
Proof. By employing Hölder’s inequality [
8], we obtain
If the constant factor
in (12) (or (14)) is the best possible, then, by comparing with the constant factors in (14) and (15), we have the following inequality:
namely
; it follows that (16) keeps the form of equality.
In view of the condition of equality for Hölder’s inequality [
8], we observe that (16) keeps the form of equality if and only if there exist constants
and
such that they are both not zero and
in
. We may assume
, which leads to
in
, hence
, namely
. This completes the proof of Theorem 3.
4. Equivalent Statements of the Main Results
Theorem 2. We have the following inequality which is equivalent to inequality (14): Proof. (i) Suppose that (17) is valid. By using Hölder’s inequality [
8], we obtain
Applying inequality (17), we acquire inequality (14).
- (ii)
On the other hand, if inequality (14) is valid, taking
then we have
. If
, (17) is naturally valid. Also, we find that
is impossible to make (17) valid, namely
.
For
, by (14), we deduce that
this yields
which is the desired inequality (17). Therefore, inequality (17) is equivalent to inequality (14). Theorem 4 is proved.
Theorem 5. The following statements (i), (ii), (iii), (iv) and (v) are equivalent:
- (i)
Both and are independent of ;
- (ii)
;
- (iii)
- (iv)
Under the condition that , is the best possible constant factor of (14);
- (v)
Under the condition that , is the best possible constant factor of (17).
Remark 3. Similarly to Theorem 4, if , then we have the following inequality with the best possible constant factor which is equivalent to (13). Proof. (i)
(ii). Since both
and
are independent of
, it follows that
(ii)(iii). If , then (16) keeps the form of equality. From the proof of Theorem 3, we deduce that .
(iii)
(i). If
, then we have
We conclude that both and are independent of . Hence, (i)(ii)(iii).
(iii)(iv). From the results of Theorem 2 and Theorem 3, we can directly deduce the conclusions.
(iv)(v). If the constant factor in (14) is the best possible, then so is the best possible constant factor in (17). Otherwise, by (18) (for ), we would reach a contradiction that the constant factor in (13) is not the best possible. On the other hand, if the constant factor in (17) is the best possible, then so is the best possible constant factor in (14). Otherwise, in view of the equivalency between (14) and (17), we would reach a contradiction that the constant in (14) is not the best possible.
Therefore, the statements (i), (ii), (iii), (iv) and (v) are equivalent. The proof of Theorem 5 is complete.
5. Special Cases of Parameterized Inequalities
As applications of the main results, below we will deduce some new inequalities of the Hardy–Littlewood–Polya type from the special values of parameters.
Remark 4. (i) Putting in (13) and (19), we obtain the following inequalities: - (ii)
Taking in (13) and (19), we derive the following inequalities: - (iii)
Taking in (22) and (23), we acquire the inequalities - (iv)
Choosing in (20) and (21), we have - (v)
Choosing in (26), we obtain the classical Hardy–Littlewood–Polya inequality (2).
6. Conclusions
Compared to the Hardy–Hilbert inequality, the Hardy–Littlewood–Polya inequality is relatively less investigated. In this work, we focus on the extension of the Hardy–Littlewood–Polya inequality in the whole plane. The main idea is to construct weight coefficients and introduce more parameters. In Theorem 1, we establish a discrete version of the Hardy–Littlewood–Polya inequality, involving multiple parameters in the whole plane. We give the equivalent forms of the parameterized Hardy–Littlewood–Polya inequality in Theorem 4. The characterization of the equivalent conditions via the parameters linked to the best possible constant factor are considered in Theorems 2, 3 and 5. At the end of the paper, in Remark 4, we show that some new inequalities of Hardy–Littlewood–Polya type can be derived from the special values of parameters. The main result provided a new generalization of the Hardy–Littlewood–Polya inequality. The limitation of the presented work is that we are unable to obtain a corresponding integral version of the generalized Hardy–Littlewood–Polya inequality. In the future, the Hardy–Littlewood–Polya will be studied in a wider range of areas, such as refined forms, reversed forms, integral forms and half-discrete forms.
Author Contributions
B.Y. carried out the mathematical studies and drafted the manuscript. S.W. participated in the design of the study and performed the numerical analysis. All authors contributed equally to the preparation of this paper. All authors have read and agreed to the published version of the manuscript.
Funding
This work is supported by the Natural Science Foundation of Fujian Province of China (No. 2020J01365).
Data Availability Statement
The data presented in this study is available on request from the corresponding authors, and the dataset was jointly completed by the team, so the data is not publicly available.
Conflicts of Interest
The authors declare no conflicts of interest.
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