1. Introduction
Let
. Then,
where the constant factor
is the best possible. Inequality (1) is known in the literature as the Hardy–Hilbert inequality ([
1], Theorem 315). It is interesting that under a similar assumption condition to that above, one has the following Hardy–Mulholland’s inequality ([
1], Theorem 343):
where the constant factor
is also the best possible.
In 1934, a half-discrete Hilbert-type inequality was presented ([
1], Theorem 351), as follows:
where
is a decreasing function,
, the function
is defined as
,
and
. Since then, some half-discrete Hilbert-type inequalities have been proposed intermittently; we refer the interested reader to monograph [
2] and the references cited therein.
In 2018, Batbold and Azar [
3] gave a new form of half-discrete Hilbert inequality for three variables. In 2021, Huang and Yang [
4] established a half-discrete Hardy–Mulholland-type inequality involving one multiple upper limit function,
, as follows:
where
,
, and function
is defined by the form:
,
. The constant factor
in (4) is the best possible. In 2023, Peng et al. [
5] provided a reverse half-discrete Hardy–Mulholland-type inequality.
It is important to note that, in 2019, Adiyasuren et al. [
6] presented a novel extension of the Hardy–Hilbert inequality by imbedding two partial sums,
and
, in the right series of the following inequality, i.e.:
where
,
. The constant factor
is the best possible.
To date, there have been lots of papers published on half-discrete Hardy–Hilbert-type and Mulholland-type inequalities (see [
2,
3,
7,
8,
9]). However, the study of half-discrete Hardy–Mulholland-type inequalities involving partial sums has not been reported yet. This is the main motivation for us to carry out this work.
Hong et al. [
10,
11,
12,
13,
14,
15] gave equivalent parameter conditions for the construction of Hilbert-type integral inequalities. What has caught our attention is that Hong, Zhang, and Xiao [
16] improve the condition for constructing a Hilbert-type integral inequality involving upper limit functions.
Inspired by the work of [
6,
16], in this paper, by replacing the term of series
with the partial sum
in the right-hand side of inequality (4), we establish a new half-discrete Hardy–Mulholland-type inequality. Our method is mainly based on the use of the Euler–Maclaurin summation formula, Abel’s partial summation formula, and the differentiation mid-value theorem.
The rest of this paper is structured as follows. Firstly, we introduce some lemmas on the construction of weight function and related identities and inequalities. Secondly, we establish a half-discrete Hardy–Mulholland-type inequality involving one multiple upper limit function and one partial sum. For the obtained inequality, we deal with the equivalent conditions for the best possible constant factor associated with the parameters. Finally, we illustrate that the main results obtained can generate some new half-discrete Hardy–Mulholland-type inequalities.
2. Preliminaries and Lemmas
For convenience, we first specify the following assumption conditions (H1) because they will be used frequently in the subsequent analysis.
(H1). ,
,
,
,
,
.
Let be a nonnegative continuous function unless at finite points in , , , inductively, for , we define the following multiple upper limit functions: , namely,which satisfies Let , which satisfies Lemma 1. (i) ([17], 2.2.13). If and , are Bernoulli functions and Bernoulli numbers of i-order, then (ii) ([17], 2.3.2) If then we have the following Euler–Maclaurin summation formula: Lemma 2. Let . We can then define the following weight coefficient: The following inequalities are valid:where Proof. For fixed
we define a positive function
as follows:
(i) For
it is easy to find
Now, using Hemite–Hadamard’s inequality [
18] and the decreasingness property of the series, we deduce that
(ii) For
in view of (8) (for
), we have
From the definition of
, we derive
. Now, setting
, integrating by parts gives
If
then we have the following inequality:
Based on the above result, for
with the aid of (8), it follows that
Hence, for
we deduce that
We still have
where
For
by means of (8), we find
Hence, we derive inequality (12).
In conclusion, for
by using (11) and setting
it follows that
where we indicate that
which satisfies
Therefore, we obtain inequality (11). The proof of Lemma 1 is complete. □
Remark 1. For , we can still obtain Thus, inequality (11) is still valid in the case that .
Lemma 3. Under assumption H1, we have the following half-discrete Hardy–Mulholland-type inequality: Proof. Setting
, for
we can also obtain the following weight function:
By using Hölder’s inequality [
18], we have
By utilizing inequalities (11) and (14), based on the condition of parameters we have , and then using (6) (for ) and (7), we derive inequality (13). This completes the proof of Lemma 3. □
Lemma 4. Under assumption H1, for , we have the following expression: Proof. For
since
, equality (15) is naturally valid; for
by applying Hölder’s inequality and assumption H1, we have
It follows that
and then, from
, we derive
. Using integration by parts, we obtain
Substituting in the above equality, by simplification, we get equality (15). Lemma 4 is proved. □
Remark 2. For , by exchanging the assumption
for the assumption , we still have equality (15).
Lemma 5. Under assumption H1, for , we have the following inequality: Proof. In view of
, by using the Abel summation by parts formula, we deduce that
We define a function
Then, we obtain
where
is decreasing in
. By applying the differentiation mid-value theorem, we have the following equality
and then, we have
which leads to the required inequality (16). This completes the proof of Lemma 5. □
Lemma 6. Under assumption H1, the following inequality holds: Proof. With the aid of the expression of the Gamma function [
19]
and using the Lebesgue term-by-term theorem [
20], from (15) and (16), we deduce that
This proves inequality (17). The proof of Lemma 6 is complete. □
Lemma 7. Let then, there exists a constant such that Proof. Let
Then, one has
where
It is easy to see that
Since
we deduce that
In view of (8), we obtain
By utilizing (9), it follows that
Substituting the above results, by simplification, we obtain (18), where
This completes the proof of Lemma 7. □
3. Main Results
Theorem 1. Under assumption H1, we have the following inequality: In particular, if , this implies thatand we have the following inequality:where the constant factor is the best possible. Proof. By applying inequalities (17) and (13), we acquire inequality (19).
For any
, we set
and for
N, we set
where
is a positive polynomial of degree
with
We denote
Then, the above expression satisfies
, with the following assumption:
For
with the aid of (18), we obtain
which satisfies
.
Note that for derivable function
we have
and then we deduce
Hence, we obtain
If there exists a positive constant
such that inequality (20) is valid when we replace
by
, then, in particular, for
we still have
In the following, we prove that
, which means that
is the least value that makes (20) valid. In view of (21) and the above results, we acquire
By using (10) and (11), and setting
,
we get
Based on the above results, we obtain
For
, with the help of the continuous property of the Beta function, we have
Consequently, is the best possible constant factor in inequality (20). The proof of Theorem 1 is complete. □
Theorem 2. Under assumption H1, if the constant factorin (19) is the best possible, then for we have Proof. Since , it follows that and .
For
one has
. Since
satisfies the condition of
in (20), then, by virtue of (20), for
we derive the following inequality:
By applying Hölder’s inequality, we have
According to the hypothesis that the constant factor
in (19) is the best possible, we observe that it is the least value that makes (19) valid. By comparing the constant factors in (19) and (22), we deduce that
which is
Therefore, we conclude that inequality (23) keeps the form of an equality. Furthermore, we observe that (23) keeps the form of an equality if, and only if, constants and exist such that they are not both zero and in . Assuming that , it follows that in , which implies that . Hence, we acquire . This completes the proof of Theorem 2. □
5. Conclusions
In this work, by embedding multiple upper limit functions and partial sums, we have provided a new approach to producing an extension of half-discrete Hardy–Mulholland-type inequalities. By the clever use of the Euler–Maclaurin summation formula, Abel’s summation by parts formula, and the differentiation mid-value theorem, we establish a new half-discrete Hardy–Mulholland-type inequality involving one multiple upper limit function and one partial sum in Theorem 1. Then, in Theorem 2, we characterize the condition for the best possible constant factor related to several parameters. Finally, in Remarks 3 and 4, we illustrate that some new half-discrete Hardy–Mulholland-type inequalities can be derived from the special values of the parameters. The main highlight of this work is that we introduce both multiple upper limit functions and partial sums in a half-discrete Hardy-Mulholland-type inequality; such a result has not been reported in previous studies. From the perspective of future research, we shall establish more half-discrete Hardy–Mulholland-type inequalities by constructing new kernel functions and improving the conditions related to the multiple upper limit function and partial sum. This will be interesting and challenging work.