On a More Accurate Half-Discrete Mulholland-Type Inequality Involving One Multiple Upper Limit Function

Abstract

By the use of the weight functions, the symmetry property, and Hermite-Hadamard’s inequality, a more accurate half-discrete Mulholland-type inequality involving one multiple upper limit function is given. The equivalent conditions of the best possible constant factor related to multiparameters are studied. Furthermore, the equivalent forms, several inequalities for the particular parameters, and the operator expressions are provided.

1. Introduction

Suppose that . The following well-known Hardy-Hilbert’s inequality was provided (cf. [1], Theorem 315): where the constant factor is the best possible. Replacing and in (1), we have a more accurate form of (1) (cf. [1], Theorem 323). With regard to the similar assumptions, the following Mulholland’s inequality was obtained (cf. [1], Theorem 343):

In 2006, by using Euler-Maclaurin’s summation formula and the technique of real analysis, Krnić and Pečarić [2] obtained an extension of (1) as follows: where , , the constant factor in (3) is the best possible, and is the beta function. For in (3), it reduces to (1); for and in (3), it reduces to Yang’s inequality in [3].

Using Abel’s partial sum formula and (2), in 2019, Adiyasuren et al. [4] provided a discrete Hardy-Hilbert’s inequality with the kernel as (3) involving two partial sums. We find that inequalities (1)–(3) played an important role in analysis and its applications (cf. [5–14]).

In 2016, Hong and Wen [15] gave an equivalent condition of the best possible constant factor related to a few parameters in the general form of (1). In the same way, the other similar works of (1) and the integral analogues were given in [16–19].

In 1934, a half-discrete Hilbert-type inequality was published in Hardy et al. [1] (Theorem 351) as follows: if is decreasing, , and , satisfying , then, we have

Some new extensions of (5) with their applications were provided by [20–24].

In this article, following the way of [4, 15], by the use of the weight functions, the symmetry property, and Hermite-Hadamard’s inequality, a more accurate half-discrete Mulholland-type inequality with the kernel as involving one multiple upper limit function is obtained. The equivalent conditions of the best possible constant factor related to multiparameters are given. Furthermore, the equivalent forms, several new inequalities for the particular parameters, and the operator expressions are considered. The lemmas and theorems of this paper provide an extensive account of this type of inequalities.

2. Some Lemmas

In what follows, we suppose that , , , , , , , , , . We also assume that is a nonnegative Lebesgue integrable function in any interval . Define a multiple upper limit function as follows: , satisfying and for any , it follows that

Lemma 1. Assuming that , define the following weight function: The following inequality is valid:

Proof. For fixed , the function is strictly decreasing and strictly convex in the interval . In fact, for , it follows that Applying Hermite-Hadamard’s inequality (cf. [25]), we have Setting , it follows that By (8) and (11), we have the following inequality: namely, (9) follows.
The lemma is prove.

Lemma 2. For and , if then we have the following more accurate half-discrete Mulholland-type inequality:

Proof. Setting , we obtain the expression of the following another weight function: Using Hölder’s inequality (cf. [25]), it follows that We show that the above inequality does not keep the form of equality. Otherwise (cf. [25]), there exist constants and , such that they are not both zero and Suppose that . There exists a , such that which contradicts the fact that based on . Then, by (9) and (17), we have (16).
The lemma is proven.

Remark 1. Replacing by and setting , , and in (16), we find , , and and obtain the following more accurate half-discrete Mulholland-type inequality with new parameters:

Lemma 3. For , the following expression is valid:

Proof. For since , (23) is naturally valid; for , integration by parts, since , we find By substitution of in the above expression, we have (23).
The lemma is proven.

3. Main Results

Theorem 1. The following more accurate half-discrete Mulholland-type inequality involving one multiple upper limit function is valid: In particular, for (), we have and the following inequality:

Proof. In view of the following expression: by the Lebesgue term by the term integration theorem (cf. [26]) and (23), we obtain Then, by (22), we have (25).
The theorem is proven.

Theorem 2. If , then the constant factor is the best possible. On the other hand, if we add and the same constant factor in (25) is the best possible, then we have .

Proof. If then (25) reduces to (27). For any , we set We have and satisfying .
If there exists a positive constant , such that (27) is valid when we replace by , then in particular, it follows that For , we define . By the decreasing property of the series, we find By (17), setting and , it follows that In virtue of the above results, we have For , by the continuity of the beta function, we have Hence, is the best possible constant factor in (27).
On the other hand, for , in view of , we find , , , and By substitution of in (27), we find By means of Hölder’s inequality (cf. [25]), it follows that Since is the best possible constant factor in (25), by (39), we have the following inequality: namely, , and then, (40) keeps the form of equality.
We observe that (40) keeps the form of equality if and only if there exist constants and (cf. [25]), such that they are not both zero and in . Suppose that . We find in , namely, . Hence, it follows that .
The theorem is proven.

4. Equivalent Forms and Some Particular Inequalities

Theorem 3. The following half-discrete more accurate Mulholland-type inequality involving a multiple upper limit function is equivalent to (25): In particular, for , we have the following Mulholland-type inequality equivalent to (27):

Proof. By assuming that (42) is valid, by Hölder’s inequality (cf. [25]), it follows that Then, by (42), we have (25).
On the other hand, suppose that (25) is valid. We set If , then (42) is naturally valid; if , then it is impossible that makes (42) valid, i.e., . By assuming that , by (25), it follows that namely, (42) follows, which is equivalent to (25).
The theorem is proven.

Theorem 4. If , then the constant factor in (42) is the best possible. On the other hand, if we add and the same constant factor in (42) is the best possible, then we have .

Proof. If , then by Theorem 2, the constant factor in (25) is the best possible. By (44), the constant factor in (42) is still the best possible. Otherwise, we would reach a contradiction that the constant factor in (25) is not the best possible.
On the other hand, if the constant factor in (42) is the best possible, then by the equivalency of (42) and (25), in view of (see the proof of Theorem 3), we still can show that the constant factor in (25) is the best possible. By the assumptions and Theorem 2, it follows that .
The theorem is proven.

Example 1. (i)For , (27) and (43) reduce to the following equivalent inequalities:(ii)For , (27) and (43) reduce to the following equivalent inequalities:Hence, (27) (resp. (43)) is a more accurate form of (51) (resp. (52)). (iii)For , (27) and (43) reduce to the following equivalent inequalities: