A new reverse Mulholland-type inequality with multi-parameters

1.   Introduction

In 1925, Hardy ^[1] gave a generalization of the Hilbert's inequality by introducing one pair of conjugated exponents $(p, q)$ which satisfies $\tfrac{1}{p} + \tfrac{1}{q} = 1$ and $p > 1$, as follows:

If ${a_m} \geqslant 0, \; {b_n} \geqslant 0,$ $0 < \sum\nolimits_{m = 1}^\infty {a_m^p} < \infty$ and $0 < \sum\nolimits_{n = 1}^\infty {b_n^q} < \infty,$ then

where the constant factor $\tfrac{\pi }{{\sin (\pi /p)}}$ is the best possible. The inequality (1) is called the Hardy-Hilbert inequality. In particular, when $p = q = 2,$ the Hardy-Hilbert inequality reduces to the Hilbert's inequality (see ^[2]). As is known to us, the Hardy-Hilbert inequality plays an important role in analysis number theory, real analysis and divergent series theory (see ^[3]).

For the continuous case, the integral version of Hardy-Hilbert inequality can be stated as follows (see ^[3], Theorem 316):

If $f(x) \geqslant 0, \, \, g(y) \geqslant 0, \; 0 < \int_0^\infty {{f^p}(x)dx < \infty }$ and $0 < \int_0^\infty {{g^q}(y)dy < \infty }$, then

where the constant factor $\tfrac{\pi }{{\sin (\pi /p)}}$ is the best possible.

The Hardy-Hilbert inequalities (1) and (2) has been studied extensively, numerous variants, generalizations, and extensions can be found in the literatures (see ^[4,5,6,7,8]).

Motivated by the Hardy-Hilbert inequality, in 1929, Mulholland ^[9] proposed a similar version of inequality (1), which contains the same best possible constant factor $\tfrac{\pi }{{\sin (\pi /p)}}$ as in (1), i.e.,

Obviously, the Mulholland's inequality (3) can be rewritten in an equivalent form as:

In recent years, the Mulholland inequality has been generalized by various methods of constructing parameters, see ^{[10,11,12,13,14,15]} and the references cited therein.

Let us recall some results which are connected with the current investigation. Hong and Wen ^[16] and Hong ^[17] studied the Hilbert type series inequalities and the Hilbert type integral inequalities with homogeneous kernel, respectively. They established the necessary and sufficient condition for which the inequalities hold under the best constant factor. Subsequently, by using the quasi-homogeneous kernels instead of the homogeneous kernel in the Hilbert type integral inequalities, Hong, He and Yang ^[18] established the necessary and sufficient condition for which the inequalities hold under the best constant factor. Recently, with the help of the Euler-Maclaurin summation formula, Yang, Wu and Liao ^[19] and gave the extension of Hardy-Hilbert's inequality and its equivalent forms. In ^[20], Yang, Wu and Chen investigated the generalization of Hardy-Littlewood-Polya's inequality and its equivalent forms.

In this paper, following the ideas of ^{[16,17,18,19,20]}, we will study the Mulholland-type inequalities. The present research objects are structurally different from Hilbert type inequalities, and this will involve new techniques in dealing with the inequalities. Specifically, we will establish a reverse Mulholland-type inequality with multi-parameters. And then, we discuss the equivalent statements of the best possible constant factor associated with several parameters. Finally, we illustrate that some new inequalities of Mulholland-type can be derived from the equivalent expressions of the reverse Mulholland-type inequality.

2.   Preliminaries

In this section, we present some preliminary results which are essential for establishing our main results in subsequent sections. We begin with introducing the notations $k_\lambda ^{(\eta)}(x, y)$ and $k_{\lambda, \eta }^{}(\gamma)$ with their associated formulas.

(i) In view of the following expression (see ^[21])

for $b \in (0, 1)$, we have

Moreover, it is easy to observe that ${A_b} > 0$ for $b \in (0, \tfrac{1}{2})$; ${A_b} < 0$ for $b \in (\tfrac{1}{2}, 1)$; ${A_b} = 0$ for $b = \tfrac{1}{2}$.

(ii) For $\lambda, \eta > 0,$ we set the homogeneous function of order $- \lambda$ as follows:

which satisfies $k_\lambda ^{(\eta)}(ux, uy) = {u^{ - \lambda }}k_\lambda ^{(\eta)}{(x, y)^{}}(u, x, y > 0)$, and $k_\lambda ^{(\eta)}(v, v): = {\tfrac{\eta }{{(\lambda + \eta){v^\lambda }}}^{}}(v > 0)$.

It follows that $k_\lambda ^{(\eta)}(x, y)$ is a positive and continuous function with respect to $x, y > 0$. For $x \ne y,$ we obtain

where $\varphi (x, y): = \lambda {x^{\lambda + \eta }} - (\lambda + \eta ){y^\eta }{x^\lambda } + \eta {y^{\lambda + \eta }}^{}(x, y > 0).$

We find that for $0 < x < y$,

for $x > y,$ $\tfrac{\partial }{{\partial x}}\varphi (x, y) > 0$. It follows that $\varphi (x, y)$ is strictly decreasing (resp. increasing) with respect to $x < y$ (resp. $x > y$). Since $\varphi (y, y) = \mathop {\min }\limits_{x > 0} \varphi (x, y) = {0^{}}(y > 0)$, then $\varphi (x, y) > {0^{}}(x \ne y)$, namely, $\tfrac{\partial }{{\partial x}}k_\lambda ^{(\eta)}(x, y) < 0$ $(x \ne y)$. Note that $k_\lambda ^{(\eta)}(x, y)$ is continuous at $x = y$, we conform that $k_\lambda ^{(\eta)}(x, y)$ is strictly decreasing with respect to $x > 0$. In the same way, we can show that $k_\lambda ^{(\eta)}(x, y)$ is also strictly decreasing with respect to $y > 0$.

(iii) For $\gamma \in (0, \lambda),$ since $k_\lambda ^{(\eta)}(x, y) > 0,$ by (i), we obtain (see ^[22])

In what follows, we suppose that

for $\gamma = {\lambda _1}, \lambda - {\lambda _2}^{}( \in (0, \lambda )),$

Also, we assume ${a_m}, {b_n} \geqslant {0^{}}(m, n \in {\rm{N\backslash \{ 1\})}}$ such that

Definition 1. We define the following weight functions:

Lemma 1. For $\beta \lambda_{2} \leq 1$, we have

For $\alpha {\lambda _1} \leqslant 1$, we have

where ${\theta _{\lambda, \eta }}({\lambda _1}, n)\; (>0)$ is indicated as

Proof. For fixed $m \in \mathrm{N} \backslash\{1\}, \beta \lambda_{2}-1 \leq 0, k_{\lambda}^{(\eta)}\left(\ln ^{\alpha} m, \ln ^{\beta} t\right) \frac{\ln \beta_{2}-1_{t}}{t}$ is a strictly decreasing function with respect to $t>1$. By the decreasing property of series, setting $u = \frac{\ln ^{\alpha} m}{\ln ^{\beta} t}$, we find that

Hence, we get the inequality (8).

For $\alpha {\lambda _1} - 1 \leqslant 0$, it is evident that $k_\lambda ^{(\eta)}({\ln ^\alpha }t, {\ln ^\beta }n)\tfrac{{{{\ln }^{\alpha {\lambda _1} - 1}}~~t}}{t}$ is a strictly decreasing function with respect to $t > 1$. By the decreasing property of series, setting $u = \tfrac{{{{\ln }^\alpha }t}}{{{{\ln }^\beta }n}}$, we find that

By the decreasing property of series and (10), we obtain

Hence, we deduce the inequality (9). The Lemma 1 is proved.

Lemma 2. We have the following inequality:

Proof. By the reverse Hӧlder inequality ^[23], we obtain

By (8) and (9), we get the inequality (11). This completes the proof of Lemma 2.

Remark 1. By (11), for $\lambda_{1}+\lambda_{2} = \lambda$, we find

and obtain the following inequality:

In particular, for $\alpha = \beta = 1,$ we have

and the following inequality:

Hence, inequality (12) is an extension of inequality (13).

Lemma 3. For any $\varepsilon>0$, we have

Proof. There exist constants $m>0, M>0$ such that

By the decreasing property of series, it follows that

Hence, inequality (14) follows. The proof of Lemma 3 is complete.

Lemma 4. For $\delta_{0}: = \frac{1}{2} \min\limits_{i = 1, 2}\left\{\lambda_{i}, \lambda-\lambda_{i}\right\}, \quad 0 < \varepsilon < q \delta_{0}$, we have

Proof. By Levi Theorem (see ^[24]), it follows that

For $0 < \varepsilon < q{\delta _0}$, we have $0 < \tfrac{\varepsilon }{q} < {\delta _0}$, and then ${\lambda _1} + {\delta _0} < \lambda, \lambda - {\lambda _1} + {\delta _0} < \lambda$,

By using Lebesgue control convergence theorem ^[24], we have

Then we find

The Lemma 4 is proved.

Lemma 5. If ${\lambda _1} + {\lambda _2} = \lambda$, then the constant factor ${k_{\lambda, \eta }}({\lambda _1})$ in (12) is the best possible.

Proof. For $0 <\varepsilon <q \delta_{0}$, we set

If there exists a constant $M \geqslant \tfrac{1}{{{\beta ^{1/p}}~~{\alpha ^{1/q}}}}{k_{\lambda, \eta }}({\lambda _1})$ such that the inequalities (12) is valid when replacing $\tfrac{1}{{{\beta ^{1/p}}~~\alpha {}^{1/q}}}{k_{\lambda, \eta }}({\lambda _1})$ by $M$, then, in particular, we have

In view of (14), we obtain

By (8), for ${\hat \lambda _2} = {\lambda _2} - \tfrac{\varepsilon }{q}{(\in (0, \tfrac{1}{\beta }))^{}}({\hat \lambda _1} = {\lambda _1} + \tfrac{\varepsilon }{q} < \lambda)$, we find

Then we have

For $\varepsilon \to {0^ + }$, by (15), we get

namely, $\tfrac{1}{{{\beta ^{1/p}}~~{\alpha ^{1/q}}}}{k_{\lambda, \eta }}({\lambda _1}) \geqslant M$. Hence, $M =$ $\tfrac{1}{{{\beta ^{1/p}}~~{\alpha ^{1/q}}}}{k_{\lambda, \eta }}({\lambda _1})$ is the best possible constant factor of (12). The proof of Lemma 5 is completed.

Remark 2. Setting $\tilde{\lambda}_{1}: = \frac{\lambda-\lambda_{2}}{p}+\frac{\lambda_{1}}{q} = \lambda_{1}+\frac{\lambda-\lambda_{1}-\lambda_{2}}{p}, \tilde{\lambda}_{2}: = \frac{\lambda-\lambda_{1}}{q}+\frac{\lambda_{2}}{p}$, we find

For $|\lambda - {\lambda _1} - {\lambda _2}| < |p|{\delta _0},$ it means that $- {\delta _0} < \tfrac{{\lambda - {\lambda _1} - {\lambda _2}}}{p} < {\delta _0}$, $0 < {\tilde \lambda _1} < \lambda, 0 < {\tilde \lambda _2} = \lambda - {\tilde \lambda _1} < \lambda$, and then ${k_{\lambda, \eta }}({\tilde \lambda _1}) \in {{\rm{R}}_ + }$. We reduce (11) as follows:

Lemma 6. If $|\lambda - {\lambda _1} - {\lambda _2}| < |p|{\delta _0},$ and the constant factor $\tfrac{1}{{{\beta ^{1/p}}~~\alpha {}^{1/q}}}k_{\lambda, \eta }^{\tfrac{1}{p}}(\lambda - {\lambda _2})k_{\lambda, \eta }^{\tfrac{1}{q}}~~({\lambda _1})$ in (11) is the best possible, then we have $\lambda = {\lambda _1} + {\lambda _2}$.

Proof. If the constant factor $k_{\lambda, \eta}^{\frac{1}{p}}\left(\lambda-\lambda_{2}\right) k_{\lambda, \eta}^{\frac{1}{q}}\left(\lambda_{1}\right)$ in (11) (or (16)) is the best possible, then by (12) (for $\lambda_{i} = \tilde{\lambda}_{i}(i = 1, 2)$), we have

namely, ${k_{\lambda, \eta }}({\tilde \lambda _1})$ $\leqslant k_{\lambda, \eta }^{\tfrac{1}{p}}(\lambda - {\lambda _2})k_{\lambda, \eta }^{\tfrac{1}{q}}~~({\lambda _1})$.

By the reverse Hӧlder inequality, we find

It follows that (17) keeps the form of equality.

We observe that (17) keeps the form of equality if and only if there exist constants $A$ and $B$ such that they are not both zero and

Assuming that $A \ne 0$, it follows that ${u^{\lambda - {\lambda _2} - {\lambda _1}}} = {\tfrac{A}{B}^{}}a.e.{}^{}$ in ${{\rm{R}}_ + }$, and then $\lambda - {\lambda _2} - {\lambda _1} = 0$. Namely, $\lambda = {\lambda _1} + {\lambda _2}$. This completes the proof of Lemma 6.

3.   Main results

Theorem 1. Inequality (11) is equivalent to the following inequalities:

Proof. Suppose that (18) is valid. By the reverse Hӧlder inequality, we have

Then by (18), we obtain (11). On the other hand, assuming that (11) is valid, we set

It follows that

If $J = \infty$, then (18) is naturally valid; if $J = 0$, then it is impossible that makes (18) valid, namely, $J > 0$. Suppose that $0 < J < \infty$. By (11), we find

thus, (18) follows, which is equivalent to (11).

Suppose that (19) is valid. By the reverse Hӧlder inequality, we have

Then by (19), we obtain (11). On the other hand, assuming that (11) is valid, we set

It follows that

If ${J_1} = \infty$, then (19) is naturally valid; if ${J_1} = 0$, then it is impossible that makes (19) valid, namely, ${J_1} > 0$. Suppose that $0 < {J_1} < \infty$. By (11), we have

thereby, (19) follows, which is equivalent to (11). Hence, inequalities (11), (18) and (19) are equivalent.

This completes the proof of Theorem 1.

Theorem 2. Suppose that $|\lambda - {\lambda _1} - {\lambda _2}| < |p|{\delta _0}.$ The following statements (i), (ii), (iii), (iv) and (v) are equivalent:

(i) Both $k_{\lambda, \eta }^{\tfrac{1}{p}}(\lambda - {\lambda _2})k_{\lambda, \eta }^{\tfrac{1}{q}}~~({\lambda _1})$ and ${k_{\lambda, \eta }}(\tfrac{{\lambda - {\lambda _2}}}{p} + \tfrac{{{\lambda _1}}}{q})$ are independent of $p, q$;

(ii) $k_{\lambda, \eta }^{\tfrac{1}{p}}(\lambda - {\lambda _2})k_{\lambda, \eta }^{\tfrac{1}{q}}~~({\lambda _1})$ $= {k_{\lambda, \eta }}(\tfrac{{\lambda - {\lambda _2}}}{p} + \tfrac{{{\lambda _1}}}{q})$;

(iii) $\lambda = {\lambda _1} + {\lambda _2};$

(iv) $\tfrac{1}{{{\beta ^{1/p}}~~{\alpha ^{1/q}}}}k_{\lambda, \eta }^{\tfrac{1}{p}}(\lambda - {\lambda _2})k_{\lambda, \eta }^{\tfrac{1}{q}}~~({\lambda _1})$ is the best possible constant factor of (11);

(v) $\tfrac{1}{{{\beta ^{1/p}}~~{\alpha ^{1/q}}}}k_{\lambda, \eta }^{\tfrac{1}{p}}(\lambda - {\lambda _2})k_{\lambda, \eta }^{\tfrac{1}{q}}~~({\lambda _1})$ is the best possible constant factor of (18) (resp. (19)).

Moreover, if the statement (iii) follows, namely, $\lambda = {\lambda _1} + {\lambda _2}$, then we have (12) and the following equivalent inequalities with the best possible constant factor $\tfrac{1}{{{\beta ^{1/p}}~~{\alpha ^{1/q}}}}$ ${k_{\lambda, \eta }}({\lambda _1})$ :

Proof. (i)$\Rightarrow$(ii). By (i), we have

Then by (i), (15) and the above result, we find

(ii) $\Rightarrow$ (iii). by (ii), (17) keeps the form of equality. In view of the proof of Lemma 6, it follows that $\lambda = {\lambda _1} + {\lambda _2}$.

(iii) $\Rightarrow$ (i). By (iii), for $\lambda = {\lambda _1} + {\lambda _2}$, both $k_{\lambda, \eta }^{\tfrac{1}{p}}(\lambda - {\lambda _2})k_{\lambda, \eta }^{\tfrac{1}{q}}~~({\lambda _1})$ and ${k_{\lambda, \eta }}(\tfrac{{\lambda - {\lambda _2}}}{p} + \tfrac{{{\lambda _1}}}{q})$ are equal to $k_{\lambda, \eta }^{}({\lambda _1})$, which are independent of $p, q$. Hence, we have (i) $\Leftrightarrow$ (ii) $\Leftrightarrow$ (iii).

(iii) $\Leftrightarrow$ (iv). By Lemma 5 and Lemma 6, we have the conclusions.

(iv) $\Leftrightarrow$ (v). If the constant factor in (11) is the best possible, then so is the constant factor in (18) (resp. (19)). Otherwise, by (20) (resp. (22)), we would reach a contradiction that the constant factor in (11) is not the best possible. On the other hand, if the constant factor in (18) (resp. (19)) is the best possible, then so is the constant factor in (11). Otherwise, by (21) (resp. (23)), we would reach a contradiction that the constant factor in (18) (resp. (19)) is not the best possible.

Hence, the statements (i), (ii), (iii), (iv) and (v) are equivalent. The proof of Theorem 2 is complete.

4.   An application

In this section, we illustrate that some novel inequalities of Mulholland type can be derived from our main results as special cases.

Remark 3. Taking $\alpha = \beta = 1$ in (24) and (25) respectively, we have the following inequalities which is equivalent to (13) with the best possible constant factor $k_{\lambda, \eta }^{}({\lambda _1})$ :

In particular, putting $\lambda = 1, {\lambda _1} = {\lambda _2} = \tfrac{1}{2}$ in (13), (26) and (27) respectively, we have the following equivalent reverse inequalities with the best possible constant factor $\tfrac{{2\pi }}{{1 + \eta }}\cot \tfrac{\pi }{{2(1 + \eta)}}$ :

where ${\tilde \theta _{1, \eta }}(\tfrac{1}{2}, n): = \tfrac{{1 + \eta }}{{2\pi }}\tan \tfrac{\pi }{{2(1 + \eta )}}\int_0^{\tfrac{{\ln 2}}{{\ln n}}} {\tfrac{{{u^\eta } - 1}}~~{{{u^{1 + \eta }} - 1}}~~} {u^{ - \tfrac{1}{2}}}du = O(\tfrac{1}{{{{\ln }^{1/2}}n}}).$

Choosing $\eta = 1$ in (28), (29) and (30) respectively, we have the reverse of inequality (3) and the equivalent forms with the best possible constant factor $\pi$ :

where $\tilde \theta (n): = {\tilde \theta _{1, 1}}(\tfrac{1}{2}, n) = \tfrac{1}{\pi }\int_0^{\tfrac{{\ln 2}}{{\ln n}}} {\tfrac{1}{{u + 1}}}~~ {u^{ - \tfrac{1}{2}}}du = O(\tfrac{1}{{{{\ln }^{1/2}}n}}) \in (0, 1).$

5.   Conclusions

In this paper, by the use of the weight coefficients and the idea of introducing parameters, a new reverse Mulholland-type inequality with multi-parameters and the equivalent forms were given in Lemma 2 and Theorem 1. The equivalent statements of the best possible constant factor related to several parameters were obtained in Theorem 2. Some other inequalities associated with reverse Mulholland-type inequality were established in Remarks 1 and 3. The lemmas and theorems presented in this paper provided an extensive account of this type of inequalities.

Acknowledgments

This work was supported by the National Natural Science Foundation (No. 61772140), the Characteristic Innovation Project of Guangdong Provincial Colleges and Universities in 2020 (No. 2020KTSCX088), and the Natural Science Foundation of Fujian Province of China (No. 2020J01365). The authors would like to express sincere appreciation to the anonymous reviewers for their helpful comments and suggestions. All authors contributed equally to the manuscript and read and approved the final manuscript.

Conflict of interest

The authors declare that they have no competing interests.