Open Access

An extension of a multidimensional Hilbert-type inequality

Journal of Inequalities and Applications20172017:78

https://doi.org/10.1186/s13660-017-1355-6

Received: 20 January 2017

Accepted: 23 March 2017

Published: 18 April 2017

Abstract

In this paper, by the use of weight coefficients, the transfer formula and the technique of real analysis, a new multidimensional Hilbert-type inequality with multi-parameters and a best possible constant factor is given, which is an extension of some published results. Moreover, the equivalent forms, the operator expressions and a few particular inequalities are considered.

Keywords

Hilbert-type inequality weight coefficient equivalent form operator norm

MSC

26D15 47A05

1 Introduction

If \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(a_{m},b_{n} \ge0\), \(a = \{ a_{m}\}_{m = 1}^{\infty} \in l^{p}\), \(b = \{ b_{n}\}_{n = 1}^{\infty} \in l^{q}\), \(\Vert a \Vert _{p} = (\sum_{m = 1}^{\infty} a_{m}^{p} )^{\frac{1}{p}} > 0\), \(\Vert b \Vert _{q} > 0\), then we have the following Hardy-Hilbert inequality with the best possible constant \(\frac{\pi}{\sin(\pi/p)}\):
$$ \sum_{m = 1}^{\infty} \sum _{n = 1}^{\infty} \frac{a_{m}b{}_{n}}{m + n} < \frac{\pi}{\sin(\pi/p)} \Vert a \Vert _{p} \Vert b \Vert _{q}, $$
(1)
and the following Hilbert-type inequality:
$$ \sum_{m = 1}^{\infty} \sum _{n = 1}^{\infty} \frac{a_{m}b{}_{n}}{\max\{ m,n\}} < pq \Vert a \Vert _{p} \Vert b \Vert _{q} $$
(2)
with the best possible constant factor pq (cf. [1], Theorem 315, Theorem 341). Inequalities (1) and (2) are important in the analysis and its applications (cf. [13]).
Assuming that \(\{ \mu_{m}\}_{m = 1}^{\infty}\), \(\{ \nu_{n}\}_{n = 1}^{\infty}\) are positive sequences,
$$U_{m} = \sum_{i = 1}^{m} \mu_{i},\quad\quad V_{n} = \sum _{j = 1}^{n} \nu_{j}\quad \bigl(m,n \in \mathbb{N} = \{ 1,2,\ldots\} \bigr), $$
we have the following Hardy-Hilbert-type inequality (cf. [1], Theorem 321):
$$ \sum_{m = 1}^{\infty} \sum _{n = 1}^{\infty} \frac{a_{m}b{}_{n}}{U_{m} + V_{n}} < \frac{\pi}{\sin(\pi/p)} \Biggl( \sum_{m = 1}^{\infty} \frac{a_{m}^{p}}{m^{p - 1}} \Biggr)^{\frac{1}{p}} \Biggl( \sum_{n = 1}^{\infty} \frac{b_{n}^{q}}{n^{q - 1}} \Biggr)^{\frac{1}{q}}. $$
(3)
For \(\mu_{i} = \nu_{j} = 1\) (\(i,j \in\mathbb{N}\)), inequality (3) reduces to (1).
In 2014, Yang and Chen [4] gave the following multidimensional Hilbert-type inequality: For \(i_{0},j_{0} \in\mathbb{N}\), \(\alpha,\beta> 0\),
$$\begin{aligned}& \Vert x \Vert _{\alpha}: = \Biggl( \sum_{k = 1}^{i_{0}} \bigl\vert x^{(k)} \bigr\vert ^{\alpha} \Biggr)^{\frac{1}{\alpha}} \quad \bigl(x = \bigl(x^{(1)},\ldots,x^{(i_{0})} \bigr) \in \mathbb{R}^{i_{0}} \bigr), \\& \Vert y \Vert _{\beta}: = \Biggl( \sum_{k = 1}^{j_{0}} \bigl\vert y^{(k)} \bigr\vert ^{\beta} \Biggr)^{\frac{1}{\beta}} \quad \bigl(y = \bigl(y^{(1)},\ldots,y^{(j_{0})} \bigr) \in \mathbb{R}^{j_{0}} \bigr), \end{aligned}$$
\(0 < \lambda_{1} + \eta\le i_{0}\), \(0 < \lambda_{2} + \eta\le j_{0}\), \(\lambda_{1} + \lambda_{2} = \lambda\), \(a_{m},b_{n} \ge0\), we have
$$ \begin{aligned}[b] &\sum_{n} \sum _{m} \frac{(\min\{ \Vert m \Vert _{\alpha}, \Vert n \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert m \Vert _{\alpha }, \Vert n \Vert _{\beta} \} )^{\lambda+ \eta}} a_{m}b_{n} \\ &\quad < K_{1}^{\frac{1}{p}}K_{2}^{\frac{1}{q}} \biggl[ \sum_{m} \Vert m \Vert _{\alpha}^{p(i_{0} - \lambda_{1}) - i_{0}}a_{m}^{p} \biggr]^{\frac{1}{p}} \biggl[ \sum_{n} \Vert n \Vert _{\beta}^{q(j_{0} - \lambda_{2}) - j_{0}}b_{n}^{q} \biggr]^{\frac{1}{q}}, \end{aligned} $$
(4)
where\(\sum_{m} = \sum_{m_{i_{0}} = 1}^{\infty}\cdots\sum_{m_{1} = 1}^{\infty}\), \(\sum_{n} = \sum_{n_{j_{0}} = 1}^{\infty}\cdots\sum_{n_{1} = 1}^{\infty}\), the series on the right-hand side are positive, and the best possible constant factor \(K_{1}^{\frac{1}{p}}K_{2}^{\frac{1}{q}}\) is indicated by
$$K_{1}^{\frac{1}{p}}K_{2}^{\frac{1}{q}} = \biggl[ \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} \biggr]^{\frac{1}{p}} \biggl[ \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \biggr]^{\frac{1}{q}}\frac{\lambda+ 2\eta}{ (\lambda_{1} + \eta)(\lambda_{2} + \eta)}. $$
For \(i_{0} = j_{0} = \lambda= 1\), \(\eta= 0\), \(\lambda_{1} = \frac{1}{q}\), \(\lambda_{2} = \frac{1}{p}\), inequality (4) reduces to (2). The other results on this type of inequalities were provided by [517].
In 2015, Shi and Yang [18] gave another extension of (2) as follows:
$$ \sum_{m = 1}^{\infty} \sum _{n = 1}^{\infty} \frac{a_{m}b{}_{n}}{\max\{ U_{m},V_{n}\}} < pq \Biggl( \sum _{m = 1}^{\infty} \frac{a_{m}^{p}}{m^{p - 1}} \Biggr)^{\frac{1}{p}} \Biggl( \sum_{n = 1}^{\infty} \frac{b_{n}^{q}}{n^{q - 1}} \Biggr)^{\frac{1}{q}}. $$
(5)
Some other results on Hardy-Hilbert-type inequalities were given by [1925].

In this paper, by the use of weight coefficients, the transfer formula and the technique of real analysis, a new multidimensional Hilbert-type inequality with multi-parameters and a best possible constant factor is given, which is an extension of (4) and (5). Moreover, the equivalent forms, the operator expressions and a few particular inequalities are considered.

2 Some lemmas

If \(\mu_{i}^{(k)} > 0\) (\(k = 1,\ldots,i_{0}\); \(i = 1,\ldots,m\)), \(\nu _{j}^{(l)} > 0\) (\(l = 1,\ldots,j_{0}\); \(j = 1,\ldots,n\)), then we set
$$\begin{aligned}& U_{m}^{(k)}: = \sum_{i = 1}^{m} \mu_{i}^{(k)}\quad (k = 1,\ldots,i_{0}),\quad \quad V_{n}^{(l)}: = \sum_{j = 1}^{n} \nu_{j}^{(l)}\quad (l = 1,\ldots,j_{0}), \\& U_{m} = \bigl(U_{m}^{(1)}, \ldots,U_{m}^{(i_{0})} \bigr),\quad\quad V_{n} = \bigl(V_{n}^{(1)}, \ldots,V_{n}^{(j_{0})} \bigr) \quad (m,n \in\mathbb{N}). \end{aligned}$$
(6)
We also set functions \(\mu_{k}(t): = \mu_{m}^{(k)}\), \(t \in(m - 1,m]\) (\(m \in \mathbb{N}\)); \(\nu_{l}(t): = \nu_{n}^{(l)}\), \(t \in(n - 1,n]\) (\(n \in \mathbb{N}\)), and
$$\begin{aligned}& U_{k}(x): = \int_{0}^{x} \mu_{k}(t)\,dt\quad (k = 1, \ldots,i_{0}), \end{aligned}$$
(7)
$$\begin{aligned}& V_{l}(y): = \int_{0}^{y} \nu_{l}(t)\,dt\quad (l = 1, \ldots,j_{0}), \end{aligned}$$
(8)
$$\begin{aligned}& U(x): = \bigl(U_{1}(x),\ldots,U_{i_{0}}(x) \bigr),\quad\quad V(y): = \bigl(V_{1}(y),\ldots,V_{j_{0}}(y) \bigr) \quad (x,y \ge0). \end{aligned}$$
(9)

It follows that \(U_{k}(m) = U_{m}^{(k)}\) (\(k = 1,\ldots,i_{0}\); \(m \in \mathbb{N}\)), \(V_{l}(n) = V_{n}^{(l)}\) (\(l = 1,\ldots,j_{0}\); \(n \in \mathbb{N}\)), and for \(x \in(m - 1,m)\), \(U_{k}'(x) = \mu_{k}(x) = \mu_{m}^{(k)}\) (\(k = 1,\ldots,i_{0}\); \(m \in\mathbb{N}\)); for \(y \in(n - 1,n)\), \(V_{l}'(y) = \nu_{l}(y) = \nu_{n}^{(l)}\) (\(l = 1, \ldots,j_{0}\); \(n \in\mathbb{N}\)).

Lemma 1

cf. [21]

Suppose that \(g(t)\) (>0) is decreasing in \(\mathbb{R}_{ +}\) and strictly decreasing in \([n_{0},\infty)\) (\(n_{0} \in \mathbb{N}\)), satisfying \(\int_{0}^{\infty} g(t)\,dt \in\mathbb{R}_{ +}\). We have
$$ \int_{1}^{\infty} g(t)\,dt < \sum _{n = 1}^{\infty} g(n) < \int_{0}^{\infty} g(t)\,dt. $$
(10)

Lemma 2

If \(i_{0} \in\mathbb{N}\), \(\alpha,M > 0\), \(\Psi(u)\) is a non-negative measurable function in \((0,1]\), and
$$ D_{M}: = \Biggl\{ x \in\mathbb{R}_{ +}^{i_{0}};u = \sum_{i = 1}^{i_{0}} \biggl( \frac{x_{i}}{M} \biggr)^{\alpha} \le1 \Biggr\} , $$
(11)
then we have the following transfer formula (cf. [26]):
$$ \int\cdots \int_{D_{M}} \Psi \Biggl( \sum_{i = 1}^{i_{0}} \biggl( \frac{x_{i}}{M} \biggr)^{\alpha} \Biggr)\,dx_{1} \cdots dx_{s} = \frac{M^{i_{0}}\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0}}\Gamma ( \frac{i_{0}}{\alpha} )} \int_{0}^{1} \Psi(u)u^{\frac{i_{0}}{\alpha} - 1}\,du. $$
(12)

Lemma 3

For \(i_{0},j_{0} \in\mathbb{N}\), \(\mu_{m}^{(k)} \ge \mu_{m + 1}^{(k)}\) (\(m \in\mathbb{N}\), \(k = 1,\ldots,i_{0}\)), \(\nu_{n}^{(l)} \ge\nu_{n + 1}^{(l)}\) (\(n \in\mathbb{N}\); \(l = 1,\ldots,j_{0}\)), \(\alpha,\beta> 0\), \(\varepsilon > 0\), we have
$$\begin{aligned}& \sum_{m} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} \le\frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\varepsilon i_{0}^{\varepsilon/\alpha} \alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} + O(1), \end{aligned}$$
(13)
$$\begin{aligned}& \sum_{n} \Vert V_{n} \Vert _{\beta}^{ - j_{0} - \varepsilon} \prod_{k = 1}^{j_{0}} \nu_{n}^{(k)} \le\frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\varepsilon j_{0}^{\varepsilon/\beta} \beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} + \tilde{O}(1) \quad \bigl( \varepsilon\to0^{ +} \bigr). \end{aligned}$$
(14)

Proof

For \(M > i_{0}^{1/\alpha}\), we set
$$\Psi(u) = \textstyle\begin{cases} 0,&0 < u < \frac{i_{0}}{M^{\alpha}}, \\ \frac{1}{(Mu^{1/\alpha} )^{i_{0} + \varepsilon}},& \frac{i_{0}}{M^{\alpha}} \le u \le1. \end{cases} $$
By (12), it follows that
$$\begin{aligned} \int_{\{ x \in\mathbb{R}_{ +}^{i_{0}};x_{i} \ge1\}} \frac{dx}{ \Vert x \Vert _{\alpha}^{i_{0} + \varepsilon}} & = \lim_{M \to\infty} \int\cdots \int_{D_{M}} \Psi \Biggl( \sum_{i = 1}^{i_{0}} \biggl( \frac{x_{i}}{M} \biggr)^{\alpha} \Biggr) \,dx_{1} \cdots dx_{i_{0}} \\ &= \lim_{M \to\infty} \frac{M^{i_{0}}\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0}}\Gamma ( \frac{i_{0}}{\alpha} )} \int_{i_{0}/M^{\alpha}}^{1} \frac{u^{\frac{i_{0}}{\alpha} - 1}}{(Mu^{1/\alpha} )^{i_{0} + \varepsilon}} \,du = \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\varepsilon i_{0}^{\varepsilon/\alpha} \alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )}. \end{aligned}$$
Then by (10) and the above result, we find
$$\begin{aligned} 0 &< \sum_{\{ m \in\mathbb{N}^{i_{0}};m_{i} \ge2\}} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} \\ &= \sum_{\{ m \in\mathbb{N}^{i_{0}};m_{i} \ge2\}} \int_{\{ x \in \mathbb{N}^{i_{0}};m - 1 \le x_{i} < m\}} \bigl\Vert U(m) \bigr\Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} \,dx \\ &< \sum_{\{ m \in\mathbb{N}^{i_{0}};m_{i} \ge2\}} \int_{\{ x \in \mathbb{N}^{i_{0}};m - 1 \le x_{i} < m\}} \bigl\Vert U(x) \bigr\Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)}(x) \,dx \\ &= \int_{\{ x \in\mathbb{N}^{i_{0}};x_{i} \ge1\}} \bigl\Vert U(x) \bigr\Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu^{(k)}(x)\,dx \mathop{ =} \limits ^{\nu= U(x)} \int_{\{ \nu\in \mathbb{R}_{ +}^{i_{0}};\nu_{i} \ge\mu_{1}^{(i)}\}} \Vert \nu \Vert _{\alpha}^{ - i_{0} - \varepsilon} \,d \nu \\ &= \int_{\{ \nu\in\mathbb{R}_{ +}^{i_{0}};\nu_{i} \ge1\}} \Vert \nu \Vert _{\alpha}^{ - i_{0} - \varepsilon} \,d \nu+ O_{i_{0}}(1) = \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\varepsilon i_{0}^{\varepsilon/\alpha} \alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} + O_{i_{0}}(1). \end{aligned}$$
For \(i_{0} = 1\), \(0 < \sum_{\{ m \in\mathbb{N}^{i_{0}};m_{i} = 1\}} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} < \infty\); for \(i_{0} \ge2\), \(\mu^{(i)} = \max_{m}\mu_{m}^{(i)}\), \(b = \sum_{i = 1}^{i_{0}} \mu^{(i)}\), in the same way, we find
$$\begin{aligned} 0 &< \sum_{\{ m \in\mathbb{N}^{i_{0}};\exists i,m_{i} = 1\}} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} \\ &\le \Vert U_{1} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{1}^{(k)} + \sum_{i = 1}^{i_{0}} \mu^{(i)} \sum_{\{ m \in\mathbb{N}^{i_{0} - 1};m_{j} \ge2(j \ne i)\}} \Vert U_{m} \Vert _{\alpha}^{ - (i_{0} - 1) - (\varepsilon+ 1)} \prod_{k = 1(k \ne i)}^{i_{0}} \mu_{m}^{(k)} \\ &= O_{1}(1) + \frac{b\Gamma^{i_{0} - 1} ( \frac{1}{\alpha} )}{(1 + \varepsilon)(i_{0} - 1)^{(1 + \varepsilon)/\alpha} \alpha^{i_{0} - 2}\Gamma ( \frac{i_{0} - 1}{\alpha} )} + bO_{i_{0} - 1}(1) < \infty. \end{aligned}$$
Then we have
$$\begin{aligned} \sum_{m} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)}& = \sum_{\{ m \in\mathbb{N}^{i_{0}};\exists i,m_{i} = 1\}} \sum _{m} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} \\ &\quad{} + \sum_{\{ m \in\mathbb{N}^{i_{0}};m_{j} \ge2\}} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} \\ &\le\frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\varepsilon i_{0}^{\varepsilon/\alpha} \alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} + O(1) \quad \bigl(\varepsilon \to0^{ +} \bigr). \end{aligned}$$
Hence, we have (13). In the same way, we have (14). □

Definition 1

For \(\alpha,\beta> 0\), \(0 < \lambda_{1} + \eta \le i_{0}\), \(0 < \lambda_{2} + \eta\le j_{0}\), \(\lambda_{1} + \lambda_{2} = \lambda\), we define two weight coefficients \(w(\lambda_{1},n)\) and \(W(\lambda_{2},m)\) as follows:
$$\begin{aligned}& w(\lambda_{1},n): = \sum_{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert U_{m} \Vert _{\alpha }^{i_{0} - \lambda_{1}}}\prod_{k = 1}^{i_{0}} \mu_{m}^{(k)}, \end{aligned}$$
(15)
$$\begin{aligned}& W(\lambda_{2},m): = \sum_{n} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{ \Vert U_{m} \Vert _{\alpha}^{\lambda_{1}}}{ \Vert V_{n} \Vert _{\beta }^{j_{0} - \lambda_{2}}}\prod_{l = 1}^{j_{0}} \nu_{n}^{(l)}. \end{aligned}$$
(16)

Example 1

With regard to the assumptions of Definition 1, we set
$$k_{\lambda} (x,y) = \frac{(\min\{ x,y\} )^{\eta}}{(\max\{ x,y\} )^{\lambda+ \eta}} \quad (x,y > 0). $$
Then, (i) for fixed \(y > 0\),
$$k_{\lambda} (x,y)\frac{1}{x^{i_{0} - \lambda_{1}}} = \textstyle\begin{cases} \frac{1}{y^{\lambda+ \eta} x^{i_{0} - \lambda_{1} - \eta}},&0 < x < y ,\\ \frac{y^{\eta}}{x^{i_{0} + \lambda_{2} + \eta}},&x \ge y, \end{cases} $$
is decreasing in \(\mathbb{R}_{ +}\) and strictly decreasing in \(([y] + 1,\infty)\). In the same way, for fixed \(x > 0\), \(k_{\lambda} (x,y)\frac{1}{y^{j_{0} - \lambda_{2}}}\) is decreasing in \(\mathbb{R}_{ +}\) and strictly decreasing in \(([x] + 1,\infty)\). We still have
$$ \begin{aligned}[b] k(\lambda_{1})&: = \int_{0}^{\infty} k_{\lambda} (u,1) \frac{du}{u^{1 - \lambda_{1}}} = \int_{0}^{\infty} \frac{(\min\{ u,1\} )^{\eta}}{(\max\{ u,1\} )^{\lambda+ \eta}} \frac{du}{u^{1 - \lambda_{1}}} \\ &= \int_{0}^{1} \frac{u^{\eta}}{u^{1 - \lambda_{1}}} \,du + \int_{1}^{\infty} \frac{1}{u^{\lambda+ \eta}} \frac{du}{u^{1 - \lambda _{1}}} = \frac{\lambda + 2\eta}{(\lambda_{1} + \eta)(\lambda_{2} + \eta)}. \end{aligned} $$
(17)
(ii) For \(b > 0\), we have
$$\frac{d}{dx} \bigl(b + x^{\alpha} \bigr)^{\frac{1}{\alpha}} = \bigl(b + x^{\alpha} \bigr)^{\frac{1}{\alpha} - 1}x^{\alpha- 1} > 0\quad (x > 0). $$
Hence, for \(m - 1 < x_{i} < m\) (\(i = 1,\ldots,i_{0}\); \(m \in\mathbb {N}\)), we have \(\Vert U(m) \Vert _{\alpha} > \Vert U(x) \Vert _{\alpha}\) and
$$\begin{aligned} &\frac{(\min\{ \Vert U(m) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert U(m) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{1}{ \Vert U(m) \Vert _{\alpha}^{i_{0} - \lambda_{1}}} \\ &\quad < \frac{(\min\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{1}{ \Vert U(x) \Vert _{\alpha}^{i_{0} - \lambda_{1}}}; \end{aligned} $$
for \(m < x_{i} < m + 1\) (\(i = 1,\ldots,i_{0}\); \(m \in\mathbb{N}\)), we have \(\Vert U(m) \Vert _{\alpha} < \Vert U(x) \Vert _{\alpha}\) and
$$\begin{aligned} & \frac{(\min\{ \Vert U(m) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert U(m) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{1}{ \Vert U(m) \Vert _{\alpha}^{i_{0} - \lambda_{1}}} \\ &\quad > \frac{(\min\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{1}{ \Vert U(x) \Vert _{\alpha}^{i_{0} - \lambda_{1}}}. \end{aligned} $$

Lemma 4

With regard to the assumptions of Definition  1, (i) we have
$$\begin{aligned}& w(\lambda_{1},n) < K_{2}(\lambda_{1}) \quad \bigl(n \in\mathbb{N}^{j_{0}} \bigr), \end{aligned}$$
(18)
$$\begin{aligned}& W(\lambda_{2},m) < K_{1}(\lambda_{1}) \quad \bigl(m \in\mathbb{N}^{i_{0}} \bigr), \end{aligned}$$
(19)
where
$$ K_{1}(\lambda_{1}) = \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )}k( \lambda_{1}),\qquad K_{2}(\lambda_{1}) = \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )}k(\lambda_{1}); $$
(20)
(ii) for \(\mu_{m}^{(k)} \ge \mu_{m + 1}^{(k)}\) (\(m \in\mathbb{N}\)), \(\nu_{n}^{(l)} \ge\nu_{n + 1}^{(l)}\) (\(n \in\mathbb{N}\)), \(U_{\infty}^{(k)} = V_{\infty}^{(l)}\) (\(k = 1,\ldots,i_{0}\), \(l = 1, \ldots,j_{0}\)), \(0 < \lambda_{1} + \eta\le i_{0}\), \(\lambda_{2} + \eta> 0\), \(0 < \varepsilon< p\lambda_{1}\) (\(p > 1\)), we have
$$ 0 < K_{2}(\lambda_{1}) \bigl(1 - \theta_{\lambda} (n) \bigr) < w(\lambda_{1},n) \quad \bigl(n \in\mathbb{N}^{j_{0}} \bigr), $$
(21)
where, for \(c: = \max_{1 \le k \le i_{0}}\{ \mu_{1}^{(k)}\}\) (>0),
$$ \theta_{\lambda} (n): = \frac{1}{k(\lambda_{1})} \int_{0}^{ci_{0}^{1/\alpha} / \Vert V_{n} \Vert _{\beta}} \frac{(\min\{ v,1\} )^{\eta} v^{\lambda_{1} - 1}}{(\max\{ v,1\} )^{\lambda+ \eta}} \,dv = O \biggl( \frac{1}{ \Vert V_{n} \Vert _{\beta}^{\lambda_{1} + \eta}} \biggr). $$
(22)

Proof

(i) By (10), (12) and Example 1(ii), for \(0 < \lambda_{1} + \eta\le i_{0}\), \(\lambda> 0\), it follows that
$$\begin{aligned} w(\lambda_{1},n) &= \sum_{m} \int_{\{ x \in\mathbb{N}^{i_{0}};m_{i} - 1 \le x_{i} \le m_{i}\}} \frac{(\min\{ \Vert U(m) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert U(m) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \\ &\quad{} \times\frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert U(m) \Vert _{\alpha}^{i_{0} - \lambda{}_{1}}}\prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} \,dx \\ &< \sum_{m} \int_{\{ x \in\mathbb{N}^{i_{0}};m_{i} - 1 \le x_{i} \le m_{i}\}} \frac{(\min\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \\ &\quad{} \times\frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert U(x) \Vert _{\alpha}^{i_{0} - \lambda{}_{1}}}\prod_{k = 1}^{i_{0}} \mu_{m}^{(k)}(x) \,dx \\ & = \int_{\mathbb{R}_{ +}^{i_{0}}} \frac{(\min\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U(x) \Vert _{\alpha}, \Vert V{}_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert U(x) \Vert _{\alpha }^{i_{0} - \lambda{}_{1}}}\prod _{k = 1}^{i_{0}} \mu^{(k)}(x) \,dx \\ & \mathop{\le}^{u = U(x)} \int_{\mathbb{R}_{ +}^{i_{0}}} \frac{(\min\{ \Vert u \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert u \Vert _{\alpha}, \Vert V{}_{n} \Vert _{\beta} \} )^{\lambda + \eta}} \frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert u \Vert _{\alpha}^{i_{0} - \lambda{}_{1}}}\,du \\ & = \lim_{M \to\infty} \int_{D_{M}} \frac{(\min\{ M [ \sum_{i = 1}^{i_{0}} ( \frac{u_{i}}{M} )^{\alpha} ]^{1/\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ M [ \sum_{i = 1}^{i_{0}} ( \frac{u_{i}}{M} )^{\alpha} ]^{1/\alpha}, \Vert V{}_{n} \Vert _{\beta} \} )^{\lambda+ \eta }} \frac{M^{\lambda_{1} - i_{0}} \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}\,du}{ [ \sum_{i = 1}^{i_{0}} ( \frac{u_{i}}{M} )^{\alpha} ]^{(i_{0} - \lambda_{1})/\alpha}} \\ & = \lim_{M \to\infty} \frac{M^{i_{0}}\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0}}\Gamma ( \frac{i_{0}}{\alpha} )} \int_{0}^{1} \frac{(\min\{ Mu^{1/\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta} \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{(\max\{ Mu^{1/\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{u^{\frac{i_{0}}{\alpha} - 1}\,du}{M^{i_{0} - \lambda _{1}}u^{(i_{0} - \lambda_{1})/\alpha}} \\ & = \lim_{M \to\infty} \frac{M^{i_{0}}\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0}}\Gamma ( \frac{i_{0}}{\alpha} )} \int_{0}^{1} \frac{(\min\{ Mu^{1/\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta} \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{(\max\{ Mu^{1/\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} u^{\frac{\lambda_{1}}{\alpha} - 1} \,du \\ & \mathop{=}^{v = \frac{Mu^{1/\alpha}}{ \Vert V_{n} \Vert _{\beta}}} \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \int_{0}^{\infty} \frac{(\min\{ v,1\} )^{\eta} v^{\lambda_{1} - 1}}{(\max \{ v,1\} )^{\lambda+ \eta}} \,dv \\ & = \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \frac{\lambda+ 2\eta}{ (\lambda_{1} + \eta)(\lambda_{2} + \eta)} = K_{2}( \lambda_{1}). \end{aligned}$$
Hence, we have (18). In the same way, we have (19).
(ii) By (10) and in the same way, for \(c = \max_{1 \le k \le i_{0}}\{ \mu_{1}^{(k)}\}\) (>0), we have
$$\begin{aligned} w(\lambda_{1},n) &\ge\sum_{m} \int_{\{ x \in\mathbb{N}^{i_{0}};m_{i} \le x_{i} \le m_{i} + 1\}} \frac{(\min\{ \Vert U(m) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U(m) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda + \eta}} \\ &\quad{} \times\frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert U(m) \Vert _{\alpha}^{i_{0} - \lambda{}_{1}}}\prod_{k = 1}^{i_{0}} \mu_{m + 1}^{(k)} \,dx \\ & > \sum_{m} \int_{\{ x \in\mathbb{N}^{i_{0}};m_{i} \le x_{i} \le m_{i} + 1\}} \frac{(\min\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \\ &\quad{} \times\frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert U(x) \Vert _{\alpha}^{i_{0} - \lambda{}_{1}}}\prod_{k = 1}^{i_{0}} \mu^{(k)}(x) \,dx \\ & = \int_{[1,\infty)^{i_{0}}} \frac{(\min\{ \Vert U(x) \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U(x) \Vert _{\alpha}, \Vert V{}_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert U(x) \Vert _{\alpha }^{i_{0} - \lambda{}_{1}}}\prod _{k = 1}^{i_{0}} \mu^{(k)}(x) \,dx \\ & \mathop{\ge}^{v = U(x)} \int_{[c,\infty)^{i_{0}}} \frac{(\min\{ \Vert v \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert v \Vert _{\alpha}, \Vert V{}_{n} \Vert _{\beta} \} )^{\lambda + \eta}} \frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert v \Vert _{\alpha}^{i_{0} - \lambda{}_{1}}}\,dv. \end{aligned}$$
For \(M > ci_{0}^{1/\alpha}\), we set
$$\Psi(u) = \textstyle\begin{cases} 0,& 0 < u < \frac{c^{\alpha} i_{0}}{M^{\alpha}}, \\ \frac{(\min\{ Mu^{1/\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ Mu^{1/\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{ \Vert V_{n} \Vert _{\beta}^{\lambda _{2}}}{(Mu^{1/\alpha} )^{i_{0} - \lambda_{1}}}, &\frac{c^{\alpha} i_{0}}{M^{\alpha}} \le u \le1. \end{cases} $$
By (12), it follows that
$$\begin{aligned}& \int_{\{ x \in\mathbb{R}_{ +}^{i_{0}};x_{i} \ge c\}} \frac{(\min\{ \Vert x \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert x \Vert _{\alpha}, \Vert V{}_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}}{ \Vert x \Vert _{\alpha}^{i_{0} - \lambda{}_{1}}}\,dx \\& \quad = \lim_{M \to\infty} \int\cdots \int_{D_{M}} \Psi \Biggl( \sum_{i = 1}^{i_{0}} \biggl( \frac{x_{i}}{M} \biggr)^{\alpha} \Biggr) \,dx_{1} \cdots dx_{i_{0}} \\& \quad = \lim_{M \to\infty} \frac{M^{i_{0}}\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0}}\Gamma ( \frac{i_{0}}{\alpha} )} \int_{c^{\alpha} i_{0}/M^{\alpha}}^{1} \frac{(\min\{ Mu^{\frac{1}{\alpha}}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ Mu^{\frac{1}{\alpha}}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \frac{ \Vert V_{n} \Vert _{\beta}^{\lambda_{2}}u^{\frac {i_{0}}{\alpha} - 1}\,du}{(Mu^{\frac{1}{\alpha}} )^{i_{0} - \lambda_{1}}} \\& \quad \mathop{ =} \limits ^{v = \frac{Mu^{1/\alpha}}{ \Vert V_{n} \Vert _{\beta}}} \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \int_{ci_{0}^{1/\alpha} / \Vert V_{n} \Vert _{\beta}}^{1} \frac {(\min\{ v,1\} )^{\eta} v^{\lambda_{1} - 1}}{(\max\{ v,1\} )^{\lambda+ \eta}} \,dv. \end{aligned}$$
Hence, we have
$$\begin{aligned} w(\lambda_{1},n) &> \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \int_{ci_{0}^{1/\alpha} / \Vert V_{n} \Vert _{\beta}}^{1} \frac {(\min\{ v,1\} )^{\eta} v^{\lambda_{1} - 1}}{(\max\{ v,1\} )^{\lambda+ \eta}} \,dv \\ & = K_{2}(\lambda_{1}) \bigl(1 - \theta_{\lambda} (n) \bigr) > 0. \end{aligned}$$
For \(\Vert V_{n} \Vert _{\beta} \ge ci_{0}^{1/\alpha}\), we obtain
$$\begin{aligned} 0 &< \theta_{\lambda} (n) = \frac{1}{k(\lambda_{1})} \int_{0}^{ci_{0}^{1/\alpha} / \Vert V_{n} \Vert _{\beta}} \frac{(\min\{ v,1\} )^{\eta} v^{\lambda_{1} - 1}}{(\max \{ v,1\} )^{\lambda+ \eta}} \,dv \\ & = \frac{1}{k(\lambda_{1})} \int_{0}^{ci_{0}^{1/\alpha} / \Vert V_{n} \Vert _{\beta}} v^{\lambda_{1} + \eta- 1} \,dv = \frac{1}{(\lambda_{1} + \eta)k(\lambda_{1})} \biggl( \frac{ci_{0}^{1/\alpha}}{ \Vert V{}_{n} \Vert _{\beta}} \biggr)^{\lambda_{1} + \eta}, \end{aligned}$$
and then (21) and (22) follow. □

3 Main results

Setting functions
$$\begin{aligned}& \Phi(m): = \frac{ \Vert U_{m} \Vert _{\alpha}^{p(i_{0} - \lambda _{1}) - i_{0}}}{(\prod_{k = 1}^{i_{0}} \mu_{m}^{(k)})^{p - 1}} \quad \bigl(m \in \mathbb{N}^{i_{0}} \bigr), \\& \Psi(n): = \frac{ \Vert V_{n} \Vert _{\beta}^{q(j_{0} - \lambda _{2}) - j_{0}}}{(\prod_{l = 1}^{j_{0}} \nu_{n}^{(l)})^{q - 1}} \quad \bigl(n \in \mathbb{N}^{j_{0}} \bigr), \end{aligned}$$
and the following normed spaces:
$$\begin{aligned}& l_{p,\Phi}: = \biggl\{ a = \{ a_{m}\}; \Vert a \Vert _{p,\Phi}: = \biggl\{ \sum_{m} \Phi(m) \vert a_{m} \vert ^{p} \biggr\} ^{\frac{1}{p}} < \infty \biggr\} , \\& l_{q,\Psi}: = \biggl\{ b = \{ b_{n}\}; \Vert b \Vert _{q,\Psi}: = \biggl\{ \sum_{n} \Psi(n) \vert b_{n} \vert ^{q} \biggr\} ^{\frac{1}{q}} < \infty \biggr\} , \\& l_{p,\Psi^{1 - p}}: = \biggl\{ c = \{ c_{n}\}; \Vert c \Vert _{p,\Psi^{1 - p}}: = \biggl\{ \sum_{n} \Psi^{1 - p}(n) \vert c_{n} \vert ^{p} \biggr\} ^{\frac{1}{p}} < \infty \biggr\} , \end{aligned}$$
we have the following.

Theorem 1

If \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(\alpha ,\beta> 0\), \(\lambda> 0\), \(0 < \lambda_{1} + \eta\le i_{0}\), \(0 < \lambda_{2} + \eta\le j_{0}\), \(\lambda_{1} + \lambda_{2} = \lambda\), then for \(a_{m},b_{n} \ge0\), \(a = \{ a_{m}\} \in l_{p, \Phi}\), \(b = \{ b_{n}\} \in l_{q, \Psi}\), \(\Vert a \Vert _{p, \Phi}, \Vert b \Vert _{q,\Psi} > 0\), we have the following equivalent inequalities:
$$\begin{aligned}& I: = \sum_{n} \sum_{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} a_{m}b_{n} < K_{1}^{\frac{1}{p}}( \lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1}) \Vert a \Vert _{p,\Phi} \Vert b \Vert _{q,\Psi}, \end{aligned}$$
(23)
$$\begin{aligned}& \begin{aligned}[b] J&: = \biggl\{ \sum_{n} \frac{\prod_{k = 1}^{j_{0}} v_{n}^{(k)}}{ \Vert V_{n} \Vert _{\beta}^{j_{0} - p\lambda_{2}}} \biggl[ \sum_{m} \frac{(\min \{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta} a_{m}}{(\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \biggr]^{p} \biggr\} ^{\frac{1}{p}} \\ & < K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1}) \Vert a \Vert _{p,\Phi}, \end{aligned} \end{aligned}$$
(24)
where
$$ K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1}) = \biggl[ \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta ^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} \biggr]^{\frac{1}{p}} \biggl[ \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \biggr]^{\frac{1}{q}}k(\lambda _{1}). $$
(25)

Proof

By Hölder’s inequality with weight (cf. [27]), we have
$$\begin{aligned} I &= \sum_{n} \sum_{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \\ &\quad{} \times \biggl[ \frac{ \Vert U_{m} \Vert _{\alpha}^{\frac{i_{0} - \lambda_{1}}{q}}}{ \Vert V_{n} \Vert _{\beta}^{\frac{j_{0} - \lambda_{2}}{p}}}\frac{(\prod_{l = 1}^{j_{0}} \nu_{n}^{(l)} )^{\frac{1}{p}}a_{m}}{(\prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} )^{\frac {1}{q}}} \biggr] \biggl[ \frac{ \Vert V_{n} \Vert _{\beta}^{\frac{j_{0} - \lambda_{2}}{p}}}{ \Vert U_{m} \Vert _{\alpha}^{\frac{i_{0} - \lambda_{1}}{q}}}\frac{(\prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} )^{\frac{1}{q}}b_{n}}{(\prod_{l = 1}^{j_{0}} \nu_{n}^{(l)} )^{\frac {1}{p}}} \biggr] \\ & \le \biggl[ \sum_{m} W(\lambda_{2},m) \frac{ \Vert U_{m} \Vert _{\alpha}^{p(i_{0} - \lambda_{1}) - i_{0}}a_{m}^{p}}{(\prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} )^{p - 1}} \biggr]^{\frac{1}{p}} \biggl[ \sum _{n} w(\lambda_{1},n) \frac{ \Vert V_{n} \Vert _{\beta}^{q(j_{0} - \lambda_{2}) - j_{0}}b_{n}^{q}}{(\prod_{l = 1}^{j_{0}} \nu_{n}^{(l)} )^{q - 1}} \biggr]^{\frac{1}{q}}. \end{aligned}$$
Then by (18) and (19), we have (23). We set
$$b_{n}: = \frac{\prod_{l = 1}^{j_{0}} \nu_{n}^{(l)}}{ \Vert V_{n} \Vert _{\beta}^{j_{0} - p\lambda_{2}}} \biggl[ \sum_{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta} a_{m}}{(\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \biggr]^{p - 1},\quad n \in\mathbb {N}^{j_{0}}. $$
Then we have \(J = \Vert b \Vert _{q,\Psi}^{q - 1}\). Since the right-hand side of (24) is finite, it follows \(J < \infty\). If \(J = 0\), then (24) is trivially valid; if \(J > 0\), then by (23), we have
$$\begin{aligned}& \Vert b \Vert _{q,\Psi}^{q} = J^{p} = I < K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1}) \Vert a \Vert _{p,\Phi} \Vert b \Vert _{q,\Psi}, \\& \Vert b \Vert _{q,\Psi}^{q - 1} = J < K_{1}^{\frac{1}{p}}( \lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1}) \Vert a \Vert _{p,\Phi}, \end{aligned}$$
namely (24) follows.
On the other hand, assuming that (24) is valid, by Hölder’s inequality (cf. [27]), we have
$$ \begin{aligned}[b] I &= \sum_{n} \frac{(\prod_{l = 1}^{j_{0}} v_{n}^{(l)} )^{1/p}}{ \Vert V_{n} \Vert _{\beta}^{(j_{0}/p) - \lambda_{2}}} \sum_{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} a_{m} \\ &\quad{} \times\frac{ \Vert V_{n} \Vert _{\beta}^{(j_{0}/p) - \lambda_{2}}}{(\prod_{l = 1}^{j_{0}} \nu_{n}^{(l)})^{1/p}} b_{n} \le J \Vert b \Vert _{q,\Psi}. \end{aligned} $$
(26)
Then by (24) we have (23), which is equivalent to (24). □

Theorem 2

With regard to the assumptions of Theorem  1, if \(\mu_{m}^{(k)} \ge\mu_{m + 1}^{(k)}\) (\(m \in\mathbb{N}\)), \(\nu _{n}^{(l)} \ge \nu_{n + 1}^{(l)}\) (\(n \in\mathbb{N}\)), \(U_{\infty}^{(k)} = V_{\infty}^{(l)} = \infty\) (\(k = 1,\ldots,i_{0}\), \(l = 1,\ldots,j_{0}\)), then the constant factor \(K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1})\) in (23) and (24) is the best possible.

Proof

For \(0 < \varepsilon< p(\lambda_{1} + \eta )\), \(\tilde{\lambda}_{1} = \lambda_{1} - \frac{\varepsilon}{p}\) (\(\in ( - \eta, - \eta+ i_{0})\)), \(\tilde{\lambda}_{2} = \lambda_{2} + \frac {\varepsilon}{ p}\) (\(> - \eta\)), we set
$$\begin{aligned}& \tilde{a} = \{ \tilde{a}_{m}\},\quad\quad \tilde{a}_{m}: = \Vert U_{m} \Vert _{\alpha}^{ - i_{0} + \tilde{\lambda}_{1}}\prod _{k = 1}^{i_{0}} \mu_{m}^{(k)}\quad \bigl(m \in\mathbb{N}^{i_{0}} \bigr), \\& \tilde{b} = \{ \tilde{b}_{n}\},\quad\quad \tilde{b}_{n}: = \Vert V_{n} \Vert _{\beta}^{ - j_{0} + \tilde{\lambda}_{2} - \varepsilon} \prod _{l = 1}^{j_{0}} \nu_{n}^{(l)}\quad \bigl(n \in\mathbb{N}^{j_{0}} \bigr). \end{aligned}$$
Then by (13) and (14), we obtain
$$\begin{aligned} \Vert \tilde{a} \Vert _{p,\Phi} \Vert \tilde{b} \Vert _{q,\Psi} &= \biggl[ \sum_{m} \frac{ \Vert U_{m} \Vert _{\alpha}^{p(i_{0} - \lambda_{1}) - i_{0}}\tilde{a}_{m}^{p}}{(\prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} )^{p - 1}} \biggr]^{\frac{1}{p}} \biggl[ \sum _{n} \frac{ \Vert V_{n} \Vert _{\beta}^{q(j_{0} - \lambda_{2}) - j_{0}}\tilde{b}_{n}^{q}}{(\prod_{l = 1}^{j_{0}} \nu_{n}^{(l)} )^{q - 1}} \biggr]^{\frac{1}{q}} \\ & = \Biggl( \sum_{m} \Vert U_{m} \Vert _{\alpha}^{ - i_{0} - \varepsilon} \prod_{k = 1}^{i_{0}} \mu_{m}^{(k)} \Biggr)^{\frac{1}{p}} \Biggl( \sum _{n} \Vert V_{n} \Vert _{\beta}^{ - j_{0} - \varepsilon} \prod_{l = 1}^{j_{0}} \nu_{n}^{(l)} \Biggr)^{\frac{1}{q}} \\ & \le\frac{1}{\varepsilon} \biggl( \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{i_{0}^{\varepsilon/\alpha} \alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} + \varepsilon O(1) \biggr)^{\frac{1}{p}} \biggl( \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{j_{0}^{\varepsilon/\beta} \beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} + \varepsilon\tilde{O}(1) \biggr)^{\frac{1}{q}}. \end{aligned}$$
By (21) and (22), we find
$$\begin{aligned} \tilde{I}&: = \sum_{n} \biggl[ \sum _{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} \tilde{a}_{m} \biggr] \tilde{b}_{n} \\ & = \sum_{n} w(\tilde{\lambda}_{1},n) \Vert V_{n} \Vert _{\beta}^{ - j_{0} - \varepsilon} \prod _{l = 1}^{j_{0}} \nu_{n}^{(l)} \\ & > K_{2}(\tilde{\lambda}_{1})\sum _{n} \biggl( 1 - O \biggl( \frac{1}{ \Vert V_{n} \Vert _{\beta}^{\lambda_{1} + \eta}} \biggr) \biggr) \Vert V_{n} \Vert _{\beta}^{ - j_{0} - \varepsilon} \prod _{l = 1}^{j_{0}} \nu_{n}^{(l)} \\ & = K_{2}(\tilde{\lambda}_{1}) \biggl( \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\varepsilon j_{0}^{\varepsilon/\beta} \beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} + \tilde{O}(1) - O_{1}(1) \biggr). \end{aligned}$$
If there exists a constant \(K \le K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1})\) such that (23) is valid when replacing \(K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1})\) by K, then we have \(\varepsilon\tilde{I} < \varepsilon K \Vert \tilde{a} \Vert _{p,\Phi} \Vert \tilde{b} \Vert _{q,\Psi}\), namely
$$\begin{aligned}& K_{2} \biggl(\lambda_{1} - \frac{\varepsilon}{p} \biggr) \biggl( \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{j_{0}^{\varepsilon /\beta} \beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} + \varepsilon \tilde{O}(1) - \varepsilon O_{1}(1) \biggr) \\& \quad < K \biggl( \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{i_{0}^{\varepsilon/\alpha} \alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} + \varepsilon O(1) \biggr)^{\frac{1}{p}} \biggl( \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{j_{0}^{\varepsilon/\beta} \beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} + \varepsilon\tilde{O}(1) \biggr)^{\frac{1}{q}}. \end{aligned}$$
For \(\varepsilon\to0^{ +}\), we find
$$\frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )k(\lambda_{1})}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \le K \biggl[ \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{\alpha^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \biggr]^{\frac{1}{p}} \biggl[ \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} \biggr]^{\frac{1}{q}}, $$
and then \(K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1}) \le K\). Hence, \(K = K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1})\) is the best possible constant factor of (23). The constant factor in (24) is still the best possible. Otherwise, we would reach a contradiction by (26) that the constant factor in (23) is not the best possible. □

4 Operator expressions

With regard to the assumptions of Theorem 2, in view of
$$\begin{aligned}& c_{n}: = \frac{\prod_{k = 1}^{j_{0}} \nu_{n}^{(k)}}{ \Vert V_{n} \Vert _{\beta}^{j_{0} - p\lambda_{2}}} \biggl[ \sum _{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} a_{m} \biggr]^{p - 1}, \quad n \in\mathbb{N}^{j_{0}}, \\& c = \{ c_{n}\}, \quad\quad \Vert c \Vert _{p,\Psi^{1 - p}} = J < K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1}) \Vert a \Vert _{p,\Phi} < \infty, \end{aligned}$$
we can set the following definition.

Definition 2

Define a multidimensional Hilbert’s operator \(T:l_{p,\Phi} \to l_{p,\Psi^{1 - p}}\) as follows: For any \(a \in l_{p,\Phi}\), there exists a unique representation \(Ta = c \in l_{p,\Psi^{1 - p}}\), satisfying
$$ Ta(n): = \sum_{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{(\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} a_{m}\quad \bigl(n \in\mathbb{N}^{j_{0}} \bigr). $$
(27)
For \(b \in l_{q,\Psi}\), we define the following formal inner product of Ta and b as follows:
$$ (Ta,b): = \sum_{n} \biggl[ \sum _{m} \frac{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\eta}}{ (\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda+ \eta}} a_{m} \biggr]b_{n}. $$
(28)
Then by Theorems 1 and 2, we have the following equivalent inequalities:
$$\begin{aligned}& (Ta,b) < K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1}) \Vert a \Vert _{p,\Phi} \Vert b \Vert _{q,\Psi}, \end{aligned}$$
(29)
$$\begin{aligned}& \Vert Ta \Vert _{p,\Psi^{1 - p}} < K_{1}^{\frac{1}{p}}( \lambda_{1})K_{2}^{\frac{1}{q}}(\lambda_{1}) \Vert a \Vert _{p,\Phi}. \end{aligned}$$
(30)
It follows that T is bounded with
$$ \Vert T \Vert : = \sup_{a( \ne\theta) \in l_{p,\Phi}} \frac{ \Vert Ta \Vert _{p,\Psi^{1 - p}}}{ \Vert a \Vert _{p,\Phi}} \le K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1}). $$
(31)
Since the constant factor \(K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}(\lambda _{1})\) in (30) is the best possible, we have
$$ \Vert T \Vert = K_{1}^{\frac{1}{p}}(\lambda_{1})K_{2}^{\frac{1}{q}}( \lambda_{1}) = \biggl[ \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta ^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} \biggr]^{\frac{1}{p}} \biggl[ \frac{\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{a^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \biggr]^{\frac{1}{q}}k(\lambda_{1}). $$
(32)

Remark 1

(i) For \(\mu_{i} = \nu_{j} = 1\) (\(i,j \in\mathbb {N}\)), (23) reduces to (4). Hence, (23) is an extension of (4).

(ii) For \(\eta= 0\), \(0 < \lambda_{1} \le i_{0}\), \(0 < \lambda_{2} \le j_{0}\), (23) reduces to the following inequality:
$$ \begin{aligned}[b] &\sum_{n} \sum _{m} \frac{1}{(\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda}} a_{m}b_{n} \\ &\quad < \biggl[ \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} \biggr]^{\frac{1}{p}} \biggl[ \frac {\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{a^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \biggr]^{\frac{1}{q}} \frac{\lambda}{ \lambda_{1}\lambda_{2}} \Vert a \Vert _{p,\Phi} \Vert b \Vert _{q,\Psi}. \end{aligned} $$
(33)

In particular, for \(i_{0} = j_{0} = \lambda= 1\), \(\lambda_{1} = \frac{1}{q}\), \(\lambda_{2} = \frac{1}{p}\), (33) reduces to (5). Hence, (33) is also an extension of (5); so is (23).

(iii) For \(\eta= - \lambda\), \(\lambda_{1},\lambda_{2} < 0\), (23) reduces to the following inequality:
$$ \begin{aligned}[b] &\sum_{n} \sum _{m} \frac{1}{(\min\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \} )^{\lambda}} a_{m}b_{n} \\ &\quad < \biggl[ \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} \biggr]^{\frac{1}{p}} \biggl[ \frac {\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{a^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \biggr]^{\frac{1}{q}} \frac{( - \lambda)}{\lambda_{1}\lambda_{2}} \Vert a \Vert _{p,\Phi} \Vert b \Vert _{q,\Psi}. \end{aligned} $$
(34)
(iv) For \(\lambda= 0\), \(\lambda_{2} = - \lambda_{1}\) (\(- \eta< \lambda_{1} < \eta\)), (23) reduces to the following inequality:
$$ \begin{aligned}[b] &\sum_{n} \sum _{m} \biggl( \frac{\min\{ \Vert U{}_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \}}{\max\{ \Vert U_{m} \Vert _{\alpha}, \Vert V_{n} \Vert _{\beta} \}} \biggr)^{\eta } a_{m}b_{n} \\ &\quad < \biggl[ \frac{\Gamma^{j_{0}} ( \frac{1}{\beta} )}{\beta^{j_{0} - 1}\Gamma ( \frac{j_{0}}{\beta} )} \biggr]^{\frac{1}{p}} \biggl[ \frac {\Gamma^{i_{0}} ( \frac{1}{\alpha} )}{a^{i_{0} - 1}\Gamma ( \frac{i_{0}}{\alpha} )} \biggr]^{\frac{1}{q}} \frac{2\eta}{ \eta^{2} - \lambda_{1}^{2}} \Vert a \Vert _{p,\Phi} \Vert b \Vert _{q,\Psi}. \end{aligned} $$
(35)

The above particular inequalities are also with the best possible constant factors.

Declarations

Acknowledgements

This work is supported by the National Natural Science Foundation (No. 61370186, No. 61640222), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). We are grateful for their help.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Guangdong University of Education

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