1 Introduction

If \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(f(x),g(y)\geq0\), \(f\in L^{p}(\mathbf{R}_{+})\), \(g\in L^{q}(\mathbf{R}_{+})\), \(\|f\|_{p}=(\int_{0}^{\infty }f^{p}(x)\,dx)^{\frac{1}{p}}>0\), \(\|g\|_{q}>0\), then we have the following Hardy-Hilbert’s integral inequality (cf. [1]):

$$ \int_{0}^{\infty} \int_{0}^{\infty}\frac{f(x)g(y)}{x+y}\,dx\,dy< \frac{\pi }{\sin(\pi/p)} \|f\|_{p}\|g\|_{q}, $$
(1)

where the constant factor \(\frac{\pi}{\sin(\pi/p)}\) is the best possible. Assuming that \(a_{m},b_{n}\geq0\), \(a=\{a_{m}\}_{m=1}^{\infty }\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\|a\|_{p}=(\sum_{m=1}^{ \infty}a_{m}^{p})^{\frac{1}{p}}>0\), \(\|b\|_{q}>0\), we have the following discrete Hardy-Hilbert’s inequality with the same best constant \(\frac {\pi}{\sin(\pi/p)}\):

$$ \sum_{m=1}^{\infty}\sum _{n=1}^{\infty}\frac{a_{m}b_{n}}{m+n}< \frac {\pi}{\sin(\pi/p)}\|a \|_{p}\|b\|_{q}. $$
(2)

Inequalities (1) and (2) are important in analysis and its applications (cf. [1–6]).

In 1998, by introducing an independent parameter \(\lambda\in(0,1]\), Yang [7] gave an extension of (1) at \(p=q=2\) with the kernel \(\frac{1}{(x+y)^{\lambda}}\). In recent years, Yang [3] and [4] gave some extensions of (1) and (2) as follows:

If \(\lambda_{1},\lambda_{2}\in\mathbf{R}\), \(\lambda_{1}+\lambda _{2}=\lambda\), \(k_{\lambda}(x,y)\) is a non-negative homogeneous function of degree −λ, with

$$ k(\lambda_{1})= \int_{0}^{\infty}k_{\lambda}(t,1)t^{\lambda _{1}-1}\,dt \in \mathbf{R}_{+}, $$

\(\phi(x)=x^{p(1-\lambda_{1})-1}\), \(\psi(x)=x^{q(1-\lambda _{2})-1}\), \(f(x),g(y)\geq0\),

$$ f\in L_{p,\phi}(\mathbf{R}_{+})= \biggl\{ f;\|f \|_{p,\phi }:=\biggl( \int_{0}^{\infty}\phi(x)\bigl|f(x)\bigr|^{p}\,dx \biggr)^{\frac{1}{p}}< \infty \biggr\} , $$

\(g\in L_{q,\psi}(\mathbf{R}_{+})\), \(\|f\|_{p,\phi},\|g\|_{q,\psi}>0\), then we have

$$ \int_{0}^{\infty} \int_{0}^{\infty}k_{\lambda }(x,y)f(x)g(y)\,dx\,dy< k( \lambda _{1})\|f\|_{p,\phi}\|g\|_{q,\psi}, $$
(3)

where the constant factor \(k(\lambda_{1})\) is the best possible. Moreover, if \(k_{\lambda}(x,y)\) is finite and \(k_{\lambda}(x,y)x^{\lambda _{1}-1}(k_{\lambda}(x,y)y^{\lambda_{2}-1})\) is decreasing with respect to \(x>0\) (\(y>0\)), then for \(a_{m},b_{n}\geq0\),

$$ a\in l_{p,\phi}= \Biggl\{ a;\|a\|_{p,\phi}:=\Biggl(\sum _{n=1}^{\infty}\phi (n)|a_{n}|^{p} \Biggr)^{\frac{1}{p}}< \infty \Biggr\} , $$

\(b=\{b_{n}\}_{n=1}^{\infty}\in l_{q,\psi}\), \(\|a\|_{p,\phi },\|b\|_{q,\psi }>0\), we have the following inequality:

$$ \sum_{m=1}^{\infty}\sum _{n=1}^{\infty}k_{\lambda }(m,n)a_{m}b_{n}< k( \lambda_{1})\|a\|_{p,\phi}\|b\|_{q,\psi}, $$
(4)

where the constant factor \(k(\lambda_{1})\) is still the best possible.

Clearly, for \(\lambda=1\), \(k_{1}(x,y)=\frac{1}{x+y}\), \(\lambda_{1}=\frac {1}{q}\), \(\lambda_{2}=\frac{1}{p}\), (3) reduces to (1), while (4) reduces to (2). Some other results including the multidimensional Hilbert-type integral, discrete, and half-discrete inequalities are provided by [8–26].

In this paper, by the use of the way of weight coefficients, the transfer formula and technique of real analysis, a multidimensional discrete Hilbert’s inequality with parameters and a best possible constant factor is given, which is an extension of (4) for

$$ k_{\lambda}(m,n)=\prod_{k=1}^{s} \frac{(\min\{m,c_{k}n\})^{\frac {\gamma}{s}}}{(\max\{m,c_{k}n\})^{\frac{\lambda+\gamma}{s}}}. $$

The equivalent form, the operator expressions with the norm, and some particular cases are also considered.

2 Some lemmas

If \(i_{0},j_{0}\in\mathbf{N}\) (N is the set of positive integers), \(\alpha ,\beta>0\), we put

$$ \begin{aligned} &\|x\|_{\alpha} := \Biggl( \sum_{k=1}^{i_{0}}|x_{k}|^{\alpha} \Biggr) ^{ \frac{1}{\alpha}}\quad\bigl(x=(x_{1},\ldots,x_{i_{0}})\in \mathbf{R}^{i_{0}}\bigr), \\ &\|y\|_{\beta} := \Biggl( \sum_{k=1}^{j_{0}}|y_{k}|^{\beta} \Biggr) ^{\frac{1}{\beta}}\quad\bigl(y=(y_{1},\ldots,y_{j_{0}})\in \mathbf{R}^{j_{0}}\bigr). \end{aligned} $$
(5)

Lemma 1

If \(g(t)\) (>0) is decreasing in \(\mathbf{R}_{+}\) and strictly decreasing in \([n_{0},\infty)\subset\mathbf{R}_{+}\) (\(n_{0}\in \mathbf{N}\)), satisfying \(\int_{0}^{\infty}g(t)\,dt\in\mathbf{R}_{+}\), then we have

$$ \int_{1}^{\infty}g(t)\,dt< \sum _{n=1}^{\infty}g(n)< \int_{0}^{\infty}g(t)\,dt. $$
(6)

Proof

Since by the assumption, we have

$$\begin{aligned}& \int_{n}^{n+1}g(t)\,dt \leq g(n)\leq \int_{n-1}^{n}g(t)\,dt\quad(n=1,\ldots,n_{0}),\\& \int_{n_{0}+1}^{n_{0}+2}g(t)\,dt < g(n_{0}+1)< \int_{n_{0}}^{n_{0}+1}g(t)\,dt, \end{aligned}$$

it follows that

$$ 0< \int_{1}^{n_{0}+2}g(t)\,dt< \sum _{n=1}^{n_{0}+1}g(n)< \sum_{n=1}^{n_{0}+1} \int_{n-1}^{n}g(t)\,dt= \int_{0}^{n_{0}+1}g(t)\,dt< \infty. $$

In the same way, we still have

$$ 0< \int_{n_{0}+2}^{\infty}g(t)\,dt\leq\sum _{n=n_{0}+2}^{\infty}g(n)\leq \int_{n_{0}+1}^{\infty}g(t)\,dt< \infty. $$

Hence, choosing plus for the above two inequalities, we have (6). □

Lemma 2

If \(s\in\mathbf{N}\), \(\gamma,M>0\), \(\Psi(u)\) is a non-negative measurable function in \((0,1]\), and

$$ D_{M}:= \Biggl\{ x\in\mathbf{R}_{+}^{s};\sum _{i=1}^{s}x_{i}^{\gamma} \leq M^{\gamma} \Biggr\} , $$

then we have the following transfer formula (cf. [27]):

$$\begin{aligned} &\int\cdots \int_{D_{M}}\Psi \Biggl( \sum_{i=1}^{s} \biggl( \frac {x_{i}}{M} \biggr) ^{\gamma} \Biggr)\,dx_{1}\cdots \,dx_{s} \\ &\quad=\frac{M^{s}\Gamma^{s}(\frac{1}{\gamma})}{\gamma^{s}\Gamma(\frac {s}{\gamma})} \int_{0}^{1}\Psi(u)u^{\frac{s}{\gamma}-1}\,du. \end{aligned}$$
(7)

Lemma 3

For \(s\in\mathbf{N}\), \(\gamma, \varepsilon>0\), we have

$$ \sum_{m}\|m\|_{\gamma}^{-s-\varepsilon}= \frac{\Gamma^{s}(\frac {1}{\gamma})}{\varepsilon s^{\varepsilon/\gamma}\gamma^{s-1}\Gamma(\frac {s}{\gamma})}+O(1) \quad\bigl(\varepsilon\rightarrow0^{+}\bigr), $$
(8)

where \(\sum_{m}=\sum_{m_{s}=1}^{\infty}\cdots\) \(\sum_{m_{1}=1}^{\infty}\).

Proof

For \(M>s^{1/\gamma}\), we set

$$ \Psi(u)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0, & 0< u< \frac{s}{M^{\gamma}}, \\ (Mu^{1/\gamma})^{-s-\varepsilon}, &\frac{s}{M^{\gamma}}\leq u\leq1.\end{array}\displaystyle \right . $$

Then by Lemma 1 and (7), it follows that

$$\begin{aligned} \sum_{m}\|m\|_{\gamma}^{-s-\varepsilon} \geq& \int_{\{x\in\mathbf{R} _{+}^{s};x_{i}\geq1\}}\|x\|_{\gamma}^{-s-\varepsilon}\,dx \\ =&\lim_{M\rightarrow\infty} \int\cdots \int_{D_{M}}\Psi \Biggl( \sum_{i=1}^{s} \biggl( \frac{x_{i}}{M} \biggr) ^{\gamma} \Biggr)\,dx_{1} \cdots \,dx_{s} \\ =&\lim_{M\rightarrow\infty}\frac{M^{s}\Gamma^{s}(\frac{1}{\gamma })}{\gamma^{s}\Gamma(\frac{s}{\gamma})} \int_{s/M^{\gamma }}^{1}\bigl(Mu^{1/\gamma } \bigr)^{-s-\varepsilon}u^{\frac{s}{\gamma}-1}\,du \\ =&\frac{\Gamma^{s}(\frac{1}{\gamma})}{\varepsilon s^{\varepsilon /\gamma }\gamma^{s-1}\Gamma(\frac{s}{\gamma})}. \end{aligned}$$

By Lemma 1 and in the above way, we still find

$$ 0< \sum_{\{m\in\mathbf{N}^{s};m_{i}\geq2\}}\|m\|_{\gamma }^{-s-\varepsilon }\leq \int_{\{x\in\mathbf{R}_{+}^{s};x_{i}\geq1\}}\|x\|_{\gamma }^{-s-\varepsilon}\,dx= \frac{\Gamma^{s}(\frac{1}{\gamma})}{\varepsilon s^{\varepsilon/\gamma}\gamma^{s-1}\Gamma(\frac{s}{\gamma})}. $$

For \(s=1\), \(0<\sum_{m=1}^{1}\|m\|_{\gamma}^{-1-\varepsilon}<\infty\); for \(s\geq2\),

$$\begin{aligned} 0 < &\sum_{\{m\in\mathbf{N}^{s};\exists i_{0},m_{i_{0}}=1\} }\|m\|_{\gamma }^{-s-\varepsilon} \leq a+\sum_{\{m\in\mathbf{N}^{s-1};m_{i}\geq 2\}}\|m\|_{\gamma}^{-(s-1)-(1+\varepsilon)} \\ \leq&a+\frac{\Gamma^{s-1}(\frac{1}{\gamma})}{(1+\varepsilon )(s-1)^{(1+\varepsilon)/\gamma}\gamma^{s-2}\Gamma(\frac{s-1}{\gamma })}< \infty\quad(a\in\mathbf{R}_{+}), \end{aligned}$$

and then

$$\begin{aligned} \sum_{m}\|m\|_{\gamma}^{-s-\varepsilon}&=\sum _{\{m\in\mathbf{N}^{s};\exists i_{0},m_{i_{0}}=1\}}\|m\|_{\gamma}^{-s-\varepsilon }+\sum _{\{m\in\mathbf{N}^{s};m_{i}\geq2\}}\|m\|_{\gamma }^{-s-\varepsilon} \\ &\leq O_{1}(1)+\frac{\Gamma^{s}(\frac{1}{\gamma})}{\varepsilon s^{\varepsilon/\gamma}\gamma^{s-1}\Gamma(\frac{s}{\gamma })}\quad\bigl(\varepsilon \rightarrow0^{+}\bigr). \end{aligned}$$
(9)

Then we have (8). □

Example 1

For \(s\in\mathbf{N}\), \(0< c_{1}\leq\cdots\leq c_{s}<\infty\), \(\lambda_{1},\lambda_{2}>-\gamma\), \(\lambda_{1}+\lambda _{2}=\lambda\), we set

$$ k_{\lambda}(x,y):=\prod_{k=1}^{s} \frac{(\min\{x,c_{k}y\})^{\frac {\gamma}{s}}}{(\max\{x,c_{k}y\})^{\frac{\lambda+\gamma}{s}}}\quad\bigl((x,y)\in\mathbf {R}_{+}^{2}= \mathbf{R}_{+}\times\mathbf{R}_{+}\bigr). $$

(a) We find

$$\begin{aligned} k_{s}(\lambda_{1}) :=& \int_{0}^{\infty}k_{\lambda}(1,u)u^{\lambda _{2}-1}\,du \overset{u=1/t}{=} \int_{0}^{\infty}k_{\lambda}(t,1)t^{\lambda _{1}-1}\,dt \\ =& \int_{0}^{\infty}\prod_{k=1}^{s} \frac{(\min\{t,c_{k}\})^{\frac {\gamma}{s}}}{(\max\{t,c_{k}\})^{\frac{\lambda+\gamma}{s}}}t^{\lambda _{1}-1}\,dt \\ =& \int_{0}^{c_{1}}\prod_{k=1}^{s} \frac{(\min\{t,c_{k}\})^{\frac {\gamma}{s}}t^{\lambda_{1}-1}}{(\max\{t,c_{k}\})^{\frac{\lambda+\gamma}{s}}}\,dt+ \int_{c_{s}}^{\infty}\prod_{k=1}^{s} \frac{(\min\{t,c_{k}\})^{\frac{ \gamma}{s}}t^{\lambda_{1}-1}}{(\max\{t,c_{k}\})^{\frac{\lambda +\gamma}{s}}}\,dt \\ &{}+\sum_{i=1}^{s-1} \int_{c_{i}}^{c_{i+1}}\prod_{k=1}^{s} \frac{(\min \{t,c_{k}\})^{\frac{\gamma}{s}}t^{\lambda_{1}-1}}{(\max\{t,c_{k}\})^{ \frac{\lambda+\gamma}{s}}}\,dt\\ =&\prod_{k=1}^{s}\frac{1}{c_{k}^{(\lambda+\gamma)/s}} \int _{0}^{c_{1}}t^{\lambda_{1}+\gamma-1}\,dt+\prod _{k=1}^{s}c_{k}^{\gamma /s} \int_{c_{s}}^{\infty}t^{-\lambda_{2}-\gamma-1}\,dt \\ &{}+\sum_{i=1}^{s-1} \int_{c_{i}}^{c_{i+1}}\prod_{k=1}^{i} \frac {c_{k}^{\frac{\gamma}{s}}}{t^{\frac{\lambda+\gamma}{s}}}\prod_{k=i+1}^{s} \frac {t^{\frac{\gamma}{s}}}{c_{k}^{\frac{\lambda+\gamma}{s}}}t^{\lambda_{1}-1}\,dt \\ =&\frac{c_{1}^{\lambda_{1}+\gamma}}{\lambda_{1}+\gamma}\frac{1}{\prod_{k=1}^{s}c_{k}^{\frac{\lambda+\gamma}{s}}}+\frac{1}{(\lambda _{2}+\gamma)c_{s}^{\lambda_{2}+\gamma}}\prod _{k=1}^{s}c_{k}^{\frac {\gamma }{s}} \\ &{}+\sum_{i=1}^{s-1}\frac{\prod_{k=1}^{i}c_{k}^{\frac{\gamma}{s}}}{\prod_{k=i+1}^{s}c_{k}^{\frac{\lambda+\gamma}{s}}}\int_{c_{i}}^{c_{i+1}}t^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac {2i}{s})\gamma-1}\,dt. \end{aligned}$$

If \(\lambda_{1}-\frac{i\lambda}{s}+(1-\frac{2i}{s})\gamma\neq0\), then

$$ \int_{c_{i}}^{c_{i+1}}t^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac {2i}{s})\gamma-1}\,dt=\frac{c_{i+1}^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac {2i}{s})\gamma}-c_{i}^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac {2i}{s})\gamma}}{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac{2i}{s})\gamma}; $$

if there exists a \(i_{0}\in\{1,\ldots,s-1\}\), such that \(\lambda_{1}- \frac{i_{0}\lambda}{s}+(1-\frac{2i_{0}}{s})\gamma=0\), then we find

$$ \int_{c_{i_{0}}}^{c_{i_{0}+1}}t^{\lambda_{1}-\frac{i_{0}\lambda }{s}+(1-\frac{2i_{0}}{s})\gamma-1}\,dt=\ln\biggl( \frac{c_{i_{0}+1}}{c_{i_{0}}}\biggr)=\lim_{i\rightarrow i_{0}} \int_{c_{i}}^{c_{i+1}}t^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac{2i}{s})\gamma-1}\,dt, $$

and we still indicate \(\ln(\frac{c_{i_{0}+1}}{c_{i_{0}}})\) by the following formal expression:

$$ \frac{c_{i_{0}+1}^{\lambda_{1}-\frac{i_{0}\lambda}{s}+(1-\frac {2i_{0}}{s})\gamma}-c_{i_{0}}^{\lambda_{1}-\frac{i_{0}\lambda}{s}+(1-\frac {2i_{0}}{s})\gamma}}{\lambda_{1}-\frac{i_{0}\lambda}{s}+(1-\frac {2i_{0}}{s})\gamma}. $$

Hence, we may set

$$\begin{aligned} k_{s}(\lambda_{1}) =&\frac{c_{1}^{\lambda_{1}+\gamma}}{\lambda _{1}+\gamma} \frac{1}{\prod_{k=1}^{s}c_{k}^{\frac{\lambda+\gamma }{s}}}+\frac{1}{(\lambda_{2}+\gamma)c_{s}^{\lambda_{2}+\gamma}}\prod _{k=1}^{s}c_{k}^{\frac{\gamma}{s}} \\ &{}+\sum_{i=1}^{s-1} \biggl[ \frac{c_{i+1}^{\lambda_{1}-\frac{i\lambda }{s}+(1-\frac{2i}{s})\gamma}-c_{i}^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac {2i}{s})\gamma}}{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac{2i}{s})\gamma }\frac{\prod_{k=1}^{i}c_{k}^{\frac{\gamma}{s}}}{\prod_{k=i+1}^{s}c_{k}^{\frac{ \lambda+\gamma}{s}}} \biggr] . \end{aligned}$$
(10)

In particular, (i) for \(s=1\) (or \(c_{s}=\cdots=c_{1}\)), we have \(k_{\lambda }(x,y)=\frac{(\min\{x,c_{1}y\})^{\gamma}}{(\max\{x,c_{1}y\})^{\lambda +\gamma}}\) and

$$ k_{1}(\lambda_{1})=\frac{\lambda+2\gamma}{(\lambda_{1}+\gamma )(\lambda _{2}+\gamma)}\frac{1}{c_{1}^{\lambda_{2}}}; $$
(11)

(ii) for \(s=2\), we have \(k_{\lambda}(x,y)=\frac{(\min\{x,c_{1}y\}\min \{x,c_{2}y\})^{\gamma/2}}{(\max\{x,c_{1}y\}\max\{x,c_{2}y\} )^{(\lambda +\gamma)/2}}\) and

$$ k_{2}(\lambda_{1})= \biggl( \frac{c_{1}}{c_{2}} \biggr) ^{\frac{\gamma }{2}} \biggl[ \frac{c_{1}^{\lambda_{1}-\frac{\lambda}{2}}}{(\lambda _{1}+\gamma )c_{2}^{\frac{\lambda}{2}}}+\frac{1}{(\lambda_{2}+\gamma )c_{2}^{\lambda _{2}}}+ \frac{c_{2}^{\lambda_{1}-\frac{\lambda}{2}}-c_{1}^{\lambda _{1}-\frac{\lambda}{2}}}{(\lambda_{1}-\frac{\lambda}{2})c_{2}^{\frac {\lambda}{2}}} \biggr] ; $$
(12)

(iii) for \(\gamma=0\), we have \(\lambda_{1},\lambda_{2}>0\), \(k_{\lambda }(x,y)=\frac{1}{\prod_{k=1}^{s}(\max\{x,c_{k}y\})^{\frac{\lambda }{s}}}\) and

$$\begin{aligned} k_{s}(\lambda_{1}) =&\widetilde{k}_{s}( \lambda_{1}):=\frac {c_{1}^{\lambda _{1}}}{\lambda_{1}}\frac{1}{\prod_{k=1}^{s}c_{k}^{\frac{\lambda}{s}}}+ \frac{1}{\lambda_{2}c_{s}^{\lambda_{2}}} \\ &{}+\sum_{i=1}^{s-1}\frac{c_{i+1}^{\lambda_{1}-\frac{i}{s}\lambda }-c_{i}^{\lambda_{1}-\frac{i}{s}\lambda}}{\lambda_{1}-\frac {i}{s}\lambda}\frac{1}{\prod_{k=i+1}^{s}c_{k}^{\frac{\lambda}{s}}}; \end{aligned}$$
(13)

(iv) for \(\gamma=-\lambda\), we have \(\lambda<\lambda_{1},\lambda_{2}<0\), \(k_{\lambda}(x,y)=\frac{1}{\prod_{k=1}^{s}(\min\{x,c_{k}y\})^{\frac{\lambda}{s}}}\) and

$$\begin{aligned} k_{s}(\lambda_{1}) =&\widehat{k}_{s}( \lambda_{1}):=\frac {c_{1}^{-\lambda _{2}}}{(-\lambda_{2})}+\frac{1}{(-\lambda_{1})c_{s}^{-\lambda_{1}}}\prod _{k=1}^{s}c_{k}^{\frac{-\lambda}{s}} +\sum_{i=1}^{s-1} \Biggl( \frac{c_{i+1}^{\lambda_{1}-\frac {s-i}{s}\lambda }-c_{i}^{\lambda_{1}-\frac{s-i}{s}\lambda}}{\lambda_{1}-\frac{s-i}{s} \lambda}\prod_{k=1}^{i}c_{k}^{\frac{-\lambda}{s}} \Biggr) ; \end{aligned}$$
(14)

(v) for \(\lambda=0\), we have \(\lambda_{2}=-\lambda_{1}\), \(|\lambda _{1}|<\gamma\) (\(\gamma>0\)),

$$ k_{0}(x,y)=\prod_{k=1}^{s} \biggl( \frac{\min\{x,c_{k}y\}}{\max\{ x,c_{k}y\}} \biggr) ^{\frac{\gamma}{s}}, $$

and

$$\begin{aligned} k_{s}(\lambda_{1}) =&k_{s}^{(0)}( \lambda_{1}):=\frac{c_{1}^{\lambda _{1}+\gamma}}{\gamma+\lambda_{1}}\frac{1}{\prod_{k=1}^{s}c_{k}^{\frac {\gamma}{s}}}+ \frac{c_{s}^{\lambda_{1}-\gamma}}{\gamma-\lambda_{1}}\prod_{k=1}^{s}c_{k}^{\frac{\gamma}{s}} \\ &{}+\sum_{i=1}^{s-1} \biggl[ \frac{c_{i+1}^{\lambda_{1}+(1-\frac {2i}{s})\gamma }-c_{i}^{\lambda_{1}+(1-\frac{2i}{s})\gamma}}{\lambda_{1}+(1-\frac {2i}{s})\gamma}\frac{\prod_{k=1}^{i}c_{k}^{\frac{\gamma}{s}}}{\prod_{k=i+1}^{s}c_{k}^{\frac{\gamma}{s}}} \biggr] . \end{aligned}$$
(15)

(b) Since for \(j_{0}\in\mathbf{N,}\) we find

$$\begin{aligned} k_{\lambda}(x,y)\frac{1}{y^{j_{0}-\lambda_{2}}}&=\frac {1}{y^{j_{0}-\lambda _{2}}}\prod _{k=1}^{s}\frac{(\min\{c_{k}^{-1}x,y\})^{\frac{\gamma }{s}}}{c_{k}^{\frac{\lambda}{s}}(\max\{c_{k}^{-1}x,y\})^{\frac{\lambda +\gamma}{s}}}\\ &=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{1}{y^{j_{0}-\lambda_{2}-\gamma}}\prod_{k=1}^{s}\frac {1}{c_{k}^{\frac{\lambda}{s}}(c_{k}^{-1}x)^{\frac{\lambda+\gamma}{s}}}, &0< y\leq c_{s}^{-1}x, \\ \frac{1}{y^{j_{0}+\lambda_{1}+\gamma-\frac{i}{s}(\lambda+2\gamma )}}\frac{\prod_{k=i+1}^{s}(c_{k}^{-1}x)^{\frac{\gamma}{s}}}{\prod_{k=1}^{s}c_{k}^{\frac{\lambda}{s}}\prod_{k=1}^{i}(c_{k}^{-1}x)^{\frac{\lambda+\gamma }{s}}},& c_{i+1}^{-1}x< y\leq c_{i}^{-1}x\ (i=1,\ldots,s-1), \\ \frac{1}{y^{j_{0}+\lambda_{1}+\gamma}}\prod_{k=1}^{s}\frac {(c_{k}^{-1}x)^{\frac{\gamma}{s}}}{c_{k}^{\frac{\lambda}{s}}(y)^{\frac{\lambda +\gamma}{s}}},& c_{1}^{-1}x< y< \infty,\end{array}\displaystyle \right . \end{aligned}$$

for \(\lambda_{2}\leq j_{0}-\gamma\) (\(\lambda_{1}>-\gamma\)), \(k_{\lambda}(x,y)\frac{1}{y^{j_{0}-\lambda_{2}}}\) is decreasing for \(y>0\) and strictly decreasing for the large enough variable y. In the same way, for \(i_{0}\in\mathbf{N,}\) we find

$$\begin{aligned} k_{\lambda}(x,y)\frac{1}{x^{i_{0}-\lambda_{1}}}&=\frac {1}{x^{i_{0}-\lambda _{1}}}\prod _{k=1}^{s}\frac{(\min\{x,c_{k}y\})^{\frac{\gamma }{s}}}{(\max \{x,c_{k}y\})^{\frac{\lambda+\gamma}{s}}}\\ &=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{1}{x^{i_{0}-\lambda_{1}-\gamma}}\prod_{k=1}^{s}\frac {1}{(c_{k}y)^{\frac{\lambda+\gamma}{s}}},& 0< x\leq c_{1}y, \\ \frac{1}{x^{i_{0}-\lambda_{1}-\gamma+\frac{i}{s}(\lambda+2\gamma )}}\frac{\prod_{k=1}^{i}(c_{k}y)^{\frac{\gamma}{s}}}{\prod_{k=i+1}^{s}(c_{k}y)^{ \frac{\lambda+\gamma}{s}}},& c_{i}y< x\leq c_{i+1}y \ (i=1,\ldots,s-1), \\ \frac{1}{x^{i_{0}+\lambda_{2}+\gamma}}\prod_{k=1}^{s}(c_{k}y)^{\frac{\gamma}{s}},& c_{s}y< x< \infty,\end{array}\displaystyle \right . \end{aligned}$$

then for \(\lambda_{1}\leq i_{0}-\gamma\) (\(\lambda_{2}>-\gamma\)), \(k_{\lambda}(x,y)\frac{1}{x^{i_{0}-\lambda_{1}}}\) is decreasing for \(x>0\) and strictly decreasing for the large enough variable x.

In view of the above results, for \(i_{0},j_{0}\in\mathbf{N}\), \(-\gamma <\lambda_{1}\leq i_{0}-\gamma\), \(-\gamma<\lambda_{2}\leq j_{0}-\gamma\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(k_{\lambda}(x,y)\frac{1}{y^{j_{0}-\lambda_{2}}}\) (\(k_{\lambda}(x,y)\frac{1}{x^{i_{0}-\lambda _{1}}}\)) is still decreasing for \(y>0\) (\(x>0\)) and strictly decreasing for the large enough variable \(y(x)\).

Definition 1

For \(s,i_{0},j_{0}\in\mathbf{N}\), \(0< c_{1}\leq \cdots\leq c_{s}<\infty\), \(-\gamma<\lambda_{1}\leq i_{0}-\gamma\), \(-\gamma <\lambda_{2}\leq j_{0}-\gamma\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(m=(m_{1},\ldots,m_{i_{0}})\in\mathbf{N}^{i_{0}}\), \(n=(n_{1},\ldots ,n_{j_{0}})\in\mathbf{N}^{j_{0}}\), define two weight coefficients \(w(\lambda_{1},n)\) and \(W(\lambda_{2},m)\) as follows:

$$\begin{aligned}& w(\lambda_{1},n) :=\sum_{m}\prod _{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}} \frac{\|n\|_{\beta }^{\lambda_{2}}}{\|m\|_{\alpha}^{i_{0}-\lambda_{1}}}, \end{aligned}$$
(16)
$$\begin{aligned}& W(\lambda_{2},m) :=\sum_{n}\prod _{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}} \frac {\|m\|_{\alpha }^{\lambda_{1}}}{\|n\|_{\beta}^{j_{0}-\lambda_{2}}}, \end{aligned}$$
(17)

where \(\sum_{m}=\sum_{m_{i_{0}}=1}^{\infty}\cdots\sum_{m_{1}=1}^{\infty}\) and \(\sum_{n}=\sum_{n_{j_{0}}=1}^{\infty}\cdots\sum_{n_{1}=1}^{\infty}\).

Lemma 4

As the assumptions of Definition  1, then (i) we have

$$\begin{aligned}& w(\lambda_{1},n) < K_{2}^{(s)}\quad\bigl(n\in \mathbf{N}^{j_{0}}\bigr), \end{aligned}$$
(18)
$$\begin{aligned}& W(\lambda_{2},m) < K_{1}^{(s)}\quad\bigl(m\in \mathbf{N}^{i_{0}}\bigr), \end{aligned}$$
(19)

where

$$ K_{1}^{(s)}=\frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta ^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})}k_{s}( \lambda_{1}),\qquad K_{2}^{(s)}=\frac{\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})}k_{s}( \lambda_{1}); $$
(20)

(ii) for \(p>1\), \(0<\varepsilon<\frac{p}{2}(\lambda_{1}+\gamma)\), setting \(\widetilde{\lambda}_{1}=\lambda_{1}-\frac{\varepsilon}{p}\) (\(\in (-\gamma ,i_{0}-\gamma)\)), \(\widetilde{\lambda}_{2}=\lambda_{2}+\frac {\varepsilon}{p}\) (\(>{-}\gamma\)), we have

$$ 0< \widetilde{K}_{2}^{(s)}\bigl(1-\widetilde{ \theta}_{\lambda }(n)\bigr)< w(\widetilde{\lambda}_{1},n), $$
(21)

where

$$\begin{aligned}& 0 < \widetilde{\theta}_{\lambda}(n)=\frac{1}{k_{s}(\widetilde {\lambda}_{1})} \int_{0}^{i_{0}^{1/\alpha}/\|n\|_{\beta}}\prod_{k=1}^{s} \frac {(\min \{v,c_{k}\})^{\frac{\gamma}{s}}v^{\widetilde{\lambda}_{1}-1}}{(\max \{v,c_{k}\})^{\frac{\lambda+\gamma}{s}}}\,dv =O \biggl( \frac{1}{\|n\|_{\beta}^{\gamma+\widetilde{\lambda}_{1}}} \biggr) , \end{aligned}$$
(22)
$$\begin{aligned}& \widetilde{K}_{2}^{(s)} =\frac{\Gamma^{i_{0}}(\frac{1}{\alpha })}{\alpha ^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})}k_{s}( \widetilde{\lambda}_{1}). \end{aligned}$$
(23)

Proof

By Lemma 1, Example 1, and (7), it follows that

$$\begin{aligned} w(\lambda_{1},n) &< \int_{\mathbf{R}_{+}^{i_{0}}}\prod_{k=1}^{s} \frac {(\min \{\|x\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max \{\|x\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma }{s}}}\frac{\|n\|_{\beta}^{\lambda_{2}}}{\|x\|_{\alpha}^{i_{0}-\lambda_{1}}}\,dx\\ &=\lim_{M\rightarrow\infty} \int_{\mathbf{D}_{M}}\prod_{k=1}^{s} \frac{ (\min\{M[\sum_{i=1}^{i_{0}}(\frac{x_{i}}{M})^{\alpha}]^{\frac {1}{\alpha}},c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{M[\sum_{i=1}^{i_{0}}(\frac{x_{i}}{M})^{\alpha}]^{\frac{1}{\alpha}},c_{k}\|n\|_{\beta}\})^{ \frac{\lambda+\gamma}{s}}}\frac{M^{\lambda_{1}-i_{0}}\|n\|_{\beta }^{\lambda_{2}}\,dx}{[\sum_{i=1}^{i_{0}}(\frac{x_{i}}{M})^{\alpha }]^{\frac{i_{0}-\lambda_{1}}{\alpha}}}\\ &=\lim_{M\rightarrow\infty}\frac{M^{i_{0}}\Gamma^{i_{0}}(\frac {1}{\alpha })}{\alpha^{i_{0}}\Gamma(\frac{i_{0}}{\alpha})} \int_{0}^{1}\prod_{k=1}^{s}\frac{(\min\{Mu^{1/\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma }{s}}}{(\max\{Mu^{1/\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma }{s}}}\frac{\|n\|_{\beta}^{\lambda_{2}}u^{\frac{i_{0}}{\alpha}-1}\,du}{M^{i_{0}-\lambda_{1}}u^{(i_{0}-\lambda_{1})/\alpha}} \\ &=\lim_{M\rightarrow\infty}\frac{M^{\lambda_{1}}\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}}\Gamma(\frac{i_{0}}{\alpha})} \int_{0}^{1}\prod _{k=1}^{s}\frac{(\min\{Mu^{1/\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\gamma}{s}}\|n\|_{\beta}^{\lambda_{2}}}{(\max\{Mu^{1/\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}u^{\frac{\lambda _{1}}{\alpha}-1}\,du \\ &\overset{u=\|n\|_{\beta}^{\alpha}M^{-\alpha}v^{\alpha}}{=}\frac {\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}-1}\Gamma(\frac {i_{0}}{\alpha})}\int_{0}^{\infty}\prod_{k=1}^{s} \frac{(\min\{v,c_{k}\})^{\frac{\gamma }{s}}v^{\lambda_{1}-1}}{(\max\{v,c_{k}\})^{\frac{\lambda+\gamma}{s}}}\,dv \\ &=\frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}-1}\Gamma (\frac{i_{0}}{\alpha})}k_{s}(\lambda_{1})=K_{2}^{(s)}. \end{aligned}$$

Hence, we have (18). In the same way, we have (19).

By Lemma 1, Example 1, and in the same way as obtaining (8), we have

$$\begin{aligned}& \begin{aligned}[b] w(\widetilde{\lambda}_{1},n)&> \int_{\{x\in\mathbf {R}_{+}^{i_{0}};x_{i}\geq1\}}\prod_{k=1}^{s} \frac{(\min\{\|x\|_{\alpha},c_{k}\|n\|_{\beta }\})^{\frac{\gamma}{s}}}{(\max\{\|x\|_{\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\lambda+\gamma}{s}}}\frac{\|n\|_{\beta}^{\widetilde{\lambda }_{2}}\,dx}{\|x\|_{\alpha}^{i_{0}-\widetilde{\lambda}_{1}}}\\ &=\frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}-1}\Gamma (\frac{i_{0}}{\alpha})} \int_{i_{0}^{1/\alpha}/\|n\|_{\beta}}^{\infty }\prod_{k=1}^{s} \frac{(\min\{v,c_{k}\})^{\frac{\gamma }{s}}v^{\widetilde{\lambda}_{1}-1}}{(\max\{v,c_{k}\})^{\frac{\lambda+\gamma}{s}}}\,dv=\widetilde{K}_{2}^{(s)} \bigl(1-\widetilde{\theta}_{\lambda}(n)\bigr)>0, \end{aligned}\\& 0< \widetilde{\theta}_{\lambda}(n)=\frac{1}{k_{s}(\widetilde{\lambda }_{1})}\int_{0}^{i_{0}^{1/\alpha}/\|n\|_{\beta}}\prod_{k=1}^{s} \frac{(\min \{v,c_{k}\})^{\frac{\gamma}{s}}v^{\widetilde{\lambda}_{1}-1}}{(\max \{v,c_{k}\})^{\frac{\lambda+\gamma}{s}}}\,dv. \end{aligned}$$

For \(\|n\|_{\beta}\geq c_{1}^{-1}i_{0}^{1/\alpha}\), we find \(v\leq i_{0}^{1/\alpha}/\|n\|_{\beta}\leq c_{1}\leq c_{k}\) (\(k=1,\ldots,s\)) and

$$ \widetilde{\theta}_{\lambda}(n)=\frac{1}{k_{s}(\widetilde{\lambda }_{1})}\int_{0}^{i_{0}^{1/\alpha}/\|n\|_{\beta}}\frac{v^{\widetilde{\lambda} _{1}+\gamma-1}\,dv}{\prod_{k=1}^{s}c_{k}{}^{\frac{\lambda+\gamma}{s}}}= \frac{(\prod_{k=1}^{s}c_{k}{}^{\frac{\lambda+\gamma}{s}})^{-1}}{(\widetilde{\lambda}_{1}+\gamma)k_{s}(\widetilde{\lambda}_{1})} \biggl( \frac{i_{0}^{1/\alpha}}{\|n\|_{\beta}} \biggr) ^{\widetilde{\lambda}_{1}+\gamma}, $$

and then (22) follows. □

3 Main results

Setting \(\Phi(m):=\|m\|_{\alpha}^{p(i_{0}-\lambda_{1})-i_{0}}\) (\(m\in \mathbf{N}^{i_{0}}\)) and \(\Psi(n):=\|n\|_{\beta}^{q(j_{0}-\lambda _{2})-j_{0}}\) (\(n\in\mathbf{N}^{j_{0}}\)), we have the following.

Theorem 1

If \(s,i_{0},j_{0}\in\mathbf{N}\), \(0< c_{1}\leq\cdots \leq c_{s}<\infty\), \(-\gamma<\lambda_{1}\leq i_{0}-\gamma\), \(-\gamma <\lambda_{2}\leq j_{0}-\gamma\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(k_{s}(\lambda_{1})\) is indicated by (10), then for \(p>1\), \(\frac {1}{p}+\frac{1}{q}=1\), \(a_{m},b_{n}\geq0\), \(0<\|a\|_{p,\Phi},\|b\|_{q,\Psi }<\infty\), we have the following inequality:

$$\begin{aligned} I :=&\sum_{n}\sum_{m} \prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}a_{m}b_{n} \\ < &\bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi} \|b\|_{q,\Psi}, \end{aligned}$$
(24)

where the constant factor

$$ \bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}}\bigl(K_{2}^{(s)} \bigr)^{\frac{1}{q}}= \biggl[ \frac {\Gamma ^{j_{0}}(\frac{1}{\beta})}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta })} \biggr] ^{\frac{1}{p}} \biggl[ \frac{\Gamma^{i_{0}}(\frac{1}{\alpha })}{\beta ^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})} \biggr] ^{\frac {1}{q}}k_{s}( \lambda _{1}) $$
(25)

is the best possible. In particular, for \(s=1\) (or \(c_{s}=\cdots=c_{1}\)), we have the following inequality:

$$ \sum_{n}\sum_{m} \frac{(\min\{\|m\|_{\alpha},c_{1}\|n\|_{\beta}\} )^{\gamma }a_{m}b_{n}}{(\max\{\|m\|_{\alpha},c_{1}\|n\|_{\beta}\})^{\lambda +\gamma }}< \bigl(K_{1}^{(1)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(1)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi } \|b\|_{q,\Psi}, $$
(26)

where

$$\begin{aligned} &\bigl(K_{1}^{(1)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(1)}\bigr)^{\frac{1}{q}} \\ &\quad= \biggl[ \frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta ^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac{1}{p}} \biggl[ \frac{\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\beta^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha })} \biggr] ^{\frac{1}{q}}\frac{(\lambda+2\gamma)c_{1}^{-\lambda_{2}}}{(\lambda _{1}+\gamma)(\lambda_{2}+\gamma)}. \end{aligned}$$
(27)

Proof

By Hölder’s inequality (cf. [28]), we have

$$\begin{aligned} I =&\sum_{n}\sum_{m} \prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}} \biggl[ \frac{\|m\|_{\alpha}^{(i_{0}-\lambda _{1})/q}}{\|n\|_{\beta }^{(j_{0}-\lambda_{2})/p}}a_{m} \biggr] \biggl[ \frac{\|n\|_{\beta }^{(j_{0}-\lambda_{2})/p}}{\|m\|_{\alpha}^{(i_{0}-\lambda _{1})/q}}b_{n} \biggr]\\ \leq& \biggl\{ \sum_{m}W(\lambda_{2},m) \|m\|_{\alpha}^{p(i_{0}-\lambda _{1})-i_{0}}a_{m}^{p} \biggr\} ^{\frac{1}{p}} \biggl\{ \sum_{n}w(\lambda_{1},n) \|n\|_{\beta }^{q(j_{0}-\lambda _{2})-j_{0}}b_{n}^{q} \biggr\} ^{\frac{1}{q}}. \end{aligned}$$

Then by (18) and (19), we have (24).

For \(0<\varepsilon<\frac{p}{2}(\lambda_{1}+\gamma)\), \(\widetilde {\lambda}_{1}=\lambda_{1}-\frac{\varepsilon}{p}\), \(\widetilde{\lambda }_{2}=\lambda _{2}+\frac{\varepsilon}{p}\), we set

$$ \widetilde{a}_{m}=\|m\|_{\alpha}^{-i_{0}+\lambda_{1}-\frac {\varepsilon}{p}}=\|m \|_{\alpha}^{\widetilde{\lambda}_{1}-i_{0}}, \qquad\widetilde{b}_{n}=\|n \|_{\beta}^{\widetilde{\lambda}_{2}-j_{0}-\varepsilon} \quad\bigl(m\in \mathbf{N}^{i_{0}},n\in \mathbf{N}^{j_{0}}\bigr). $$

Then by (8) and (21), we obtain

$$\begin{aligned}& \begin{aligned}[b] \|\widetilde{a}\|_{p,\Phi}\|\widetilde{b}\|_{q,\Psi}={}& \biggl[ \sum _{m}\|m\|_{\alpha}^{p(i_{0}-\lambda_{1})-i_{0}}\widetilde {a}_{m}^{p} \biggr] ^{\frac{1}{p}} \biggl[ \sum _{n}\|n\|_{\beta}^{q(j_{0}-\lambda _{2})-j_{0}} \widetilde{b}_{n}^{q} \biggr] ^{\frac{1}{q}} \\ ={}& \biggl( \sum_{m}\|m\|_{\alpha}^{-i_{0}-\varepsilon} \biggr) ^{\frac {1}{p}} \biggl( \sum_{n}\|n \|_{\beta}^{-j_{0}-\varepsilon} \biggr) ^{\frac{1}{q}} \\ ={}&\frac{1}{\varepsilon} \biggl( \frac{\Gamma^{i_{0}}(\frac{1}{\alpha })}{i_{0}^{\varepsilon/\alpha}\alpha^{i_{0}-1}\Gamma(\frac {i_{0}}{\alpha})}+\varepsilon O(1) \biggr) ^{\frac{1}{p}} \\ &{}\times \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta })}{j_{0}^{\varepsilon /\beta}\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} +\varepsilon \widetilde{O}(1) \biggr) ^{\frac{1}{q}}, \end{aligned} \end{aligned}$$
(28)
$$\begin{aligned}& \begin{aligned}[b] \widetilde{I} &:=\sum_{n} \Biggl[ \sum _{m}\prod_{k=1}^{s} \frac{(\min \{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max \{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}\widetilde{a}_{m} \Biggr] \widetilde{b}_{n} \\ &=\sum_{n}w(\widetilde{\lambda}_{1},n) \|n\|_{\beta }^{-j_{0}-\varepsilon}>\widetilde{K}_{2}^{(s)} \sum_{n} \biggl( 1-O\biggl(\frac{1}{\|n\|_{\beta }^{\gamma+\widetilde{\lambda}_{1}}} \biggr) \biggr) \|n\|_{\beta}^{-j_{0}-\varepsilon} \\ &=\widetilde{K}_{2}^{(s)} \biggl( \sum _{n}\|n\|_{\beta }^{-j_{0}-\varepsilon }-\sum _{n}O\biggl(\frac{1}{\|n\|_{\beta}^{\gamma+\lambda_{1}+j_{0}+\frac{\varepsilon}{q}}}\biggr) \biggr) \\ &=\widetilde{K}_{2}^{(s)} \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta })}{\varepsilon j_{0}^{\varepsilon/\beta}\beta^{j_{0}-1}\Gamma(\frac {j_{0}}{\beta})}+ \widetilde{O}(1)-O(1) \biggr) . \end{aligned} \end{aligned}$$
(29)

If there exists a constant \(K\leq(K_{1}^{(s)})^{\frac {1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\), such that (24) is valid as we replace \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) by K, then using (28) and (29) we have

$$\begin{aligned} &\bigl(K_{2}^{(s)}+o(1)\bigr) \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{j_{0}^{\varepsilon/\beta}\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta })}+ \varepsilon\widetilde{O}(1)-\varepsilon O(1) \biggr)< \varepsilon \widetilde{I} < \varepsilon K\|\widetilde{a}\|_{p,\varphi}\|\widetilde{b} \|_{q,\psi } \\ &\quad=K \biggl( \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{i_{0}^{\varepsilon /\alpha}\alpha^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})}+\varepsilon O(1) \biggr) ^{\frac{1}{p}} \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta })}{j_{0}^{\varepsilon /\beta}\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})}+\varepsilon \widetilde{O}(1) \biggr) ^{\frac{1}{q}}. \end{aligned}$$

For \(\varepsilon\rightarrow0^{+}\), we find

$$\begin{aligned} &\frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta^{j_{0}-1}\Gamma(\frac {j_{0}}{\beta})}\frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\alpha ^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})}k_{s}(\lambda_{1}) \leq K \biggl( \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\alpha ^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})} \biggr) ^{\frac{1}{p}} \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{ \beta})} \biggr) ^{\frac{1}{q}}, \end{aligned}$$

and then \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\leq K\). Hence, \(K=(K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) is the best possible constant factor of (24). □

Theorem 2

As regards the assumptions of Theorem  1, for \(0<\|a\|_{p,\Phi }<\infty\), we have the following inequality with the best constant factor \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\):

$$\begin{aligned} J :=& \Biggl\{ \sum_{n}\|n\|_{\beta}^{p\lambda_{2}-j_{0}} \Biggl[ \sum_{m}\prod _{k=1}^{s}\frac{(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta }\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\lambda+\gamma}{s}}}a_{m} \Biggr] ^{p} \Biggr\} ^{\frac{1}{p}} \\ < &\bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi}, \end{aligned}$$
(30)

which is equivalent to (24). In particular, for \(s=1\) (or \(c_{s}=\cdots=c_{1}\)), we have the following inequality equivalent to (26):

$$\begin{aligned} & \biggl\{ \sum_{n}\|n\|_{\beta}^{p\lambda_{2}-j_{0}} \biggl[ \sum_{m}\frac{(\min\{\|m\|_{\alpha},c_{1}\|n\|_{\beta}\})^{\gamma}}{(\max \{\|m\|_{\alpha},c_{1}\|n\|_{\beta}\})^{\lambda+\gamma}}a_{m} \biggr] ^{p} \biggr\} ^{\frac{1}{p}} < \bigl(K_{1}^{(1)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(1)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi}. \end{aligned}$$
(31)

Proof

We set \(b_{n}\) as follows:

$$ b_{n}:=\|n\|_{\beta}^{p\lambda_{2}-j_{0}} \Biggl( \sum _{m}\prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma }{s}}}{(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma }{s}}}a_{m} \Biggr) ^{p-1},\quad n\in \mathbf{N}^{j_{0}}. $$

Then it follows that \(J^{p}=\|b\|_{q,\Psi}^{q}\). If \(J=0\), then (30) is trivially valid for \(0<\|a\|_{p,\Phi}<\infty\); if \(J=\infty\), then it is impossible since the right hand side of (30) is finite. Suppose that \(0< J<\infty\). Then by (24), we find

$$ \|b\|_{q,\Psi}^{q}=J^{p}=I< \bigl(K_{1}^{(s)} \bigr)^{\frac {1}{p}}\bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}} \|a\|_{p,\Phi}\|b\|_{q,\Psi}, $$

namely,

$$ \|b\|_{q,\Psi}^{q-1}=J< \bigl(K_{1}^{(s)} \bigr)^{\frac{1}{p}}\bigl(K_{2}^{(s)}\bigr)^{\frac {1}{q}} \|a\|_{p,\Phi}, $$

and then (30) follows.

On the other hand, assuming that (30) is valid, by Hölder’s inequality, we have

$$\begin{aligned} I =&\sum_{n}\bigl(\Psi(n)\bigr)^{\frac{-1}{q}} \Biggl[ \sum_{m}\prod_{k=1}^{s} \frac{(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma }{s}}a_{m}}{(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma }{s}}} \Biggr] \bigl[\bigl(\Psi(n)\bigr)^{\frac{1}{q}}b_{n} \bigr] \\ \leq&J\|b\|_{q,\Psi}. \end{aligned}$$
(32)

Then by (30), we have (24). Hence (30) and (24) are equivalent.

By the equivalency, the constant factor \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) in (30) is the best possible. Otherwise, we would reach a contradiction by (32) that the constant factor \((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) in (24) is not the best possible. □

4 Operator expressions and some particular cases

For \(p>1\), we define two real weight normal discrete spaces \(l_{p,\varphi}\) and \(l_{q,\psi}\) as follows:

$$\begin{aligned}& l_{p,\varphi} := \biggl\{ a=\{a_{m}\};\|a\|_{p,\Phi}= \biggl(\sum_{m}\Phi (m)a_{m}^{p} \biggr)^{\frac{1}{p}}< \infty \biggr\} , \\& l_{q,\psi} := \biggl\{ b=\{b_{n}\};\|b\|_{q,\Psi}= \biggl(\sum_{n}\Psi (n)b_{n}^{q} \biggr)^{\frac{1}{q}}< \infty \biggr\} . \end{aligned}$$

As regards the assumptions of Theorem 1, in view of \(J<(K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\|a\|_{p,\Phi}\), we give the following definition.

Definition 2

Define a multidimensional Hilbert-type operator \(T:l_{p,\Phi}\rightarrow l_{p,\Psi^{1-p}}\) as follows: For \(a\in l_{p,\Phi }\), there exists an unique representation \(Ta\in l_{p,\Psi^{1-p}}\), satisfying

$$ Ta(n):=\sum_{m}\prod _{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}a_{m}\quad\bigl(n \in\mathbf {N}^{j_{0}}\bigr). $$
(33)

For \(b\in l_{q,\Psi}\), we define the following formal inner product of Ta and b as follows:

$$ (Ta,b):=\sum_{n}\sum_{m} \prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}a_{m}b_{n}. $$
(34)

Then by Theorem 1 and Theorem 2, for \(0<\|a\|_{p,\varphi},\|b\|_{q,\psi }<\infty\), we have the following equivalent inequalities:

$$\begin{aligned}& (Ta,b) < \bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi} \|b\|_{q,\Psi}, \end{aligned}$$
(35)
$$\begin{aligned}& \|Ta\|_{p,\Psi^{1-p}} < \bigl(K_{1}^{(s)} \bigr)^{\frac {1}{p}}\bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}} \|a\|_{p,\Phi}. \end{aligned}$$
(36)

It follows that T is bounded with

$$ \|T\|:=\sup_{a(\neq\theta)\in l_{p,\Phi}}\frac{\|Ta\|_{p,\Psi ^{1-p}}}{\|a\|_{p,\Phi}}\leq\bigl(K_{1}^{(s)} \bigr)^{\frac{1}{p}}\bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}}. $$
(37)

Since the constant factor \((K_{1}^{(s)})^{\frac {1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\) in (36) is the best possible, we have the following.

Corollary 1

As regards the assumptions of Theorem  2, T is defined by Definition  2, it follows that

$$\begin{aligned} \|T\| =&\bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}} \\ =& \biggl[ \frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta ^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac{1}{p}} \biggl[ \frac{\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha })} \biggr] ^{\frac{1}{q}}k_{s}(\lambda_{1}). \end{aligned}$$
(38)

Remark 1

(i) For \(i_{0}=j_{0}=1\) in (24), we have the inequality

$$ \sum_{m=1}^{\infty}\sum _{n=1}^{\infty}\prod_{k=1}^{s} \frac{(\min \{m,c_{k}n\})^{\frac{\gamma}{s}}}{(\max\{m,c_{k}n\})^{\frac{\lambda +\gamma}{s}}}a_{m}b_{n}< k_{s}( \lambda_{1})\|a\|_{p,\phi }\|b\|_{q,\psi}. $$
(39)

Hence, (24) is an extension of (4) for

$$ k_{\lambda}(m,n)=\prod_{k=1}^{s} \frac{(\min\{m,c_{k}n\})^{\frac {\gamma}{s}}}{(\max\{m,c_{k}n\})^{\frac{\lambda+\gamma}{s}}}. $$

(ii) For \(\gamma=0\) in (24) and (30), we have \(0<\lambda _{1}\leq i_{0}\), \(0<\lambda_{2}\leq j_{0}\) and the following equivalent inequalities:

$$\begin{aligned}& \sum_{n}\sum_{m} \frac{a_{m}b_{n}}{\prod_{k=1}^{s}(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda}{s}}}< \widetilde{K}_{s}(\lambda _{1})\|a \|_{p,\Phi}\|b\|_{q,\Psi}, \end{aligned}$$
(40)
$$\begin{aligned}& \biggl\{ \sum_{n}\|n\|_{\beta}^{p\lambda_{2}-j_{0}} \biggl[ \sum_{m}\frac {a_{m}}{\prod_{k=1}^{s}(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\lambda}{s}}} \biggr] ^{p} \biggr\} ^{\frac{1}{p}}< \widetilde {K}_{s}(\lambda _{1})\|a\|_{p,\Phi}, \end{aligned}$$
(41)

where the best possible constant factor is defined by

$$ \widetilde{K}_{s}(\lambda_{1}):= \biggl[ \frac{\Gamma^{j_{0}}(\frac {1}{\beta })}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac{1}{p}} \biggl[ \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\beta^{i_{0}-1}\Gamma (\frac{i_{0}}{\alpha})} \biggr] ^{\frac{1}{q}}\widetilde{k}_{s}(\lambda_{1}), $$
(42)

and \(\widetilde{k}_{s}(\lambda_{1})\) is indicated by (13).

(iii) For \(\gamma=-\lambda\) in (24) and (30), we have \(\lambda<\lambda_{1}\leq i_{0}+\lambda\), \(\lambda<\lambda_{2}\leq j_{0}+\lambda\), \(\lambda_{1},\lambda_{2}<0\) and the following equivalent inequalities:

$$\begin{aligned}& \sum_{n}\sum_{m} \frac{a_{m}b_{n}}{\prod_{k=1}^{s}(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda}{s}}}< \widehat{K}_{s}(\lambda _{1})\|a \|_{p,\Phi}\|b\|_{q,\Psi}, \end{aligned}$$
(43)
$$\begin{aligned}& \biggl\{ \sum_{n}\|n\|_{\beta}^{p\lambda_{2}-j_{0}} \biggl[ \sum_{m}\frac {a_{m}}{\prod_{k=1}^{s}(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\lambda}{s}}} \biggr] ^{p} \biggr\} ^{\frac{1}{p}}< \widetilde {K}_{s}(\lambda _{1})\|a\|_{p,\Phi}, \end{aligned}$$
(44)

where the best possible constant factor is defined by

$$ \widehat{K}_{s}(\lambda_{1}):= \biggl[ \frac{\Gamma^{j_{0}}(\frac {1}{\beta})}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac {1}{p}} \biggl[ \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\beta^{i_{0}-1}\Gamma(\frac {i_{0}}{\alpha})} \biggr] ^{\frac{1}{q}} \widehat{k}_{s}(\lambda_{1}), $$
(45)

and \(\widehat{k}_{s}(\lambda_{1})\) is indicated by (14).

(iv) For \(\lambda=0\) in (24) and (30), we have \(\lambda _{2}=-\lambda_{1}\), \(0<\gamma+\lambda_{1}\leq i_{0}\), \(0<\gamma-\lambda _{1}\leq j_{0}\) (\(\gamma>0\)), and the following equivalent inequalities:

$$\begin{aligned}& \sum_{n}\sum_{m} \prod_{k=1}^{s} \biggl( \frac{\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\}}{\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\}} \biggr) ^{\frac{\gamma}{s}}a_{m}b_{n}< K_{s}^{(0)}( \lambda _{1})\|a\|_{p,\Phi}\|b\|_{q,\Psi}, \end{aligned}$$
(46)
$$\begin{aligned}& \begin{aligned}[b] & \Biggl\{ \sum_{n}\frac{1}{\|n\|_{\beta}^{p\lambda_{1}+j_{0}}} \Biggl[ \sum_{m}\prod_{k=1}^{s} \biggl( \frac{\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta }\}}{\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\}} \biggr) ^{\frac {\gamma}{s}}a_{m} \Biggr] ^{p} \Biggr\} ^{\frac{1}{p}} \\ &\quad< K_{s}^{(0)}(\lambda_{1})\|a\|_{p,\Phi}, \end{aligned} \end{aligned}$$
(47)

where the best possible constant factor is defined by

$$ K_{s}^{(0)}(\lambda_{1}):= \biggl[ \frac{\Gamma^{j_{0}}(\frac{1}{\beta })}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac {1}{p}} \biggl[ \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\beta^{i_{0}-1}\Gamma(\frac {i_{0}}{\alpha})} \biggr] ^{\frac{1}{q}}k_{s}^{(0)}(\lambda_{1}), $$
(48)

and \(k_{s}^{(0)}(\lambda_{1})\) is indicated by (15).