Open Access

On a multidimensional Hilbert-type inequality with parameters

Journal of Inequalities and Applications20152015:371

https://doi.org/10.1186/s13660-015-0898-7

Received: 17 July 2015

Accepted: 10 November 2015

Published: 1 December 2015

Abstract

In this paper, by the use of the way of weight coefficients, the transfer formula, and the technique of real analysis, we introduce some proper parameters and obtain a multidimensional Hilbert-type inequality with the following kernel:
$$ \prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac {\lambda +\gamma}{s}}} $$
and a best possible constant factor. The equivalent form, the operator expressions with the norm, and some particular cases are also considered. The lemmas and theorems provide an extensive account of this type of inequalities.

Keywords

Hilbert-type inequality weight coefficient equivalent form operator norm

MSC

26D15 47A07

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. [16]).

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 [826].

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).

Declarations

Acknowledgements

This work is supported by Hunan Province Natural Science Foundation (No. 2015JJ4041), and Science Research General Foundation Item of Hunan Institution of Higher Learning College and University (No. 14C0938).

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)
Normal College of Jishou University
(2)
Department of Mathematics, Guangdong University of Education

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