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]): 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 \(\frac {\pi}{\sin(\pi/p)}\): Inequalities (1) and (2) are important in analysis and its applications (cf. [1‐6]).
$$ \int_{0}^{\infty} \int_{0}^{\infty}\frac{f(x)g(y)}{x+y}\,dx\,dy< \frac{\pi }{\sin(\pi/p)} \|f\|_{p}\|g\|_{q}, $$
(1)
$$ \sum_{m=1}^{\infty}\sum _{n=1}^{\infty}\frac{a_{m}b_{n}}{m+n}< \frac {\pi}{\sin(\pi/p)}\|a \|_{p}\|b\|_{q}. $$
(2)
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:
Anzeige
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
\(\phi(x)=x^{p(1-\lambda_{1})-1}\), \(\psi(x)=x^{q(1-\lambda _{2})-1}\), \(f(x),g(y)\geq0\),
\(g\in L_{q,\psi}(\mathbf{R}_{+})\), \(\|f\|_{p,\phi},\|g\|_{q,\psi}>0\), then we have 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 \(a_{m},b_{n}\geq0\),
\(b=\{b_{n}\}_{n=1}^{\infty}\in l_{q,\psi}\), \(\|a\|_{p,\phi },\|b\|_{q,\psi }>0\), we have the following inequality: where the constant factor \(k(\lambda_{1})\) is still the best possible.
$$ k(\lambda_{1})= \int_{0}^{\infty}k_{\lambda}(t,1)t^{\lambda _{1}-1}\,dt \in \mathbf{R}_{+}, $$
$$ f\in L_{p,\phi}(\mathbf{R}_{+})= \biggl\{ f;\|f \|_{p,\phi }:=\biggl( \int_{0}^{\infty}\phi(x)\bigl|f(x)\bigr|^{p}\,dx \biggr)^{\frac{1}{p}}< \infty \biggr\} , $$
$$ \int_{0}^{\infty} \int_{0}^{\infty}k_{\lambda }(x,y)f(x)g(y)\,dx\,dy< k( \lambda _{1})\|f\|_{p,\phi}\|g\|_{q,\psi}, $$
(3)
$$ a\in l_{p,\phi}= \Biggl\{ a;\|a\|_{p,\phi}:=\Biggl(\sum _{n=1}^{\infty}\phi (n)|a_{n}|^{p} \Biggr)^{\frac{1}{p}}< \infty \Biggr\} , $$
$$ \sum_{m=1}^{\infty}\sum _{n=1}^{\infty}k_{\lambda }(m,n)a_{m}b_{n}< k( \lambda_{1})\|a\|_{p,\phi}\|b\|_{q,\psi}, $$
(4)
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 The equivalent form, the operator expressions with the norm, and some particular cases are also considered.
$$ k_{\lambda}(m,n)=\prod_{k=1}^{s} \frac{(\min\{m,c_{k}n\})^{\frac {\gamma}{s}}}{(\max\{m,c_{k}n\})^{\frac{\lambda+\gamma}{s}}}. $$
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)
Anzeige
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 it follows that In the same way, we still have Hence, choosing plus for the above two inequalities, we have (6). □
$$\begin{aligned}& \int_{n}^{n+1}g(t)\,dt \leq g(n)\leq \int_{n-1}^{n}g(t)\,dt\quad(n=1,\ldots,n_{0}),\\& \int_{n_{0}+1}^{n_{0}+2}g(t)\,dt < g(n_{0}+1)< \int_{n_{0}}^{n_{0}+1}g(t)\,dt, \end{aligned}$$
$$ 0< \int_{1}^{n_{0}+2}g(t)\,dt< \sum _{n=1}^{n_{0}+1}g(n)< \sum_{n=1}^{n_{0}+1} \int_{n-1}^{n}g(t)\,dt= \int_{0}^{n_{0}+1}g(t)\,dt< \infty. $$
$$ 0< \int_{n_{0}+2}^{\infty}g(t)\,dt\leq\sum _{n=n_{0}+2}^{\infty}g(n)\leq \int_{n_{0}+1}^{\infty}g(t)\,dt< \infty. $$
Lemma 2
If
\(s\in\mathbf{N}\), \(\gamma,M>0\), \(\Psi(u)\)
is a non-negative measurable function in
\((0,1]\), and
then we have the following transfer formula (cf. [27]):
$$ D_{M}:= \Biggl\{ x\in\mathbf{R}_{+}^{s};\sum _{i=1}^{s}x_{i}^{\gamma} \leq M^{\gamma} \Biggr\} , $$
$$\begin{aligned} &\int\cdots \int_{D_{M}}\Psi \Biggl( \sum_{i=1}^{s} \biggl( \frac {x_{i}}{M} \biggr) ^{\gamma} \Biggr)\,dx_{1}\cdots \,dx_{s} \\ &\quad=\frac{M^{s}\Gamma^{s}(\frac{1}{\gamma})}{\gamma^{s}\Gamma(\frac {s}{\gamma})} \int_{0}^{1}\Psi(u)u^{\frac{s}{\gamma}-1}\,du. \end{aligned}$$
(7)
Lemma 3
For
\(s\in\mathbf{N}\), \(\gamma, \varepsilon>0\), we have
where
\(\sum_{m}=\sum_{m_{s}=1}^{\infty}\cdots\)
\(\sum_{m_{1}=1}^{\infty}\).
$$ \sum_{m}\|m\|_{\gamma}^{-s-\varepsilon}= \frac{\Gamma^{s}(\frac {1}{\gamma})}{\varepsilon s^{\varepsilon/\gamma}\gamma^{s-1}\Gamma(\frac {s}{\gamma})}+O(1) \quad\bigl(\varepsilon\rightarrow0^{+}\bigr), $$
(8)
Proof
For \(M>s^{1/\gamma}\), we set Then by Lemma 1 and (7), it follows that
$$ \Psi(u)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0, & 0< u< \frac{s}{M^{\gamma}}, \\ (Mu^{1/\gamma})^{-s-\varepsilon}, &\frac{s}{M^{\gamma}}\leq u\leq1.\end{array}\displaystyle \right . $$
$$\begin{aligned} \sum_{m}\|m\|_{\gamma}^{-s-\varepsilon} \geq& \int_{\{x\in\mathbf{R} _{+}^{s};x_{i}\geq1\}}\|x\|_{\gamma}^{-s-\varepsilon}\,dx \\ =&\lim_{M\rightarrow\infty} \int\cdots \int_{D_{M}}\Psi \Biggl( \sum_{i=1}^{s} \biggl( \frac{x_{i}}{M} \biggr) ^{\gamma} \Biggr)\,dx_{1} \cdots \,dx_{s} \\ =&\lim_{M\rightarrow\infty}\frac{M^{s}\Gamma^{s}(\frac{1}{\gamma })}{\gamma^{s}\Gamma(\frac{s}{\gamma})} \int_{s/M^{\gamma }}^{1}\bigl(Mu^{1/\gamma } \bigr)^{-s-\varepsilon}u^{\frac{s}{\gamma}-1}\,du \\ =&\frac{\Gamma^{s}(\frac{1}{\gamma})}{\varepsilon s^{\varepsilon /\gamma }\gamma^{s-1}\Gamma(\frac{s}{\gamma})}. \end{aligned}$$
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 \(s\geq2\), and then Then we have (8). □
$$\begin{aligned} 0 < &\sum_{\{m\in\mathbf{N}^{s};\exists i_{0},m_{i_{0}}=1\} }\|m\|_{\gamma }^{-s-\varepsilon} \leq a+\sum_{\{m\in\mathbf{N}^{s-1};m_{i}\geq 2\}}\|m\|_{\gamma}^{-(s-1)-(1+\varepsilon)} \\ \leq&a+\frac{\Gamma^{s-1}(\frac{1}{\gamma})}{(1+\varepsilon )(s-1)^{(1+\varepsilon)/\gamma}\gamma^{s-2}\Gamma(\frac{s-1}{\gamma })}< \infty\quad(a\in\mathbf{R}_{+}), \end{aligned}$$
$$\begin{aligned} \sum_{m}\|m\|_{\gamma}^{-s-\varepsilon}&=\sum _{\{m\in\mathbf{N}^{s};\exists i_{0},m_{i_{0}}=1\}}\|m\|_{\gamma}^{-s-\varepsilon }+\sum _{\{m\in\mathbf{N}^{s};m_{i}\geq2\}}\|m\|_{\gamma }^{-s-\varepsilon} \\ &\leq O_{1}(1)+\frac{\Gamma^{s}(\frac{1}{\gamma})}{\varepsilon s^{\varepsilon/\gamma}\gamma^{s-1}\Gamma(\frac{s}{\gamma })}\quad\bigl(\varepsilon \rightarrow0^{+}\bigr). \end{aligned}$$
(9)
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 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 and we still indicate \(\ln(\frac{c_{i_{0}+1}}{c_{i_{0}}})\) by the following formal expression:
$$ \int_{c_{i}}^{c_{i+1}}t^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac {2i}{s})\gamma-1}\,dt=\frac{c_{i+1}^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac {2i}{s})\gamma}-c_{i}^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac {2i}{s})\gamma}}{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac{2i}{s})\gamma}; $$
$$ \int_{c_{i_{0}}}^{c_{i_{0}+1}}t^{\lambda_{1}-\frac{i_{0}\lambda }{s}+(1-\frac{2i_{0}}{s})\gamma-1}\,dt=\ln\biggl( \frac{c_{i_{0}+1}}{c_{i_{0}}}\biggr)=\lim_{i\rightarrow i_{0}} \int_{c_{i}}^{c_{i+1}}t^{\lambda_{1}-\frac{i\lambda}{s}+(1-\frac{2i}{s})\gamma-1}\,dt, $$
$$ \frac{c_{i_{0}+1}^{\lambda_{1}-\frac{i_{0}\lambda}{s}+(1-\frac {2i_{0}}{s})\gamma}-c_{i_{0}}^{\lambda_{1}-\frac{i_{0}\lambda}{s}+(1-\frac {2i_{0}}{s})\gamma}}{\lambda_{1}-\frac{i_{0}\lambda}{s}+(1-\frac {2i_{0}}{s})\gamma}. $$
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 \(\lambda_{2}=-\lambda_{1}\), \(|\lambda _{1}|<\gamma\) (\(\gamma>0\)), and
$$ k_{0}(x,y)=\prod_{k=1}^{s} \biggl( \frac{\min\{x,c_{k}y\}}{\max\{ x,c_{k}y\}} \biggr) ^{\frac{\gamma}{s}}, $$
$$\begin{aligned} k_{s}(\lambda_{1}) =&k_{s}^{(0)}( \lambda_{1}):=\frac{c_{1}^{\lambda _{1}+\gamma}}{\gamma+\lambda_{1}}\frac{1}{\prod_{k=1}^{s}c_{k}^{\frac {\gamma}{s}}}+ \frac{c_{s}^{\lambda_{1}-\gamma}}{\gamma-\lambda_{1}}\prod_{k=1}^{s}c_{k}^{\frac{\gamma}{s}} \\ &{}+\sum_{i=1}^{s-1} \biggl[ \frac{c_{i+1}^{\lambda_{1}+(1-\frac {2i}{s})\gamma }-c_{i}^{\lambda_{1}+(1-\frac{2i}{s})\gamma}}{\lambda_{1}+(1-\frac {2i}{s})\gamma}\frac{\prod_{k=1}^{i}c_{k}^{\frac{\gamma}{s}}}{\prod_{k=i+1}^{s}c_{k}^{\frac{\gamma}{s}}} \biggr] . \end{aligned}$$
(15)
(b) Since for \(j_{0}\in\mathbf{N,}\) we find 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 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.
$$\begin{aligned} k_{\lambda}(x,y)\frac{1}{y^{j_{0}-\lambda_{2}}}&=\frac {1}{y^{j_{0}-\lambda _{2}}}\prod _{k=1}^{s}\frac{(\min\{c_{k}^{-1}x,y\})^{\frac{\gamma }{s}}}{c_{k}^{\frac{\lambda}{s}}(\max\{c_{k}^{-1}x,y\})^{\frac{\lambda +\gamma}{s}}}\\ &=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{1}{y^{j_{0}-\lambda_{2}-\gamma}}\prod_{k=1}^{s}\frac {1}{c_{k}^{\frac{\lambda}{s}}(c_{k}^{-1}x)^{\frac{\lambda+\gamma}{s}}}, &0< y\leq c_{s}^{-1}x, \\ \frac{1}{y^{j_{0}+\lambda_{1}+\gamma-\frac{i}{s}(\lambda+2\gamma )}}\frac{\prod_{k=i+1}^{s}(c_{k}^{-1}x)^{\frac{\gamma}{s}}}{\prod_{k=1}^{s}c_{k}^{\frac{\lambda}{s}}\prod_{k=1}^{i}(c_{k}^{-1}x)^{\frac{\lambda+\gamma }{s}}},& c_{i+1}^{-1}x< y\leq c_{i}^{-1}x\ (i=1,\ldots,s-1), \\ \frac{1}{y^{j_{0}+\lambda_{1}+\gamma}}\prod_{k=1}^{s}\frac {(c_{k}^{-1}x)^{\frac{\gamma}{s}}}{c_{k}^{\frac{\lambda}{s}}(y)^{\frac{\lambda +\gamma}{s}}},& c_{1}^{-1}x< y< \infty,\end{array}\displaystyle \right . \end{aligned}$$
$$\begin{aligned} k_{\lambda}(x,y)\frac{1}{x^{i_{0}-\lambda_{1}}}&=\frac {1}{x^{i_{0}-\lambda _{1}}}\prod _{k=1}^{s}\frac{(\min\{x,c_{k}y\})^{\frac{\gamma }{s}}}{(\max \{x,c_{k}y\})^{\frac{\lambda+\gamma}{s}}}\\ &=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{1}{x^{i_{0}-\lambda_{1}-\gamma}}\prod_{k=1}^{s}\frac {1}{(c_{k}y)^{\frac{\lambda+\gamma}{s}}},& 0< x\leq c_{1}y, \\ \frac{1}{x^{i_{0}-\lambda_{1}-\gamma+\frac{i}{s}(\lambda+2\gamma )}}\frac{\prod_{k=1}^{i}(c_{k}y)^{\frac{\gamma}{s}}}{\prod_{k=i+1}^{s}(c_{k}y)^{ \frac{\lambda+\gamma}{s}}},& c_{i}y< x\leq c_{i+1}y \ (i=1,\ldots,s-1), \\ \frac{1}{x^{i_{0}+\lambda_{2}+\gamma}}\prod_{k=1}^{s}(c_{k}y)^{\frac{\gamma}{s}},& c_{s}y< x< \infty,\end{array}\displaystyle \right . \end{aligned}$$
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:
where \(\sum_{m}=\sum_{m_{i_{0}}=1}^{\infty}\cdots\sum_{m_{1}=1}^{\infty}\) and \(\sum_{n}=\sum_{n_{j_{0}}=1}^{\infty}\cdots\sum_{n_{1}=1}^{\infty}\).
$$\begin{aligned}& w(\lambda_{1},n) :=\sum_{m}\prod _{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}} \frac{\|n\|_{\beta }^{\lambda_{2}}}{\|m\|_{\alpha}^{i_{0}-\lambda_{1}}}, \end{aligned}$$
(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)
Lemma 4
As the assumptions of Definition
1, then (i) we have
where
$$\begin{aligned}& w(\lambda_{1},n) < K_{2}^{(s)}\quad\bigl(n\in \mathbf{N}^{j_{0}}\bigr), \end{aligned}$$
(18)
$$\begin{aligned}& W(\lambda_{2},m) < K_{1}^{(s)}\quad\bigl(m\in \mathbf{N}^{i_{0}}\bigr), \end{aligned}$$
(19)
$$ K_{1}^{(s)}=\frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta ^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})}k_{s}( \lambda_{1}),\qquad K_{2}^{(s)}=\frac{\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})}k_{s}( \lambda_{1}); $$
(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
where
$$ 0< \widetilde{K}_{2}^{(s)}\bigl(1-\widetilde{ \theta}_{\lambda }(n)\bigr)< w(\widetilde{\lambda}_{1},n), $$
(21)
$$\begin{aligned}& 0 < \widetilde{\theta}_{\lambda}(n)=\frac{1}{k_{s}(\widetilde {\lambda}_{1})} \int_{0}^{i_{0}^{1/\alpha}/\|n\|_{\beta}}\prod_{k=1}^{s} \frac {(\min \{v,c_{k}\})^{\frac{\gamma}{s}}v^{\widetilde{\lambda}_{1}-1}}{(\max \{v,c_{k}\})^{\frac{\lambda+\gamma}{s}}}\,dv =O \biggl( \frac{1}{\|n\|_{\beta}^{\gamma+\widetilde{\lambda}_{1}}} \biggr) , \end{aligned}$$
(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 Hence, we have (18). In the same way, we have (19).
$$\begin{aligned} w(\lambda_{1},n) &< \int_{\mathbf{R}_{+}^{i_{0}}}\prod_{k=1}^{s} \frac {(\min \{\|x\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max \{\|x\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma }{s}}}\frac{\|n\|_{\beta}^{\lambda_{2}}}{\|x\|_{\alpha}^{i_{0}-\lambda_{1}}}\,dx\\ &=\lim_{M\rightarrow\infty} \int_{\mathbf{D}_{M}}\prod_{k=1}^{s} \frac{ (\min\{M[\sum_{i=1}^{i_{0}}(\frac{x_{i}}{M})^{\alpha}]^{\frac {1}{\alpha}},c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{M[\sum_{i=1}^{i_{0}}(\frac{x_{i}}{M})^{\alpha}]^{\frac{1}{\alpha}},c_{k}\|n\|_{\beta}\})^{ \frac{\lambda+\gamma}{s}}}\frac{M^{\lambda_{1}-i_{0}}\|n\|_{\beta }^{\lambda_{2}}\,dx}{[\sum_{i=1}^{i_{0}}(\frac{x_{i}}{M})^{\alpha }]^{\frac{i_{0}-\lambda_{1}}{\alpha}}}\\ &=\lim_{M\rightarrow\infty}\frac{M^{i_{0}}\Gamma^{i_{0}}(\frac {1}{\alpha })}{\alpha^{i_{0}}\Gamma(\frac{i_{0}}{\alpha})} \int_{0}^{1}\prod_{k=1}^{s}\frac{(\min\{Mu^{1/\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma }{s}}}{(\max\{Mu^{1/\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma }{s}}}\frac{\|n\|_{\beta}^{\lambda_{2}}u^{\frac{i_{0}}{\alpha}-1}\,du}{M^{i_{0}-\lambda_{1}}u^{(i_{0}-\lambda_{1})/\alpha}} \\ &=\lim_{M\rightarrow\infty}\frac{M^{\lambda_{1}}\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}}\Gamma(\frac{i_{0}}{\alpha})} \int_{0}^{1}\prod _{k=1}^{s}\frac{(\min\{Mu^{1/\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\gamma}{s}}\|n\|_{\beta}^{\lambda_{2}}}{(\max\{Mu^{1/\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}u^{\frac{\lambda _{1}}{\alpha}-1}\,du \\ &\overset{u=\|n\|_{\beta}^{\alpha}M^{-\alpha}v^{\alpha}}{=}\frac {\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}-1}\Gamma(\frac {i_{0}}{\alpha})}\int_{0}^{\infty}\prod_{k=1}^{s} \frac{(\min\{v,c_{k}\})^{\frac{\gamma }{s}}v^{\lambda_{1}-1}}{(\max\{v,c_{k}\})^{\frac{\lambda+\gamma}{s}}}\,dv \\ &=\frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\alpha^{i_{0}-1}\Gamma (\frac{i_{0}}{\alpha})}k_{s}(\lambda_{1})=K_{2}^{(s)}. \end{aligned}$$
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 and then (22) follows. □
$$ \widetilde{\theta}_{\lambda}(n)=\frac{1}{k_{s}(\widetilde{\lambda }_{1})}\int_{0}^{i_{0}^{1/\alpha}/\|n\|_{\beta}}\frac{v^{\widetilde{\lambda} _{1}+\gamma-1}\,dv}{\prod_{k=1}^{s}c_{k}{}^{\frac{\lambda+\gamma}{s}}}= \frac{(\prod_{k=1}^{s}c_{k}{}^{\frac{\lambda+\gamma}{s}})^{-1}}{(\widetilde{\lambda}_{1}+\gamma)k_{s}(\widetilde{\lambda}_{1})} \biggl( \frac{i_{0}^{1/\alpha}}{\|n\|_{\beta}} \biggr) ^{\widetilde{\lambda}_{1}+\gamma}, $$
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:
where the constant factor
is the best possible. In particular, for
\(s=1\) (or
\(c_{s}=\cdots=c_{1}\)), we have the following inequality:
where
$$\begin{aligned} I :=&\sum_{n}\sum_{m} \prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}a_{m}b_{n} \\ < &\bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi} \|b\|_{q,\Psi}, \end{aligned}$$
(24)
$$ \bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}}\bigl(K_{2}^{(s)} \bigr)^{\frac{1}{q}}= \biggl[ \frac {\Gamma ^{j_{0}}(\frac{1}{\beta})}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta })} \biggr] ^{\frac{1}{p}} \biggl[ \frac{\Gamma^{i_{0}}(\frac{1}{\alpha })}{\beta ^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})} \biggr] ^{\frac {1}{q}}k_{s}( \lambda _{1}) $$
(25)
$$ \sum_{n}\sum_{m} \frac{(\min\{\|m\|_{\alpha},c_{1}\|n\|_{\beta}\} )^{\gamma }a_{m}b_{n}}{(\max\{\|m\|_{\alpha},c_{1}\|n\|_{\beta}\})^{\lambda +\gamma }}< \bigl(K_{1}^{(1)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(1)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi } \|b\|_{q,\Psi}, $$
(26)
$$\begin{aligned} &\bigl(K_{1}^{(1)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(1)}\bigr)^{\frac{1}{q}} \\ &\quad= \biggl[ \frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta ^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac{1}{p}} \biggl[ \frac{\Gamma ^{i_{0}}(\frac{1}{\alpha})}{\beta^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha })} \biggr] ^{\frac{1}{q}}\frac{(\lambda+2\gamma)c_{1}^{-\lambda_{2}}}{(\lambda _{1}+\gamma)(\lambda_{2}+\gamma)}. \end{aligned}$$
(27)
Proof
By Hölder’s inequality (cf. [28]), we have Then by (18) and (19), we have (24).
$$\begin{aligned} I =&\sum_{n}\sum_{m} \prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}} \biggl[ \frac{\|m\|_{\alpha}^{(i_{0}-\lambda _{1})/q}}{\|n\|_{\beta }^{(j_{0}-\lambda_{2})/p}}a_{m} \biggr] \biggl[ \frac{\|n\|_{\beta }^{(j_{0}-\lambda_{2})/p}}{\|m\|_{\alpha}^{(i_{0}-\lambda _{1})/q}}b_{n} \biggr]\\ \leq& \biggl\{ \sum_{m}W(\lambda_{2},m) \|m\|_{\alpha}^{p(i_{0}-\lambda _{1})-i_{0}}a_{m}^{p} \biggr\} ^{\frac{1}{p}} \biggl\{ \sum_{n}w(\lambda_{1},n) \|n\|_{\beta }^{q(j_{0}-\lambda _{2})-j_{0}}b_{n}^{q} \biggr\} ^{\frac{1}{q}}. \end{aligned}$$
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 Then by (8) and (21), we obtain
$$ \widetilde{a}_{m}=\|m\|_{\alpha}^{-i_{0}+\lambda_{1}-\frac {\varepsilon}{p}}=\|m \|_{\alpha}^{\widetilde{\lambda}_{1}-i_{0}}, \qquad\widetilde{b}_{n}=\|n \|_{\beta}^{\widetilde{\lambda}_{2}-j_{0}-\varepsilon} \quad\bigl(m\in \mathbf{N}^{i_{0}},n\in \mathbf{N}^{j_{0}}\bigr). $$
$$\begin{aligned}& \begin{aligned}[b] \|\widetilde{a}\|_{p,\Phi}\|\widetilde{b}\|_{q,\Psi}={}& \biggl[ \sum _{m}\|m\|_{\alpha}^{p(i_{0}-\lambda_{1})-i_{0}}\widetilde {a}_{m}^{p} \biggr] ^{\frac{1}{p}} \biggl[ \sum _{n}\|n\|_{\beta}^{q(j_{0}-\lambda _{2})-j_{0}} \widetilde{b}_{n}^{q} \biggr] ^{\frac{1}{q}} \\ ={}& \biggl( \sum_{m}\|m\|_{\alpha}^{-i_{0}-\varepsilon} \biggr) ^{\frac {1}{p}} \biggl( \sum_{n}\|n \|_{\beta}^{-j_{0}-\varepsilon} \biggr) ^{\frac{1}{q}} \\ ={}&\frac{1}{\varepsilon} \biggl( \frac{\Gamma^{i_{0}}(\frac{1}{\alpha })}{i_{0}^{\varepsilon/\alpha}\alpha^{i_{0}-1}\Gamma(\frac {i_{0}}{\alpha})}+\varepsilon O(1) \biggr) ^{\frac{1}{p}} \\ &{}\times \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta })}{j_{0}^{\varepsilon /\beta}\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} +\varepsilon \widetilde{O}(1) \biggr) ^{\frac{1}{q}}, \end{aligned} \end{aligned}$$
(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 For \(\varepsilon\rightarrow0^{+}\), we find 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). □
$$\begin{aligned} &\bigl(K_{2}^{(s)}+o(1)\bigr) \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{j_{0}^{\varepsilon/\beta}\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta })}+ \varepsilon\widetilde{O}(1)-\varepsilon O(1) \biggr)< \varepsilon \widetilde{I} < \varepsilon K\|\widetilde{a}\|_{p,\varphi}\|\widetilde{b} \|_{q,\psi } \\ &\quad=K \biggl( \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{i_{0}^{\varepsilon /\alpha}\alpha^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})}+\varepsilon O(1) \biggr) ^{\frac{1}{p}} \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta })}{j_{0}^{\varepsilon /\beta}\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})}+\varepsilon \widetilde{O}(1) \biggr) ^{\frac{1}{q}}. \end{aligned}$$
$$\begin{aligned} &\frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta^{j_{0}-1}\Gamma(\frac {j_{0}}{\beta})}\frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\alpha ^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})}k_{s}(\lambda_{1}) \leq K \biggl( \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\alpha ^{i_{0}-1}\Gamma(\frac{i_{0}}{\alpha})} \biggr) ^{\frac{1}{p}} \biggl( \frac{\Gamma^{j_{0}}(\frac{1}{\beta})}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{ \beta})} \biggr) ^{\frac{1}{q}}, \end{aligned}$$
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
\((K_{1}^{(s)})^{\frac{1}{p}}(K_{2}^{(s)})^{\frac{1}{q}}\):
which is equivalent to (24). In particular, for
\(s=1\) (or
\(c_{s}=\cdots=c_{1}\)), we have the following inequality equivalent to (26):
$$\begin{aligned} J :=& \Biggl\{ \sum_{n}\|n\|_{\beta}^{p\lambda_{2}-j_{0}} \Biggl[ \sum_{m}\prod _{k=1}^{s}\frac{(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta }\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\} )^{\frac{\lambda+\gamma}{s}}}a_{m} \Biggr] ^{p} \Biggr\} ^{\frac{1}{p}} \\ < &\bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi}, \end{aligned}$$
(30)
$$\begin{aligned} & \biggl\{ \sum_{n}\|n\|_{\beta}^{p\lambda_{2}-j_{0}} \biggl[ \sum_{m}\frac{(\min\{\|m\|_{\alpha},c_{1}\|n\|_{\beta}\})^{\gamma}}{(\max \{\|m\|_{\alpha},c_{1}\|n\|_{\beta}\})^{\lambda+\gamma}}a_{m} \biggr] ^{p} \biggr\} ^{\frac{1}{p}} < \bigl(K_{1}^{(1)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(1)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi}. \end{aligned}$$
(31)
Proof
We set \(b_{n}\) as follows: 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 namely, and then (30) follows.
$$ b_{n}:=\|n\|_{\beta}^{p\lambda_{2}-j_{0}} \Biggl( \sum _{m}\prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma }{s}}}{(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma }{s}}}a_{m} \Biggr) ^{p-1},\quad n\in \mathbf{N}^{j_{0}}. $$
$$ \|b\|_{q,\Psi}^{q}=J^{p}=I< \bigl(K_{1}^{(s)} \bigr)^{\frac {1}{p}}\bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}} \|a\|_{p,\Phi}\|b\|_{q,\Psi}, $$
$$ \|b\|_{q,\Psi}^{q-1}=J< \bigl(K_{1}^{(s)} \bigr)^{\frac{1}{p}}\bigl(K_{2}^{(s)}\bigr)^{\frac {1}{q}} \|a\|_{p,\Phi}, $$
On the other hand, assuming that (30) is valid, by Hölder’s inequality, we have Then by (30), we have (24). Hence (30) and (24) are equivalent.
$$\begin{aligned} I =&\sum_{n}\bigl(\Psi(n)\bigr)^{\frac{-1}{q}} \Biggl[ \sum_{m}\prod_{k=1}^{s} \frac{(\min\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\gamma }{s}}a_{m}}{(\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma }{s}}} \Biggr] \bigl[\bigl(\Psi(n)\bigr)^{\frac{1}{q}}b_{n} \bigr] \\ \leq&J\|b\|_{q,\Psi}. \end{aligned}$$
(32)
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 For \(b\in l_{q,\Psi}\), we define the following formal inner product of Ta and b as follows:
$$ Ta(n):=\sum_{m}\prod _{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}a_{m}\quad\bigl(n \in\mathbf {N}^{j_{0}}\bigr). $$
(33)
$$ (Ta,b):=\sum_{n}\sum_{m} \prod_{k=1}^{s}\frac{(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\gamma}{s}}}{(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda+\gamma}{s}}}a_{m}b_{n}. $$
(34)
Then by Theorem 1 and Theorem 2, for \(0<\|a\|_{p,\varphi},\|b\|_{q,\psi }<\infty\), we have the following equivalent inequalities:
It follows that T is bounded with 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.
$$\begin{aligned}& (Ta,b) < \bigl(K_{1}^{(s)}\bigr)^{\frac{1}{p}} \bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}}\|a\|_{p,\Phi} \|b\|_{q,\Psi}, \end{aligned}$$
(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)
$$ \|T\|:=\sup_{a(\neq\theta)\in l_{p,\Phi}}\frac{\|Ta\|_{p,\Psi ^{1-p}}}{\|a\|_{p,\Phi}}\leq\bigl(K_{1}^{(s)} \bigr)^{\frac{1}{p}}\bigl(K_{2}^{(s)}\bigr)^{\frac{1}{q}}. $$
(37)
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 Hence, (24) is an extension of (4) for
$$ \sum_{m=1}^{\infty}\sum _{n=1}^{\infty}\prod_{k=1}^{s} \frac{(\min \{m,c_{k}n\})^{\frac{\gamma}{s}}}{(\max\{m,c_{k}n\})^{\frac{\lambda +\gamma}{s}}}a_{m}b_{n}< k_{s}( \lambda_{1})\|a\|_{p,\phi }\|b\|_{q,\psi}. $$
(39)
$$ k_{\lambda}(m,n)=\prod_{k=1}^{s} \frac{(\min\{m,c_{k}n\})^{\frac {\gamma}{s}}}{(\max\{m,c_{k}n\})^{\frac{\lambda+\gamma}{s}}}. $$
(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:
where the best possible constant factor is defined by and \(\widetilde{k}_{s}(\lambda_{1})\) is indicated by (13).
$$\begin{aligned}& \sum_{n}\sum_{m} \frac{a_{m}b_{n}}{\prod_{k=1}^{s}(\max\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda}{s}}}< \widetilde{K}_{s}(\lambda _{1})\|a \|_{p,\Phi}\|b\|_{q,\Psi}, \end{aligned}$$
(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)
$$ \widetilde{K}_{s}(\lambda_{1}):= \biggl[ \frac{\Gamma^{j_{0}}(\frac {1}{\beta })}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac{1}{p}} \biggl[ \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\beta^{i_{0}-1}\Gamma (\frac{i_{0}}{\alpha})} \biggr] ^{\frac{1}{q}}\widetilde{k}_{s}(\lambda_{1}), $$
(42)
(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:
where the best possible constant factor is defined by and \(\widehat{k}_{s}(\lambda_{1})\) is indicated by (14).
$$\begin{aligned}& \sum_{n}\sum_{m} \frac{a_{m}b_{n}}{\prod_{k=1}^{s}(\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\})^{\frac{\lambda}{s}}}< \widehat{K}_{s}(\lambda _{1})\|a \|_{p,\Phi}\|b\|_{q,\Psi}, \end{aligned}$$
(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)
$$ \widehat{K}_{s}(\lambda_{1}):= \biggl[ \frac{\Gamma^{j_{0}}(\frac {1}{\beta})}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac {1}{p}} \biggl[ \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\beta^{i_{0}-1}\Gamma(\frac {i_{0}}{\alpha})} \biggr] ^{\frac{1}{q}} \widehat{k}_{s}(\lambda_{1}), $$
(45)
(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:
where the best possible constant factor is defined by and \(k_{s}^{(0)}(\lambda_{1})\) is indicated by (15).
$$\begin{aligned}& \sum_{n}\sum_{m} \prod_{k=1}^{s} \biggl( \frac{\min\{\|m\|_{\alpha },c_{k}\|n\|_{\beta}\}}{\max\{\|m\|_{\alpha},c_{k}\|n\|_{\beta}\}} \biggr) ^{\frac{\gamma}{s}}a_{m}b_{n}< K_{s}^{(0)}( \lambda _{1})\|a\|_{p,\Phi}\|b\|_{q,\Psi}, \end{aligned}$$
(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)
$$ K_{s}^{(0)}(\lambda_{1}):= \biggl[ \frac{\Gamma^{j_{0}}(\frac{1}{\beta })}{\beta^{j_{0}-1}\Gamma(\frac{j_{0}}{\beta})} \biggr] ^{\frac {1}{p}} \biggl[ \frac{\Gamma^{i_{0}}(\frac{1}{\alpha})}{\beta^{i_{0}-1}\Gamma(\frac {i_{0}}{\alpha})} \biggr] ^{\frac{1}{q}}k_{s}^{(0)}(\lambda_{1}), $$
(48)
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. YS participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.