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A Hilbert-type operator with a symmetric homogeneous kernel of two parameters and its applications
Journal of Inequalities and Applications volume 2015, Article number: 266 (2015)
Abstract
We introduce a general homogeneous kernel whose degree is given by two parameters to establish the equivalent inequalities with the norm of a new Hilbert-type operator. As applications, we provide new extended Hilbert-type inequalities with the best possible constant factors.
1 Introduction
Let \(\{a_{n}\}\) and \(\{b_{m}\}\) be two sequences of nonnegative real numbers. The well-known Hilbert’s inequality says that if \(p > 1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(0< \sum_{m=1}^{\infty}a_{m} ^{p} < \infty\) and \(0 < \sum_{n=1}^{\infty}b_{n} ^{q} < \infty\), then
where the constant factor \(\frac{\pi}{\sin(\frac{\pi}{p})}\) is the best possible [1]. This inequality has been generalized in numerous ways with introducing suitable parameters and weight coefficients. (For example, see [2–13] and the references therein.) In particular, by introducing a Hilbert-type linear operator with a symmetric homogeneous kernel, one can obtain various Hilbert-type inequalities with the best constant factors. For this purpose, let \(k(x, y)\) be a nonnegative symmetric function defined on \((0,\infty )\times(0,\infty)\), i.e., \(k(x,y)=k(y,x)\). For \(p>1\) and \(\frac {1}{p} +\frac{1}{q}=1\), let \(\ell^{r}\) (\(r=p, q\)) be two normed spaces. If T is a bounded self-adjoint semi-positive definite operator defined by
for \(a=\{a_{m}\}_{m=1}^{\infty}\in\ell^{p}\), or similarly,
for \(b=\{b_{n}\}_{n=1}^{\infty}\in\ell^{q}\). The operator T is called the Hilbert-type operator and the function \(k(x,y)\) is called the symmetric kernel of T. In view of this point, Hilbert’s inequality (1) can be expressed by
where the kernel \(k(x,y)=\frac{1}{x+y}\) and the formal inner product \((Ta, b)\) between Ta and b is given by \((Ta, b):= \sum_{n=1}^{\infty}(Ta)(n)b_{n}\). Motivated by this observation, Yang [14] defined a Hilbert-type linear operator \(T: \ell^{r} \rightarrow\ell^{r}\) (\(r=p,q\)) with the kernel \(k(x,y)=\frac{(xy)^{\frac{\lambda -1}{2}}}{(x+y)^{\lambda}}\) of degree −1. As a consequence, he was able to prove that if \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(a_{m}, b_{n} \geq0\), \(1-2\min\{\frac{1}{p}, \frac{1}{q}\} <\lambda< 1+2\min\{\frac{1}{p}, \frac{1}{q}\}\), then the following two inequalities are equivalent:
where \(B(u,v)\) denotes the beta function defined by
Moreover, the constant factor \(\frac{1}{\lambda}B (\frac{q(\lambda +1)-2}{2q\lambda}, \frac{p(\lambda+1)-2}{2p\lambda} )\) is the best possible. In 2010, Jin and Debnath [15] generalized the Hilbert-type linear operator whose kernel is symmetric and homogeneous of degree −1. In fact, they obtained several extended Hilbert-type inequalities by using the kernel \(k(x,y)=\frac{1}{(x^{\frac{1}{\lambda }}+y^{\frac{1}{\lambda}})^{\lambda}} \) (\(\lambda>0\)). For instance, they proved that if \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\alpha, \beta>0\), \(0 <\lambda\leq\min\{\frac{q}{\alpha}, \frac{p}{\beta}\}\), then the following two inequalities are equivalent:
where the constant factor \(\frac{B(\frac{\lambda}{p}, \frac{\lambda }{q})}{\alpha^{\frac{1}{q}}\beta^{\frac{1}{p}}}\) is the best possible. See [16–23] for other Hilbert-type operators and the corresponding extended Hilbert-type inequalities with the best factors.
Most of the previous results were, however, obtained by using the Hilbert-type operator with the symmetric homogeneous kernel of −λ-order, which depends on a parameter \(\lambda>0\). In this paper, we introduce a more general homogeneous kernel whose degree is given by two parameters (Definition 2.3). We establish the equivalent inequalities with the norm of a new Hilbert-type operator (Theorem 3.1). As applications, we provide new extended Hilbert-type inequalities with the best possible constant factors (Corollary 4.1 and Cases 1-3).
2 Hilbert-type operator with a symmetric homogeneous kernel whose degree is given by two parameters
For completeness, we begin with the following definitions and notations.
Definition 2.1
Let \(p>1\), \(n_{0} \in\mathbb{Z}\), \(w(n)\geq0 \) (\(n \geq n_{0}\), \(n \in\mathbb{Z}\)). Define the normed space \(\ell_{w,n_{0}}^{p}\) by
Definition 2.2
Let \(\lambda_{1}, \lambda_{2}, \lambda>0\) satisfying that \(\lambda= \lambda_{1}+\lambda_{2}\). Denote by \(F_{n_{0}}(r)\) (\(n_{0} \in\mathbb{Z}\)) the set of all real-valued \(C^{1}\)-functions \(\phi(x)\) satisfying the following conditions:
-
(1)
\(\phi(x)\) is strictly increasing in \((n_{0}-1,\infty)\) with \(\phi((n_{0}-1)+)=0\), \(\phi(\infty) =\infty\).
-
(2)
For \(\alpha>0\), \(\frac{\phi'(x)}{\phi(x)^{\alpha+1-\lambda _{i}}}\) is decreasing in \((n_{0}-1,\infty)\).
Let \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\lambda= \lambda _{1}+\lambda_{2}\), \(\lambda_{1}, \lambda_{2}, \lambda>0\). For \(\phi(x) \in F_{m_{0}}(r)\) and \(\psi(y) \in F_{n_{9}}(s)\), \(r,s>1\), we define the following weight functions:
Definition 2.3
Let \(\lambda_{1}, \lambda_{2}, \lambda>0\) satisfying that \(\lambda= \lambda_{1}+\lambda_{2}\). For \(\alpha>0\) and \(x, y >0\), \(K_{\alpha, \lambda}(x,y)\) is a continuous real-valued function on \((0, \infty) \times(0,\infty)\) satisfying the following properties:
-
(1)
\(K_{\alpha, \lambda}(x,y)\) is a symmetric homogeneous function of degree \(2\alpha-\lambda\), that is,
$$\begin{aligned} &K_{\alpha, \lambda}(x,y) = K_{\alpha, \lambda}(y,x),\\ &K_{\alpha, \lambda}(tx,ty) = t^{2\alpha-\lambda} K_{\alpha, \lambda }(x,y) \quad\mbox{for any } t>0. \end{aligned}$$ -
(2)
\(K_{\alpha, \lambda}(x,y)\) is decreasing with respect to x and y, respectively.
-
(3)
For sufficiently small \(\varepsilon\geq0\), the following integral
$$\widetilde{K}_{\alpha, \lambda}(\lambda_{i},\varepsilon) := \int _{0}^{\infty}K_{\alpha, \lambda}(1,t)t^{-1+\lambda_{i}-\alpha-\varepsilon}\,dt $$exists for \(i=1,2\). Moreover, assume that \(\widetilde{K}_{\alpha, \lambda}(\lambda_{i},0):=K_{\alpha}(\lambda_{i})>0\) and \(\widetilde{K}_{\alpha, \lambda}(\lambda_{i},\varepsilon) = K_{\alpha }(\lambda_{i}) + o(1)\) as \(\varepsilon\rightarrow0+\).
-
(4)
Given \(p>1\), \(\phi(x) \in F_{m_{0}} (r)\), and \(\psi(y) \in F_{n_{0}} (s)\) (\(r,s>1\)),
$$\sum_{n=n_{0}}^{\infty}\frac{\psi'(n)}{\psi(n)^{1+\varepsilon}}\int _{0}^{\frac{\phi(m_{0})}{\psi(n)}} K_{\alpha, \lambda} (1, t) t^{-1+\lambda _{i}-\alpha-\frac{\varepsilon}{p}}\,dt=O(1) $$as \(\varepsilon\rightarrow0+\).
Lemma 2.4
Let \(\lambda_{1}, \lambda_{2}, \lambda>0\) satisfying that \(\lambda=\lambda _{1} + \lambda_{2}\). For any \(\alpha>0\), we have
Proof
Since
letting \(t=\frac{1}{s}\) gives
□
In view of Lemma 2.4, we may assume that
Lemma 2.5
Let \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\lambda_{1}, \lambda_{2} >0\), \(\alpha>0\). For \(\phi(x) \in F_{m_{0}}(r)\) and \(\psi(y) \in F_{n_{0}}(s)\), \(r,s >1\), define the weight coefficients \(W_{1}(m)\) and \(W_{2}(n)\) by
Then
for any \(m \geq m_{0}\), \(n \geq n_{0} \) (\(m,n \in\mathbb{Z}\)).
Proof
We have
Setting \(t=\frac{\psi(x)}{\phi(m)}\), we get
Similarly, one can obtain \(W_{2}(n) < K_{\alpha}(\lambda)\). □
Lemma 2.6
Let \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\lambda_{1}, \lambda_{2} >0\). For \(a_{m}, b_{n} \geq0 \) (\(m_{0}, n_{0} \in\mathbb{Z}\)), let \(a=\{a_{m}\} _{m=m_{0}}^{\infty}\in\ell_{w_{1},m_{0}}^{p}\) and \(b=\{b_{n}\}_{n=n_{0}}^{\infty}\in\ell_{w_{2}, n_{0}}^{q}\). Then, for \(\phi(x) \in F_{m_{0}}(r)\) and \(\psi(y) \in F_{n_{0}}(s)\) (\(r,s >1\)), we have
and hence
Proof
Applying Hölder’s inequality, we observe
By Definition 2.2, we get
Therefore, by using Lemma 2.5, we get
In the same manner, one can obtain
□
In view of Lemma 2.6, we can define a Hilbert-type operator \(T: \ell_{w_{1}, m_{0}}^{p} \rightarrow\ell_{\widetilde {w}_{1}, n_{0}}^{p}\) by
Similarly, define \(T: \ell_{w_{2}, n_{0}}^{q} \rightarrow\ell_{\widetilde {w}_{2}, m_{0}}^{q}\) by
It immediately follows from Lemma 2.6 that
and
Hence the operator T is bounded. The formal inner product \((Ta, b)\) of Ta and b is defined by
Lemma 2.7
Let \(p>1\), \(\frac{1}{p} +\frac{1}{q}=1\). Let \(\widetilde{a}=\{\widetilde {a}_{m}\}_{m=m_{0}}^{\infty}\) and \(\widetilde{b}=\{\widetilde{b}_{n}\} _{n=n_{0}}^{\infty}\) with \(\widetilde{a}_{m}= \frac{\phi'(m)}{\phi(m)^{\alpha +1-\lambda_{2}+\frac{\varepsilon}{p}}}\) and \(\widetilde{b}_{n}= \frac{\psi '(n)}{\psi(n)^{\alpha+1-\lambda_{1}+\frac{\varepsilon}{q}}}\) for \(0<\varepsilon< p\lambda_{i}\), \(i=1,2\). Then, as \(\varepsilon \rightarrow0+\),
Proof
We have
Setting \(t=\frac{\phi(x)}{\psi(n)}\), we get
Moreover,
Note that the definition of \(K_{\alpha, \lambda} (x,y)\) implies that
and
Thus, using the fact that for \(a>0\),
as \(\varepsilon\rightarrow0+\), we obtain
which completes the proof. □
Theorem 2.8
Let \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\lambda_{1}, \lambda_{2} >0\). For \(a_{m}, b_{n} \geq0 \) (\(m_{0}, n_{0} \in\mathbb{Z}\)), let \(a=\{a_{m}\} _{m=m_{0}}^{\infty}\in\ell_{w_{1},m_{0}}^{p}\) and \(b=\{b_{n}\}_{n=n_{0}}^{\infty}\in\ell_{w_{2}, n_{0}}^{q}\). Then, for \(\phi(x) \in F_{m_{0}}(r)\) and \(\psi(y) \in F_{n_{0}}(s)\) (\(r,s >1\)),
Proof
Suppose that \(\|T\|_{p}< K_{\alpha}(\lambda)\). Consider \(\widetilde{a}_{m} = \phi' (m)\phi(m)^{-1+\lambda_{2}-\alpha-\frac{\varepsilon}{p}}\) and \(\widetilde{b}_{n} = \phi' (n) \psi(n)^{-1+\lambda_{1}-\alpha-\frac {\varepsilon}{q}}\), where \(m\geq m_{0}\), \(n\geq n_{0}\), \(m,n\in\mathbb {Z}\), \(0<\varepsilon<p\lambda_{i}\), \(i=1,2\). A simple computation shows that \(\widetilde{a} \in\ell_{w_{1},m_{0}}^{p}\) and \(\widetilde{b} \in\ell _{w_{2}, n_{0}}^{q}\) with \(\|\widetilde{a}\|_{p, w_{1}}>0\) and \(\|\widetilde {b}\|_{q,w_{2}}>0\). Then
Moreover, we have
On the other hand, from Lemma 2.7 it follows
Therefore, combining these inequalities (2) and (3),
Since
as \(\varepsilon\rightarrow0+\), we obtain that \(K_{\alpha}(\lambda) \leq\|T\|_{p}\), which is a contradiction. Thus we conclude that \(\|T\|_{p} = K_{\alpha}(\lambda)\). Applying the same argument, we have \(\|T\|_{q} = K_{\alpha}(\lambda)\), which completes the proof. □
3 Two equivalent inequalities for the Hilbert-type operator
Equipped with the Hilbert-type operator defined as above, we have the following theorem.
Theorem 3.1
Let \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\lambda _{1}, \lambda_{2} >0\). For \(a_{m}, b_{n} \geq0 \) (\(m_{0}, n_{0} \in\mathbb{Z}\)), let \(a=\{a_{m}\} _{m=m_{0}}^{\infty}\in\ell_{w_{1},m_{0}}^{p}\), \(b=\{b_{n}\}_{n=n_{0}}^{\infty}\in \ell_{w_{2}, n_{0}}^{q}\) \(\|a\|_{p,w_{1}}>0\), \(\|b\|_{q,w_{2}}>0\). Then, for \(\phi (x) \in F_{m_{0}}(r)\) and \(\psi(y) \in F_{n_{0}}(s)\) (\(r,s >1\)), we have the following equivalent inequalities:
Furthermore, the constant factor \(K_{\alpha}(\lambda)\) is the best possible.
Proof
It follows from Hölder’s inequality that
Applying Lemma 2.5, we see that
In order to prove that inequality (4) implies inequality (5), we define b̃ as follows:
for \(n\geq n_{0}\), \(n\in\mathbb{Z}\). Then we see that \(\widetilde{b} \in \ell_{w_{2},n_{0}}^{q}\) and \(\|\widetilde{b}\|_{q,w_{2}}>0\) as before. Thus using inequality (4) shows that
which gives \(\|Ta\|_{p,\widetilde{w}_{1}} = \|\widetilde{b}\|_{q,w_{2}}^{q-1}< K_{\alpha}(\lambda) \|a\|_{p,w_{1}} \). Hence inequality (4) implies inequality (5).
Now suppose that inequality (5) holds for any \(a \in\ell _{w_{1},m_{0}}^{p}\).
which means that inequality (5) implies inequality (4). Therefore inequality (4) is equivalent to inequality (5). Furthermore, Theorem 2.8 implies that the constant factor \(K_{\alpha}(\lambda)\) in inequalities (4) and (5) is the best possible, which completes the proof. □
4 Applications to various Hilbert-type inequalities
In this section, we apply our previous theorems to obtain several Hilbert-type inequalities. Recall that the beta function \(B(u,v)\) is defined by
Define the function \(K_{\alpha,\lambda} (x,y)\) by
for \(\lambda>\alpha\geq0\). Then \(K_{\alpha,\lambda} (x,y)\) is a symmetric homogeneous function of degree \(2\alpha-\lambda\) and is decreasing with respect to x and y, respectively. Moreover,
To see this, for \(0< \varepsilon< p\lambda_{2}\),
Note that since
we see that
as \(\varepsilon\rightarrow0+\). Therefore from Theorem 3.1 we observe the following.
Corollary 4.1
Let \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\lambda_{1}, \lambda_{2} >0\), \(\lambda>\alpha\geq0\). For \(a_{m}, b_{n} \geq0 \) (\(m_{0}, n_{0} \in\mathbb{Z}\)), let \(a=\{a_{m}\} _{m=m_{0}}^{\infty}\in\ell_{w_{1},m_{0}}^{p}\), \(b=\{b_{n}\}_{n=n_{0}}^{\infty}\in \ell_{w_{2}, n_{0}}^{q}\) and \(\|a\|_{p,w_{1}}>0\), \(\|b\|_{q,w_{2}}>0\). Then, for \(\phi(x) \in F_{m_{0}}(r)\) and \(\psi(y) \in F_{n_{0}}(s)\) (\(r,s >1\)), we have the following equivalent inequalities:
Furthermore, the constant factor \(B(\lambda_{1}, \lambda_{2}) \) is the best possible.
As applications, we have the following.
Case 1. Let \(\phi(x)=x^{\beta}\) and \(\psi(x)=x^{\gamma}\) (\(\beta, \gamma>0\)) for \(m_{0}=n_{0}=1\). For \(0<\lambda_{i} <\alpha+\min\{ \frac{1}{\beta}, \frac{1}{\gamma}\}\) and \(0\leq\alpha<\lambda\), one has the following equivalent inequalities:
where \(w_{1} (m)=m^{p(1-\lambda_{2}\beta+\alpha\beta)-1}\) and \(w_{2}(n)=n^{q(1-\lambda_{1}\gamma+\alpha\gamma)-1}\).
-
(I)
For \(\lambda_{1}=\frac{\lambda}{p}\) and \(\lambda _{2}=\frac{\lambda}{q}\) with \(0<\lambda_{i}<\alpha+\min\{\frac{1}{\beta}, \frac{1}{\gamma}\}\) and \(0 \leq\alpha< \lambda\), one has the following equivalent inequalities:
$$\begin{aligned}& \sum_{n=1}^{\infty}\sum _{m=1}^{\infty}\frac{(m^{\beta}n^{\gamma})^{\alpha}}{(m^{\beta}+ n^{\gamma})^{\lambda}} a_{m} b_{n} < \frac{B(\frac{\lambda }{p},\frac{\lambda}{q})}{\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a\| _{p,w_{1}} \|b \|_{q,w_{2}},\\& \Biggl\{ \sum_{n=1}^{\infty}n^{\gamma(\lambda-p\alpha)-1} \Biggl(\sum_{m=1}^{\infty}\frac{(m^{\beta}n^{\gamma})^{\alpha}}{(m^{\beta}+ n^{\gamma})^{\lambda}}a_{m} \Biggr)^{p} \Biggr\} ^{\frac{1}{p}} < \frac{B(\frac{\lambda }{p},\frac{\lambda}{q})}{\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a \|_{p,w_{1}}, \end{aligned}$$where \(w_{1} (m)=m^{(p-1)(1-\lambda\beta)+p\alpha\beta}\) and \(w_{2}(n)=n^{(q-1)(1-\lambda\gamma)+q\alpha\gamma}\).
-
(II)
For \(\lambda_{1}=\frac{\lambda}{q}\) and \(\lambda _{2}=\frac{\lambda}{p}\) with \(0<\lambda_{i}<\alpha+\min\{\frac{1}{\beta}, \frac{1}{\gamma}\}\) and \(0 \leq\alpha< \lambda\), one has the following equivalent inequalities:
$$\begin{aligned}& \sum_{n=1}^{\infty}\sum _{m=1}^{\infty}\frac{(m^{\beta}n^{\gamma})^{\alpha}}{(m^{\beta}+ n^{\gamma})^{\lambda}} a_{m} b_{n} < \frac{B(\frac{\lambda }{p},\frac{\lambda}{q})}{\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a\| _{p,w_{1}} \|b \|_{q,w_{2}},\\& \Biggl\{ \sum_{n=1}^{\infty}n^{\gamma\lambda(p-1)-p\alpha\gamma-1} \Biggl(\sum_{m=1}^{\infty}\frac{(m^{\beta}n^{\gamma})^{\alpha}}{(m^{\beta}+ n^{\gamma})^{\lambda}}a_{m} \Biggr)^{p} \Biggr\} ^{\frac{1}{p}} < \frac{B(\frac{\lambda }{p},\frac{\lambda}{q})}{\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a \|_{p,w_{1}}, \end{aligned}$$where \(w_{1} (m)=m^{p-1-\beta\lambda+p\alpha\beta}\) and \(w_{2}(n)=n^{q-1-\gamma\lambda+q\alpha\gamma}\).
-
(III)
Let \(\lambda_{1}=\frac{p+\lambda-2}{p}\), \(\lambda _{2}=\frac{q+\lambda-2}{q}\), \(\lambda>\max\{2-p, 2-q\}\), \(0 < \beta< \frac{p}{p+\lambda-2-p\alpha}\), \(0 < \gamma< \frac{q}{q+\lambda -2-q\alpha}\), \(0\leq\alpha< \min\{\frac{p+\lambda-2}{p}, \frac {q+\lambda-2}{q}\}\). Then one has the following equivalent inequalities:
$$\begin{aligned}& \sum_{n=1}^{\infty}\sum _{m=1}^{\infty}\frac{(m^{\beta}n^{\gamma})^{\alpha}}{(m^{\beta}+ n^{\gamma})^{\lambda}} a_{m} b_{n} < \frac{B(\frac{p+\lambda -2}{p}, \frac{q+\lambda-2}{q})}{\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \| a\|_{p,w_{1}} \|b \|_{q,w_{2}},\\& \Biggl\{ \sum_{n=1}^{\infty}n^{\gamma(p+\lambda-2)-p\alpha\gamma-1} \Biggl(\sum_{m=1}^{\infty}\frac{(m^{\beta}n^{\gamma})^{\alpha}}{(m^{\beta}+ n^{\gamma})^{\lambda}}a_{m} \Biggr)^{p} \Biggr\} ^{\frac{1}{p}} < \frac{B(\frac{p+\lambda -2}{p}, \frac{q+\lambda-2}{q})}{\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \| a \|_{p,w_{1}}, \end{aligned}$$where \(w_{1} (m)=m^{(p-1)(1-\beta(q+\lambda-2))+p\alpha\beta}\) and \(w_{2}(n)=n^{(q-1)(1-\gamma(p+\lambda-2))+q\alpha\gamma}\).
-
(IV)
Let \(\lambda_{1}=\frac{q+\lambda-2}{q}\), \(\lambda _{2}=\frac{p+\lambda-2}{p}\), \(\lambda>\max\{2-p, 2-q\}\), \(0 < \beta< \frac{q}{q+\lambda-2-q\alpha}\), \(0 < \gamma< \frac{p}{p+\lambda -2-p\alpha}\), \(0\leq\alpha< \min\{\frac{p+\lambda-2}{p}, \frac {q+\lambda-2}{q}\}\). Then one has the following equivalent inequalities:
$$\begin{aligned}& \sum_{n=1}^{\infty}\sum _{m=1}^{\infty}\frac{(m^{\beta}n^{\gamma})^{\alpha}}{(m^{\beta}+ n^{\gamma})^{\lambda}} a_{m} b_{n} < \frac{B(\frac{p+\lambda -2}{p}, \frac{q+\lambda-2}{q})}{\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \| a\|_{p,w_{1}} \|b \|_{q,w_{2}},\\& \Biggl\{ \sum_{n=1}^{\infty}n^{\gamma(p-1)(q+\lambda-2)-p\alpha\gamma-1} \Biggl(\sum_{m=1}^{\infty}\frac{(m^{\beta}n^{\gamma})^{\alpha}}{(m^{\beta}+ n^{\gamma})^{\lambda}}a_{m} \Biggr)^{p} \Biggr\} ^{\frac{1}{p}} < \frac{B(\frac {p+\lambda-2}{p}, \frac{q+\lambda-2}{q})}{\beta^{\frac{1}{q}}\gamma^{\frac {1}{p}}} \|a \|_{p,w_{1}}, \end{aligned}$$where \(w_{1} (m)=m^{p-1-\beta(p+\lambda-2)+p\alpha\beta}\) and \(w_{2}(n)=n^{q-1-\gamma(q+\lambda-2)+q\alpha\gamma}\).
Case 2. For \(A, B>0\), let \(\phi(x)=A(\ln x)^{\beta}\) and \(\psi(x)=B(\ln x)^{\gamma}\) (\(\beta, \gamma>0\)), \(m_{0}=n_{0}=2\). For \(0<\lambda_{i}< \alpha+\min\{\frac{1}{\beta}, \frac{1}{\gamma}\}\) and \(0\leq\alpha< \lambda\), one has the following equivalent inequalities:
where \(w_{1} (m)=m^{p-1}(\ln m)^{p(1-\lambda_{2}\beta+\alpha\beta)-1}\) and \(w_{2}(n)=n^{q-1}(\ln n)^{q(1-\lambda_{1} \gamma+\alpha\gamma)-1}\).
-
(I)
For \(\lambda_{1}=\frac{\lambda}{p}\) and \(\lambda _{2}=\frac{\lambda}{q}\) with \(0<\lambda_{i}<\alpha+\min\{\frac{1}{\beta}, \frac{1}{\gamma}\}\) and \(0 \leq\alpha< \lambda\), one has the following equivalent inequalities:
$$\begin{aligned}& \sum_{n=2}^{\infty}\sum _{m=2}^{\infty}\frac{((\ln m)^{\beta}(\ln n)^{\gamma})^{\alpha}}{(A(\ln m)^{\beta}+ B(\ln n)^{\gamma})^{\lambda}} a_{m} b_{n} < \frac {B(\frac{\lambda}{p}, \frac{\lambda}{q})}{A^{\lambda_{2}}B^{\lambda _{1}}\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a\|_{p,w_{1}} \|b \|_{q,w_{2}},\\& \Biggl\{ \sum_{n=2}^{\infty}\frac{1}{n}(\ln n)^{\gamma-p\alpha\gamma-1} \Biggl(\sum _{m=2}^{\infty}\frac{((\ln m)^{\beta}(\ln n)^{\gamma})^{\alpha}}{(A(\ln m)^{\beta}+ B(\ln n)^{\gamma})^{\lambda}} a_{m} \Biggr)^{p} \Biggr\} ^{\frac{1}{p}} \\& \quad< \frac{B(\frac{\lambda}{p}, \frac{\lambda}{q})}{A^{\lambda _{2}}B^{\lambda_{1}}\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a \|_{p,w_{1}}, \end{aligned}$$where \(w_{1} (m)=m^{p-1}(\ln m)^{(p-1)(1-\lambda\beta)+p\alpha\beta}\) and \(w_{2}(n)=n^{q-1}(\ln n)^{(q-1)(1-\lambda\gamma)+q\alpha\gamma}\).
-
(II)
Let \(\lambda_{1}=\frac{p+\lambda-2}{p}\), \(\lambda _{2}=\frac{q+\lambda-2}{q}\), \(\lambda>\max\{2-p, 2-q\}\), \(0 < \beta< \frac{p}{p+\lambda-2-p\alpha}\), \(0 < \gamma< \frac{q}{q+\lambda -2-q\alpha}\), \(0\leq\alpha< \min\{\frac{p+\lambda-2}{p}, \frac {q+\lambda-2}{q}\}\). Then one has the following equivalent inequalities:
$$\begin{aligned}& \sum_{n=2}^{\infty}\sum _{m=2}^{\infty}\frac{((\ln m)^{\beta}(\ln n)^{\gamma})^{\alpha}}{(A(\ln m)^{\beta}+ B(\ln n)^{\gamma})^{\lambda}} a_{m} b_{n} < \frac {B(\frac{p+\lambda-2}{p}, \frac{q+\lambda-2}{q})}{A^{\lambda _{2}}B^{\lambda_{1}}\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a\|_{p,w_{1}} \| b \|_{q,w_{2}},\\& \Biggl\{ \sum_{n=2}^{\infty}\frac{1}{n}(\ln n)^{\gamma(p+\lambda-2)-p\alpha \gamma-1} \Biggl(\sum _{m=2}^{\infty}\frac{((\ln m)^{\beta}(\ln n)^{\gamma})^{\alpha}}{(A(\ln m)^{\beta}+ B(\ln n)^{\gamma})^{\lambda}} a_{m} \Biggr)^{p} \Biggr\} ^{\frac{1}{p}} \\& \quad< \frac{B(\frac{p+\lambda-2}{p}, \frac{q+\lambda-2}{q})}{A^{\lambda _{2}}B^{\lambda_{1}}\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a\|_{p,w_{1}}, \end{aligned}$$where \(w_{1} (m)=m^{p-1}(\ln m)^{(p-1)(1-\beta(q+\lambda-2))+p\alpha\beta }\) and \(w_{2}(n)=n^{q-1}(\ln n)^{(q-1)(1-\gamma(p+\lambda-2))+q\alpha \gamma}\).
Case 3. For \(A, B>0\), let \(\phi(x)=A(\ln x)^{\beta}\) and \(\psi(x)=Bx^{\gamma}\) (\(\beta, \gamma>0\)), \(m_{0}=2\), \(n_{0}=1\). For \(0<\lambda_{i}< \alpha+\min\{\frac{1}{\beta}, \frac{1}{\gamma}\}\) and \(0\leq\alpha< \lambda\), one has the following equivalent inequalities:
where \(w_{1} (m)=m^{p-1}(\ln m)^{p(1-\lambda_{2}\beta+\alpha\beta)-1}\) and \(w_{2}(n)=n^{q(1-\lambda_{1} \gamma+\alpha\gamma)-1}\).
-
(I)
For \(\lambda_{1}=\frac{\lambda}{p}\) and \(\lambda _{2}=\frac{\lambda}{q}\) with \(0<\lambda_{i}<\alpha+\min\{\frac{1}{\beta}, \frac{1}{\gamma}\}\) and \(0 \leq\alpha< \lambda\), one has the following equivalent inequalities:
$$\begin{aligned}& \sum_{n=1}^{\infty}\sum _{m=2}^{\infty}\frac{((\ln m)^{\beta}n^{\gamma})^{\alpha}}{(A(\ln m)^{\beta}+ Bn^{\gamma})^{\lambda}} a_{m} b_{n} < \frac{B(\frac {\lambda}{p}, \frac{\lambda}{q})}{A^{\lambda_{2}}B^{\lambda_{1}}\beta^{\frac {1}{q}}\gamma^{\frac{1}{p}}} \|a\|_{p,w_{1}} \|b \|_{q,w_{2}}, \\& \Biggl\{ \sum_{n=1}^{\infty}n^{\gamma(1-p\alpha)-1} \Biggl(\sum_{m=2}^{\infty}\frac{((\ln m)^{\beta}n^{\gamma})^{\alpha}}{(A(\ln m)^{\beta}+ Bn^{\gamma})^{\lambda}} a_{m} \Biggr)^{p} \Biggr\} ^{\frac{1}{p}} < \frac{B(\frac {\lambda}{p}, \frac{\lambda}{q})}{A^{\lambda_{2}}B^{\lambda_{1}}\beta^{\frac {1}{q}}\gamma^{\frac{1}{p}}} \|a\|_{p,w_{1}}, \end{aligned}$$where \(w_{1} (m)=m^{p-1}(\ln m)^{(p-1)(1-\lambda\beta)+p\alpha\beta}\) and \(w_{2}(n)=n^{(q-1)(1-\lambda\gamma)+q\alpha\gamma}\).
-
(II)
Let \(\lambda_{1}=\frac{p+\lambda-2}{p}\), \(\lambda _{2}=\frac{q+\lambda-2}{q}\), \(\lambda>\max\{2-p, 2-q\}\), \(0 < \beta< \frac{p}{p+\lambda-2-p\alpha}\), \(0 < \gamma< \frac{q}{q+\lambda -2-q\alpha}\), \(0\leq\alpha< \min\{\frac{p+\lambda-2}{p}, \frac {q+\lambda-2}{q}\}\). Then one has the following equivalent inequalities:
$$\begin{aligned}& \sum_{n=1}^{\infty}\sum _{m=2}^{\infty}\frac{((\ln m)^{\beta}n^{\gamma})^{\alpha}}{(A(\ln m)^{\beta}+ Bn^{\gamma})^{\lambda}} a_{m} b_{n} < \frac{B(\frac {p+\lambda-2}{p}, \frac{q+\lambda-2}{q})}{A^{\lambda_{2}}B^{\lambda _{1}}\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a\|_{p,w_{1}} \|b \|_{q,w_{2}},\\& \Biggl\{ \sum_{n=1}^{\infty}n^{\gamma(p+\lambda-2)-p\alpha\gamma-1} \Biggl(\sum_{m=2}^{\infty}\frac{((\ln m)^{\beta}n^{\gamma})^{\alpha}}{(A(\ln m)^{\beta}+ Bn^{\gamma})^{\lambda}} a_{m} \Biggr)^{p} \Biggr\} ^{\frac{1}{p}} < \frac {B(\frac{p+\lambda-2}{p}, \frac{q+\lambda-2}{q})}{A^{\lambda _{2}}B^{\lambda_{1}}\beta^{\frac{1}{q}}\gamma^{\frac{1}{p}}} \|a\|_{p,w_{1}}, \end{aligned}$$where \(w_{1} (m)=m^{p-1}(\ln m)^{(p-1)(1-\beta(q+\lambda-2))+p\alpha\beta }\) and \(w_{2}(n)=n^{(q-1)(1-\gamma(p+\lambda-2))+q\alpha\gamma}\).
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This research was supported by the Sookmyung Women’s University Research Grants 2012.
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Seo, K. A Hilbert-type operator with a symmetric homogeneous kernel of two parameters and its applications. J Inequal Appl 2015, 266 (2015). https://doi.org/10.1186/s13660-015-0788-z
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DOI: https://doi.org/10.1186/s13660-015-0788-z