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Journal of Inequalities and Applications
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Equivalent property of a half-discrete Hilbert’s inequality with parameters

Journal of Inequalities and Applications20182018:333

https://doi.org/10.1186/s13660-018-1926-1

©  The Author(s) 2018

  • Received: 6 September 2018
  • Accepted: 27 November 2018
  • Published:

Abstract

By using the weight functions and the idea of introducing parameters, a half-discrete Hilbert inequality with a nonhomogeneous kernel and its equivalent form are given. The equivalent statements of the constant factor are best possible related to parameters, and some particular cases are considered. The cases of a homogeneous kernel are also deduced.

Keywords

  • Weight function
  • Half-discrete Hilbert’s inequality
  • Equivalent form
  • Best possible constant factor

MSC

  • 26D15

1 Introduction

If \(0 < \sum_{m = 1}^{\infty } a_{m}^{2} < \infty ,0 < \sum_{n = 1}^{\infty } b_{n}^{2} < \infty \), then we have the following discrete Hilbert inequality with the best possible constant factor π [1]:
$$ \sum_{m = 1}^{\infty } \sum _{n = 1}^{\infty } \frac{a_{m}b_{n}}{m + n} < \pi \Biggl(\sum _{m = 1}^{\infty } a_{m}^{2} \sum_{n = 1}^{\infty } b_{n}^{2} \Biggr)^{1/2}. $$
(1)
Assuming that \(0 < \int _{0}^{\infty } f^{2}(x)\,dx < \infty \), \(0 < \int _{0}^{\infty } g^{2}(y)\,dy < \infty \), we still have the following Hilbert integral inequality [1]:
$$ \int _{0}^{\infty } \int _{0}^{\infty } \frac{f(x)g(y)}{x + y}\,dx\,dy < \pi \biggl( \int _{0}^{\infty } f^{2}(x)\,dx \int _{0}^{\infty } g^{2}(y)\,dy \biggr)^{1/2}, $$
(2)
where the constant factor π is the best possible. Inequalities (1) and (2) are important in analysis and its applications (cf. [2–13]).
We still have the following half-discrete Hilbert-type inequalities (cf. [1], Theorem 351): If \(K(x)(x > 0)\) is decreasing, \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(0 < \phi (s) = \int _{0}^{\infty } K(x)x^{s - 1}\,dx <\infty \), then
$$\begin{aligned}& \int _{0}^{\infty } x^{p - 2} \Biggl(\sum _{n = 1}^{\infty } K(nx)a_{n} \Biggr)^{p} \,dx < \phi ^{p} \biggl(\frac{1}{q} \biggr)\sum _{n = 1}^{\infty } a_{n}^{p}, \end{aligned}$$
(3)
$$\begin{aligned}& \sum_{n = 1}^{\infty } n^{p - 2} \biggl( \int _{0}^{\infty } K (nx)f(x)\,dx \biggr)^{p} < \phi ^{p} \biggl(\frac{1}{q} \biggr) \int _{0}^{\infty } f^{p} (x)\,dx. \end{aligned}$$
(4)

In recent years, some new extensions of (3) and (4) were provided by [14–19].

In 2016, Hong [20, 21] also considered some equivalent statements of the extensions of (1) and (2) with a few parameters. For the following work we refer to [22–24].

In this paper, following [20], by the use of the weight functions and the idea of introducing parameters, a half-discrete Hilbert inequality with the nonhomogeneous kernel and its equivalent form are given. The equivalent statements of the constant factor are best possible related to parameters, and some particular cases are considered. The cases of a homogeneous kernel are also deduced.

2 Some lemmas

In what follows, we assume that \(p > 1\), \(\frac{1}{p} + \frac{1}{q} =1\), \(\lambda > 0\), σ, \(\sigma _{1} \le 1\), \(\sigma ,\sigma _{1} \in (0,\lambda ),f(x)\) is a nonnegative measurable function in \(\mathbb{R}_{ +} = (0,\infty )\), \(a_{n} \ge 0\) (\(n \in \mathbb{N} = \{ 1,2, \ldots \} \)), such that
$$ 0 < \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx< \infty ,\qquad 0 < \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1} a_{n}^{q} < \infty . $$

Lemma 1

Define the following weight functions:
$$\begin{aligned}& \omega _{\sigma } (\sigma _{1},n): = n^{\sigma } \int _{0}^{\infty } \frac{1}{(1 + xn)^{\lambda }} x^{\sigma _{1} - 1}\,dx \quad (n \in \mathbb{N}), \end{aligned}$$
(5)
$$\begin{aligned}& \varpi _{\sigma _{1}}(\sigma ,x): = x^{\sigma _{1}}\sum _{n = 1}^{\infty } \frac{1}{(1 + xn)^{\lambda }} n^{\sigma - 1} \quad (x \in \mathbb{R}_{ +} ). \end{aligned}$$
(6)
We have the following equality and inequalities:
$$\begin{aligned}& \omega _{\sigma } (\sigma _{1},n) = B(\sigma _{1}, \lambda - \sigma _{1})n^{\sigma - \sigma _{1}}\quad (n \in \mathbb{N}), \end{aligned}$$
(7)
$$\begin{aligned}& \biggl(B(\sigma ,\lambda - \sigma ) - \frac{x^{\sigma }}{\sigma } \biggr)x^{\sigma _{1} - \sigma } < \varpi _{\sigma _{1}}(\sigma ,x) < B(\sigma ,\lambda - \sigma ) \cdot x^{\sigma _{1} - \sigma } \quad (x \in \mathbb{R}_{ +} ), \end{aligned}$$
(8)
where \(B(u,v): = \int _{0}^{\infty } \frac{t^{u - 1}}{(1 + t)^{u + v}}\,dt\) (\(u,v> 0\)) is the Beta function.

Proof

Setting \(u = xn\), we have
$$\begin{aligned} \omega _{\sigma } (\sigma _{1},n) &= n^{\sigma } \int _{0}^{\infty } \frac{1}{(1 + u)^{\lambda }} \biggl( \frac{u}{n} \biggr)^{\sigma _{1} - 1} \frac{1}{n}\,du \\ &= n^{\sigma - \sigma _{1}} \int _{0}^{\infty } \frac{u^{\sigma _{1} - 1}}{(1 + u)^{\lambda }}\,du \\ &= B(\sigma _{1},\lambda - \sigma _{1})n^{\sigma -\sigma _{1}}, \end{aligned}$$
and then (7) follows. In view of the decreasing property, we find
$$\begin{aligned} \varpi _{\sigma _{1}}(\sigma ,x) &< x^{\sigma _{1}} \int _{0}^{\infty } \frac{t^{\sigma - 1}}{(1 + xt)^{\lambda }}\,dt \\ &= x^{\sigma _{1} - \sigma } \int _{0}^{\infty } \frac{u^{\sigma - 1}}{(1 + u)^{\lambda }}\,du \\ &= x^{\sigma _{1} - \sigma } B(\sigma ,\lambda - \sigma ), \\ \varpi _{\sigma _{1}}(\sigma ,x)&> x^{\sigma _{1}} \int _{1}^{\infty } \frac{t^{\sigma - 1}}{(1 + xt)^{\lambda }}\,dt \\ &= x^{\sigma _{1} - \sigma } \int _{x}^{\infty } \frac{u^{\sigma - 1}}{(1 + u)^{\lambda }}\,du \\ &= x^{\sigma _{1} - \sigma } \biggl[ \int _{0}^{\infty } \frac{u^{\sigma - 1}\,du}{(1 + u)^{\lambda }} - \int _{0}^{x} \frac{u^{\sigma - 1}\,du}{(1 + u)^{\lambda }} \biggr] \ge x^{\sigma _{1} - \sigma } \biggl(B(\sigma ,\lambda - \sigma ) - \int _{0}^{x} u^{\sigma - 1}\,du \biggr) \\ &= x^{\sigma _{1} - \sigma } \biggl(B(\sigma ,\lambda - \sigma ) - \frac{x^{\sigma }}{\sigma } \biggr). \end{aligned}$$
Hence, (8) follows. □

Lemma 2

Setting \(k_{\lambda } (\eta ): = B(\eta ,\lambda - \eta )(\eta = \sigma ,\sigma _{1})\), we have the following inequality:
$$\begin{aligned} I&: = \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} f(x)\,dx \\ &= \sum_{n = 1}^{\infty } \int _{0}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} f(x)\,dx \\ &< k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}( \sigma _{1}) \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{p}} \Biggl\{ \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{q}}. \end{aligned}$$
(9)

Proof

By Hölder’s inequality (cf. [25]), we have
$$\begin{aligned} I &= \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{1}{(1 + xn)^{\lambda }} \biggl[\frac{x^{(1 - \sigma _{1})/q}}{n^{(1 - \sigma )/p}}f(x) \biggr] \biggl[ \frac{n^{(1 - \sigma )/p}}{x^{(1 - \sigma _{1})/q}}a_{n} \biggr]\,dx \\ &\le \Biggl[ \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{1}{(1 + xn)^{\lambda }} \frac{x^{(1 - \sigma _{1})p/q}}{n^{1 - \sigma }} f^{p}(x)\,dx \Biggr]^{\frac{1}{p}} \Biggl[\sum_{n = 1}^{\infty } \int _{0}^{\infty } \frac{1}{(1 + xn)^{\lambda }} \frac{n^{(1 - \sigma )q/p}}{x^{1 - \sigma _{1}}} \,dx\,a_{n}^{q} \Biggr]^{\frac{1}{q}} \\ &= \biggl[\int _{0}^{\infty } \varpi _{\sigma _{1}}(\sigma ,x)x^{p(1 - \sigma _{1}) - 1}f^{p}(x)\,dx \biggr]^{\frac{1}{p}} \Biggl[\sum _{n = 1}^{\infty } \omega _{\sigma } (\sigma _{1},n) n^{q(1 - \sigma ) - 1}a_{n}^{q}\Biggr]^{\frac{1}{q}}. \end{aligned}$$
Then, by (7) and (8), we have (9). □
By (9), for \(\sigma _{1} = \sigma \), we find \(0 < \int _{0}^{\infty } x^{p(1 - \sigma ) - 1}f^{p}(x)\,dx < \infty \), \(0 < \sum_{n = 1}^{\infty } n^{q(1 - \sigma ) - 1} a_{n}^{q} < \infty \), and
$$ \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} f(x)\,dx< k_{\lambda } (\sigma ) \biggl[ \int _{0}^{\infty } x^{p(1 - \sigma ) - 1}f^{p}(x)\,dx \biggr]^{\frac{1}{p}} \Biggl[\sum_{n = 1}^{\infty } n^{q(1 - \sigma ) - 1} a_{n}^{q} \Biggr]^{\frac{1}{q}}. $$
(10)

Lemma 3

The constant factor \(k_{\lambda } (\sigma ) = B(\sigma ,\lambda - \sigma )\) in (10) is the best possible.

Proof

For \(0 < \varepsilon < q\sigma \), we set
$$ \tilde{a}_{n} = n^{\sigma - \frac{\varepsilon }{q} - 1}\quad (n \in \mathbb{N}),\qquad \tilde{f}(x) = \textstyle\begin{cases} \sigma + \frac{\varepsilon }{p} - 1,&0 < x \le 1, \\ 0,&x > 1. \end{cases} $$
If there exists a constant \(M \le k_{\lambda } (\sigma )\), such that (10) is valid when replacing \(k_{\lambda } (\sigma )\) by M, then, for \(a_{n} = \tilde{a}_{n}\), \(f = \tilde{f}\), we have
$$ \tilde{I}: = \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{\tilde{a}_{n}}{(1 + xn)^{\lambda }} \tilde{f}(x)\,dx< M \biggl[ \int _{0}^{\infty } x^{p(1 - \sigma ) - 1}\tilde{f}^{p}(x) \,dx \biggr]^{\frac{1}{p}} \Biggl[\sum_{n = 1}^{\infty } n^{q(1 - \sigma ) - 1} \tilde{a}_{n}^{q} \Biggr]^{\frac{1}{q}}. $$
We obtain
$$\begin{aligned} \tilde{I} &< M\biggl[ \int _{0}^{1} x^{p(1 - \sigma ) - 1}x^{p(\sigma + \frac{\varepsilon }{p} - 1)}\,dx \biggr]^{\frac{1}{p}}\Biggl[\sum_{n = 1}^{\infty } n^{q(1 - \sigma ) - 1} n^{q(\sigma - \frac{\varepsilon }{q} - 1)}\Biggr]^{\frac{1}{q}} \\ &= M\biggl( \int _{0}^{1} x^{\varepsilon - 1}\,dx \biggr)^{\frac{1}{p}}\Biggl(1 + \sum_{n = 2}^{\infty } n^{ - \varepsilon - 1} \Biggr)^{\frac{1}{q}} \\ &< M\biggl( \int _{0}^{1} x^{\varepsilon - 1}\,dx \biggr)^{\frac{1}{p}}\biggl(1 + \int _{1}^{\infty } x^{ - \varepsilon - 1}\,dx \biggr)^{\frac{1}{q}} \\ &= \frac{M}{\varepsilon } (\varepsilon + 1)^{\frac{1}{q}}. \end{aligned}$$
In view of (8), we find
$$\begin{aligned} \tilde{I} &= \int _{0}^{1} x^{\varepsilon - 1} \Biggl[x^{(\sigma - \frac{\varepsilon }{q})} \sum_{n = 1}^{\infty } \frac{1}{(1 + xn)^{\lambda }} n^{(\sigma - \frac{\varepsilon }{q}) - 1} \Biggr]\,dx \\ &> \int _{0}^{1} x^{\varepsilon - 1} \biggl(B \biggl( \sigma - \frac{\varepsilon }{q},\lambda - \sigma + \frac{\varepsilon }{q} \biggr) - \frac{x^{\sigma - \frac{\varepsilon }{q}}}{\sigma - \frac{\varepsilon }{q}} \biggr)\,dx \\ & = \frac{1}{\varepsilon } B\biggl(\sigma - \frac{\varepsilon }{q},\lambda - \sigma + \frac{\varepsilon }{q}\biggr) - \frac{1}{\sigma - \frac{\varepsilon }{q}} \int _{0}^{1} x^{\sigma + \frac{\varepsilon }{p} - 1}\,dx \\ &= \frac{1}{\varepsilon } \biggl[B\biggl(\sigma - \frac{\varepsilon }{q},\lambda - \sigma + \frac{\varepsilon }{q}\biggr) - \frac{\varepsilon }{(\sigma - \frac{\varepsilon }{q})(\sigma + \frac{\varepsilon }{p})}\biggr]. \end{aligned}$$
Then we have
$$ B \biggl(\sigma - \frac{\varepsilon }{q},\lambda - \sigma + \frac{\varepsilon }{q} \biggr) - \frac{\varepsilon }{(\sigma - \frac{\varepsilon }{q})(\sigma + \frac{\varepsilon }{p})} < M(\varepsilon + 1)^{\frac{1}{q}}. $$
For \(\varepsilon \to 0^{ +} \), in view of the continuous property of the Beta function, we find
$$\begin{aligned} B(\sigma ,\lambda - \sigma ) &= \lim_{\varepsilon \to 0^{ +}} \biggl[B\biggl( \sigma - \frac{\varepsilon }{q},\lambda - \sigma + \frac{\varepsilon }{q}\biggr) - \frac{\varepsilon }{(\sigma - \frac{\varepsilon }{q})(\sigma + \frac{\varepsilon }{p})}\biggr] \\ &\le \lim_{\varepsilon \to 0^{ +}} M(\varepsilon + 1)^{\frac{1}{q}} = M. \end{aligned}$$

Hence, \(M = B(\sigma ,\lambda - \sigma )\) is the best possible constant factor of (10). □

Setting \(\tilde{\sigma } = \frac{\sigma }{p} + \frac{\sigma _{1}}{q}\) (\(\sigma ,\sigma _{1} \le 1,\sigma ,\sigma _{1} \in (0,\lambda )\)), we may rewrite (9) as follows:
$$ I < k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}( \sigma _{1}) \biggl[ \int _{0}^{\infty } x^{p(1 - \tilde{\sigma } ) - 1}f^{p}(x)\,dx \biggr]^{\frac{1}{p}} \Biggl[\sum_{n = 1}^{\infty } n^{q(1 - \tilde{\sigma } ) - 1} a_{n}^{q} \Biggr]^{\frac{1}{q}}. $$
(11)
The parameter σ̃ in (11) also satisfies
$$\begin{aligned} 0 &< k_{\lambda } (\tilde{\sigma } ) = k_{\lambda } \biggl( \frac{\sigma }{p} + \frac{\sigma _{1}}{q} \biggr) \\ &= \int _{0}^{\infty } \frac{1}{(1 + u)^{\lambda }} \bigl(u^{\frac{\sigma - 1}{p}} \bigr) \bigl(u^{\frac{\sigma _{1} - 1}{q}} \bigr)\,du \\ &\le \biggl[ \int _{0}^{\infty } \frac{u^{\sigma - 1}}{(1 + u)^{\lambda }}\,du \biggr]^{\frac{1}{p}} \biggl[ \int _{0}^{\infty } \frac{u^{\sigma _{1} - 1}}{(1 + u)^{\lambda }}\,du \biggr]^{\frac{1}{q}} \\ &= k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) \\ &< \infty , \end{aligned}$$
(12)
by Hölder’s inequality, and \(\tilde{\sigma } \le \frac{1}{p} + \frac{1}{q} = 1\), \(\tilde{\sigma } \in (0,\lambda )\), such that
$$ k_{\lambda } (\tilde{\sigma } ) - \frac{x^{\tilde{\sigma }}}{\tilde{\sigma }} < x^{\tilde{\sigma }} \sum _{n = 1}^{\infty } \frac{n^{\tilde{\sigma } - 1}}{(1 + xn)^{\lambda }} < k_{\lambda } (\tilde{\sigma } ). $$

Lemma 4

If the constant factor \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1})\) in (11) is the best possible, then we have \(\sigma _{1} = \sigma \).

Proof

If the constant factor \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1})\) in (11) is the best possible, then, by (10), the unique best possible constant factor must be \(k_{\lambda } (\tilde{\sigma } )\), namely, \(k_{\lambda } (\tilde{\sigma } ) = k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1})\). Since the condition of (12) keeps the form of equality is that there exist constants A and B, such that they are not all zero and \(Au^{\sigma - 1} =Bu^{\sigma _{1} - 1}\) a.e. in \(\mathbb{R}_{ +}\). Assuming that \(A \ne0 \), it follows that \(u^{\sigma - \sigma _{1}} = \frac{B}{A}\) a.e. in \(\mathbb{R}_{ +} \), and then \(\sigma - \sigma _{1} = 0\), namely, \(\sigma _{1} = \sigma\). □

3 Main results and some corollaries

Theorem 1

Inequality (9) is equivalent to the following inequalities:
$$\begin{aligned} J_{1}&: = \Biggl\{ \sum_{n = 1}^{\infty } n^{p(\frac{\sigma }{p} + \frac{\sigma _{1}}{q}) - 1} \biggl[ \int _{0}^{\infty } \frac{f(x)}{(1 + xn)^{\lambda }}\,dx \biggr]^{p} \Biggr\} ^{\frac{1}{p}} \\ &< k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}( \sigma _{1}) \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
(13)
$$\begin{aligned} J_{2}&: = \Biggl\{ \int _{0}^{\infty } x^{q(\frac{\sigma }{p} + \frac{\sigma _{1}}{q}) - 1} \Biggl[\sum _{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} \Biggr]^{q} \,dx \Biggr\} ^{\frac{1}{q}} \\ &< k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}( \sigma _{1}) \Biggl\{ \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{q}}. \end{aligned}$$
(14)

If the constant factor in (9) is the best possible, then so is the constant factor in (13) and (14).

Proof

Suppose that (13) (or (14)) is valid. By Hölder’s inequality, we have
$$\begin{aligned} I &= \sum_{n = 1}^{\infty } \biggl[n^{\frac{ - 1}{p} + (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})} \int _{0}^{\infty } \frac{f(x)}{(1 + xn)^{\lambda }}\,dx \biggr] \bigl[n^{\frac{1}{p} - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})}a_{n} \bigr] \\ &\le J_{1} \Biggl\{ \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{q}}, \end{aligned}$$
(15)
$$\begin{aligned} I &= \int _{0}^{\infty } \bigl[x^{\frac{1}{q} - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})}f(x) \bigr] \Biggl[x^{\frac{ - 1}{q} + (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})}\sum_{n = 1}^{\infty } \frac{1}{(1 + xn)^{\lambda }} a_{n} \Biggr]\,dx \\ &\le \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{p}}J_{2}. \end{aligned}$$
(16)
Then, by (13) (or (14)), we have (9). On the other hand, assuming (9) is valid, we set
$$ a_{n}: = n^{p(\frac{\sigma }{p} + \frac{\sigma _{1}}{q}) - 1} \biggl[ \int _{0}^{\infty } \frac{f(x)}{(1 + xn)^{\lambda }}\,dx \biggr]^{p - 1}\quad (n \in \mathbb{N}). $$
If \(J_{1} = 0\), then (13) is naturally valid; if \(J_{1} = \infty \), then it is impossible that it makes (13) valid. Suppose that \(0 < J_{1} < \infty \). By (9) we have
$$\begin{aligned}& \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}a_{n}^{q} \\& \quad = J_{1}^{p} = I < k_{\lambda }^{\frac{1}{p}}( \sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{p}} \Biggl\{ \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{q}}, \\& \Biggl\{ \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{p}} = J_{1} < k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
namely, (13) follows.
In the same way, assuming (9) is valid, we set
$$ f(x): = x^{q(\frac{\sigma }{p} + \frac{\sigma _{1}}{q}) - 1} \Biggl[\sum_{n = 1}^{\infty } \frac{1}{(1 + xn)^{\lambda }} a_{n} \Biggr]^{q - 1}\quad (x \in \mathbb{R}). $$
If \(J_{2} = 0\), then (14) is naturally valid; if \(J_{2} = \infty \), then it is impossible that makes (14) valid. Suppose that \(0 < J_{2} < \infty \). By (9) we have
$$\begin{aligned}& \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx \\& \quad =J_{2}^{q} = I< k_{\lambda }^{\frac{1}{p}}( \sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{p}} \Biggl\{ \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{q}}, \\& \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{q}} = J_{2} < k_{\lambda }^{\frac{1}{p}}( \sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) \Biggl\{ \sum _{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{q}}, \end{aligned}$$
namely, (14) follows. Hence, inequalities (9), (13) and (14) are equivalent.

If the constant factor in (9) is the best possible, then so is constant factor in (13) (or (14)). Otherwise, by (15) (or (16)), we would reach the contradiction that the constant factor in (9) is not the best possible. □

Theorem 2

The statements (i), (ii), (iii) and (iv) are equivalent:
  1. (i)

    \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1})\) is independent of \(p,q\);

     
  2. (ii)

    \(k_{\lambda }^{\frac{1}{p}}(\sigma)k_{\lambda }^{\frac{1}{q}}(\sigma _{1})\) is expressed as a single integral;

     
  3. (iii)

    \(k_{\lambda }^{\frac{1}{p}}(\sigma)k_{\lambda }^{\frac{1}{q}}(\sigma _{1})\) in (9) is the best possible constant;

     
  4. (iv)

    \(\sigma _{1} = \sigma\).

     
If the statement (iv) follows, then we have the following equivalent inequalities with the best possible constant factor \(B(\sigma ,\lambda - \sigma )\):
$$\begin{aligned}& \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} f(x)\,dx \\& \quad < B(\sigma ,\lambda - \sigma ) \biggl[ \int _{0}^{\infty } x^{p(1 - \sigma ) - 1}f^{p}(x)\,dx \biggr]^{\frac{1}{p}} \Biggl[\sum_{n = 1}^{\infty } n^{q(1 - \sigma ) - 1} a_{n}^{q} \Biggr]^{\frac{1}{q}}. \end{aligned}$$
(17)
$$\begin{aligned}& \Biggl[\sum_{n = 1}^{\infty } n^{p\sigma - 1} \biggl( \int _{0}^{\infty } \frac{f(x)}{(1 + xn)^{\lambda }}\,dx \biggr)^{p} \Biggr]^{\frac{1}{p}} \\& \quad < B(\sigma ,\lambda - \sigma ) \biggl[ \int _{0}^{\infty } x^{p(1 - \sigma ) - 1}f^{p}(x)\,dx \biggr]^{\frac{1}{p}}, \end{aligned}$$
(18)
$$\begin{aligned}& \Biggl[ \int _{0}^{\infty } x^{q\sigma - 1} \Biggl(\sum _{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} \Biggr)^{q} \,dx \Biggr]^{\frac{1}{q}} \\& \quad < B(\sigma ,\lambda - \sigma ) \Biggl[\sum_{n = 1}^{\infty } n^{q(1 - \sigma ) - 1} a_{n}^{q} \Biggr]^{\frac{1}{q}}. \end{aligned}$$
(19)

Proof

(i)⇒(ii). By (i) we have
$$ k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) = \lim_{p \to 1^{ +}} k_{\lambda }^{\frac{1}{p}}( \sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) = k_{\lambda } (\sigma ), $$
namely, \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1})\) is expressed as a single integral.

(ii)⇒(iv). In (12), if \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1})\) is expressed as a single integral \(k_{\lambda } (\frac{\sigma }{p} + \frac{\sigma _{1}}{q})\), then (12) keeps the form of equality. In view of the proof of Lemma 4, if and only if \(\sigma _{1} = \sigma \), (12) keeps the form of equality.

(iv)⇒(i). If \(\sigma _{1} = \sigma \), then \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) = k_{\lambda } (\sigma )\), which is independent of \(p,q\). Hence, we have (i) ↔ (ii) ⇔ (iv).

(iii)⇒(iv). By Lemma 4, we have \(\sigma _{1} = \sigma \). (iv)⇒(iii). By Lemma 3, \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\sigma _{1}) = k_{\lambda } (\sigma )\) in (9) (for \(\sigma _{1} = \sigma \)) is the best possible constant. Therefore, we have (iii) ⇔ (iv).

Hence, the statements (i), (ii), (iii) and (iv) are equivalent. □

Replacing x by \(\frac{1}{x}\), and then \(x^{\lambda - 2}f(\frac{1}{x})\) by \(f(x)\) in Theorem 1, setting \(\sigma _{1} = \lambda - \mu \), we have the following.

Corollary 1

The following inequalities with the homogeneous kernel are equivalent:
$$\begin{aligned} &\int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{(x + n)^{\lambda }} f(x)\,dx \\ &\quad < k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}( \lambda - \mu ) \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\lambda - \sigma }{p} + \frac{\mu }{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{p}} \Biggl\{ \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\lambda - \mu }{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{q}}. \end{aligned}$$
(20)
$$\begin{aligned} &\Biggl\{ \sum_{n = 1}^{\infty } n^{p(\frac{\sigma }{p} + \frac{\lambda - \mu }{q}) - 1} \biggl[ \int _{0}^{\infty } \frac{f(x)}{(x + n)^{\lambda }}\,dx \biggr]^{p} \Biggr\} ^{\frac{1}{p}} \\ &\quad < k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}( \lambda - \mu ) \biggl\{ \int _{0}^{\infty } x^{p[1 - (\frac{\lambda - \sigma }{p} + \frac{\mu }{q})] - 1}f^{p}(x)\,dx \biggr\} ^{\frac{1}{p}}, \end{aligned}$$
(21)
$$\begin{aligned} &\Biggl\{ \int _{0}^{\infty } x^{q(\frac{\lambda - \sigma }{p} + \frac{\mu }{q}) - 1} \Biggl[\sum _{n = 1}^{\infty } \frac{a_{n}}{(x + n)^{\lambda }} \Biggr]^{q} \,dx \Biggr\} ^{\frac{1}{q}} \\ &\quad < k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}( \lambda - \mu ) \Biggl\{ \sum_{n = 1}^{\infty } n^{q[1 - (\frac{\sigma }{p} + \frac{\lambda - \mu }{q})] - 1} a_{n}^{q} \Biggr\} ^{\frac{1}{q}}. \end{aligned}$$
(22)

If the constant factor in (20) is the best possible, then so is the constant factor in (21) and (22).

Corollary 2

The statements (I), (II), (III) and (IV) are equivalent:
  1. (I)

    \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\lambda - \mu )\) is independent of \(p,q\);

     
  2. (II)

    \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\lambda - \mu )\) is expressed as a single integral;

     
  3. (III)

    \(k_{\lambda }^{\frac{1}{p}}(\sigma )k_{\lambda }^{\frac{1}{q}}(\lambda - \mu )\) in (20) is the best possible constant;

     
  4. (IV)

    \(\mu + \sigma = \lambda\).

     
If the statement (IV) follows, then we have the following equivalent inequalities with the best possible constant factor \(B(\mu ,\sigma )\):
$$ \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{(x + n)^{\lambda }} f(x)\,dx< B(\mu ,\sigma ) \biggl[ \int _{0}^{\infty } x^{p(1 - \mu ) - 1}f^{p}(x)\,dx \biggr]^{\frac{1}{p}} \Biggl[\sum_{n = 1}^{\infty } n^{q(1 - \sigma ) - 1} a_{n}^{q} \Biggr]^{\frac{1}{q}}. $$
(23)
$$ \Biggl\{ \sum_{n = 1}^{\infty } n^{p\sigma - 1} \biggl[ \int _{0}^{\infty } \frac{f(x)}{(x + n)^{\lambda }}\,dx \biggr]^{p} \Biggr\} ^{\frac{1}{p}}< B(\mu ,\sigma ) \biggl[ \int _{0}^{\infty } x^{p(1 - \sigma ) - 1}f^{p}(x)\,dx \biggr]^{\frac{1}{p}}, $$
(24)
$$ \Biggl\{ \int _{0}^{\infty } x^{q\mu - 1} \Biggl[\sum _{n = 1}^{\infty } \frac{a_{n}}{(x + n)^{\lambda }} \Biggr]^{q} \,dx \Biggr\} ^{\frac{1}{q}}< B(\mu ,\sigma ) \Biggl[\sum _{n = 1}^{\infty } n^{q(1 - \sigma ) - 1} a_{n}^{q} \Biggr]^{\frac{1}{q}}. $$
(25)

Remark 1

(i) For \(\sigma = \frac{1}{p}( < \lambda )\) in (17), (18) and (19), we have the following equivalent inequalities with the nonhomogeneous kernel and the best possible constant factor \(B(\frac{1}{p},\lambda - \frac{1}{p})\):
$$ \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} f(x)\,dx< B \biggl(\frac{1}{p},\lambda - \frac{1}{p} \biggr) \biggl( \int _{0}^{\infty } x^{p - 2}f^{p}(x)\,dx \biggr)^{\frac{1}{p}} \Biggl(\sum_{n = 1}^{\infty } a_{n}^{q} \Biggr)^{\frac{1}{q}}. $$
(26)
$$ \Biggl\{ \sum_{n = 1}^{\infty } \biggl[ \int _{0}^{\infty } \frac{f(x)}{(1 + xn)^{\lambda }}\,dx \biggr]^{p} \Biggr\} ^{\frac{1}{p}}< B \biggl(\frac{1}{p},\lambda - \frac{1}{p} \biggr) \biggl( \int _{0}^{\infty } x^{p - 2}f^{p}(x)\,dx \biggr)^{\frac{1}{p}}, $$
(27)
$$ \Biggl\{ \int _{0}^{\infty } x^{q - 2} \Biggl[\sum _{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} \Biggr]^{q} \,dx \Biggr\} ^{\frac{1}{q}}< B \biggl(\frac{1}{p},\lambda - \frac{1}{p} \biggr) \Biggl(\sum_{n = 1}^{\infty } a_{n}^{q} \Biggr)^{\frac{1}{q}}. $$
(28)
(ii) For \(\sigma = \frac{1}{q}( < \lambda )\) in (17), (18) and (19), we have the following equivalent inequalities with the best possible constant factor \(B(\frac{1}{q},\lambda - \frac{1}{q})\):
$$ \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} f(x)\,dx< B \biggl(\frac{1}{q},\lambda - \frac{1}{q} \biggr) \biggl( \int _{0}^{\infty } f^{p}(x)\,dx \biggr)^{\frac{1}{p}} \Biggl(\sum_{n = 1}^{\infty } n^{q - 2}a_{n}^{q} \Biggr)^{\frac{1}{q}}. $$
(29)
$$ \Biggl\{ \sum_{n = 1}^{\infty } n^{p - 2} \biggl[ \int _{0}^{\infty } \frac{f(x)}{(1 + xn)^{\lambda }}\,dx \biggr]^{p} \Biggr\} ^{\frac{1}{p}}< B \biggl(\frac{1}{q},\lambda - \frac{1}{q} \biggr) \biggl( \int _{0}^{\infty } f^{p}(x)\,dx \biggr)^{\frac{1}{p}}, $$
(30)
$$ \Biggl\{ \int _{0}^{\infty } \Biggl[ \sum _{n = 1}^{\infty } \frac{a_{n}}{(1 + xn)^{\lambda }} \Biggr]^{q} \,dx \Biggr\} ^{\frac{1}{q}}< B \biggl(\frac{1}{q},\lambda - \frac{1}{q} \biggr) \Biggl(\sum_{n = 1}^{\infty } n^{q - 2}a_{n}^{q} \Biggr)^{\frac{1}{q}}. $$
(31)
(iii) For \(\lambda = 1\), \(\sigma = \frac{1}{p}\), \(\mu = \frac{1}{q}\) in (23), (24) and (25), we have the following equivalent inequalities with the homogeneous kernel and the best possible constant factor \(\frac{\pi }{\sin (\frac{\pi }{p})}\):
$$ \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{x + n} f(x)\,dx< \frac{\pi }{\sin (\frac{\pi }{p})} \biggl( \int _{0}^{\infty } f^{p}(x)\,dx \biggr)^{\frac{1}{p}} \Biggl(\sum_{n = 1}^{\infty } a_{n}^{q} \Biggr)^{\frac{1}{q}}, $$
(32)
$$ \Biggl[\sum_{n = 1}^{\infty } \biggl( \int _{0}^{\infty } \frac{f(x)}{x + n}\,dx \biggr)^{p} \Biggr]^{\frac{1}{p}}< \frac{\pi }{\sin (\frac{\pi }{p})} \biggl( \int _{0}^{\infty } f^{p}(x)\,dx \biggr)^{\frac{1}{p}}, $$
(33)
$$ \Biggl[ \int _{0}^{\infty } \Biggl( \sum _{n = 1}^{\infty } \frac{a_{n}}{x + n} \Biggr)^{q} \,dx \Biggr]^{\frac{1}{q}}< \frac{\pi }{\sin (\frac{\pi }{p})} \Biggl(\sum _{n = 1}^{\infty } a_{n}^{q} \Biggr)^{\frac{1}{q}}. $$
(34)
(iv) For \(\lambda = 1\), \(\sigma = \frac{1}{q}\), \(\mu = \frac{1}{p}\) in (23), (24) and (25), we have the following equivalent inequalities with the best possible constant factor \(\frac{\pi }{\sin (\frac{\pi }{p})}\):
$$ \int _{0}^{\infty } \sum_{n = 1}^{\infty } \frac{a_{n}}{x + n} f(x)\,dx< \frac{\pi }{\sin (\frac{\pi }{p})} \biggl( \int _{0}^{\infty } x^{p - 2}f^{p}(x)\,dx \biggr)^{\frac{1}{p}} \Biggl(\sum_{n = 1}^{\infty } n^{q - 2}a_{n}^{q} \Biggr)^{\frac{1}{q}}, $$
(35)
$$ \Biggl[\sum_{n = 1}^{\infty } n^{p - 2} \biggl( \int _{0}^{\infty } \frac{f(x)}{x + n}\,dx \biggr)^{p} \Biggr]^{\frac{1}{p}}< \frac{\pi }{\sin (\frac{\pi }{p})} \biggl( \int _{0}^{\infty } x^{p - 2}f^{p}(x)\,dx \biggr)^{\frac{1}{p}}, $$
(36)
$$ \Biggl[ \int _{0}^{\infty } x^{q - 2} \Biggl( \sum _{n = 1}^{\infty } \frac{a_{n}}{x + n} \Biggr)^{q} \,dx \Biggr]^{\frac{1}{q}}< \frac{\pi }{\sin (\frac{\pi }{p})} \Biggl(\sum _{n = 1}^{\infty } n^{q - 2}a_{n}^{q} \Biggr)^{\frac{1}{q}}. $$
(37)

4 Conclusions

In this paper, by using the weight functions and the idea of introducing parameters, a half-discrete Hilbert inequality with the nonhomogeneous kernel and its equivalent form are given in Theorem 1. The equivalent statements of the constant factor being best possible related to parameters, and some particular cases are considered in Theorem 2 and Remark 1. The cases of homogeneous kernel are deduced in Corollary 1 and Corollary 2. The lemmas and theorems provide an extensive account of this type of inequalities.

Declarations

Funding

This work is supported by the National Natural Science Foundation (No. 61772140), and Science and Technology Planning Project Item of Guangzhou City (No. 201707010229).

Authors’ contributions

BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. ZH participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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)
Zhangjiang Preschool Education College, Suixi, P.R. China
(2)
Department of Mathematics, Guangdong University of Education, Guangzhou, P.R. China

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