Abstract
By introducing independent parameters and interval variables, applying the weight functions and the technique of real analysis, an extended Hilbert’s integral inequality in the whole plane with parameters and a best possible constant factor is provided. The equivalent forms, the reverses, and the related homogeneous forms with particular parameters are considered. Meanwhile, an extended Hilbert’s integral operator in the whole plane is defined, and the operator expressions for the equivalent inequalities are obtained.
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1 Introduction
Assuming that \(0 < \int_{0}^{\infty} f^{2}(x) \,dx < \infty \) and \(0 < \int_{0}^{\infty} g^{2}(y) \,dy < \infty \), we have the following well-known Hilbert’s integral inequality with the best possible constant factor π [1]:
In 1925, Hardy gave an extension of (1) as follows [2]: If \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1 \), \(f(x) \ge 0\), satisfying \(0 < \int_{0}^{\infty} f^{p}(x) \,dx < \infty \), and \(g(y) \ge 0 \), satisfying \(0 < \int_{0}^{\infty} g^{q}(y) \,dy < \infty \), then we have
where the constant factor \(\pi /\sin (\frac{\pi}{p}) \) is still the best possible. We call (2) Hardy–Hilbert’s integral inequality, which with (1) is important in analysis and its applications (cf. [1, 3]). In 1934, Hardy et al. gave an extension of (2) with the general homogeneous kernel of degree −1 (see [1], Theorem 319). Meanwhile, a Hilbert-type integral inequality with the general nonhomogeneous kernel is provided (see [1], Theorem 350): If \(h(x) > 0\), \(\int_{0}^{\infty} h(x)x^{s - 1} \,dx = \phi (s) \in \textbf{R}_{ +} = (0,\infty ) \), then
By introducing an independent parameter \(\lambda \in (0,\infty ) \) and the beta function, in 1998, Yang [4] gave an extension of (1) as follows:
where the constant factor \(B(\frac{\lambda}{2},\frac{\lambda}{2}) \) is the best possible, and
is the beta function (cf. [5]).
In 2007, Li [6] gave an extension of (4) and Yang [7] provided the following Hilbert-type integral inequality with the nonhomogeneous kernel:
Since then, a lot of authors have continued to discuss this topic (cf. [8–14]).
In this paper, by introducing independent parameters and interval variables, applying the weight functions and the technique of real analysis, a Hilbert-type integral inequality in the whole plane with parameters and a best possible constant factor is provided as follows:
(\(\mu,\sigma > 0\), \(\mu + \sigma = \lambda \)), which is an extension of (4). The more general form of (6) with parameters, the equivalent inequalities, the reverses, and the related homogeneous form with the particular parameter are considered. Meanwhile, an extended Hilbert’s integral operator in the whole plane is defined, and the operator expressions for the equivalent inequalities are obtained.
2 Weight functions and an initial inequality
Definition 1
Suppose that \(\delta \in \{ - 1,1\} \), \(- 1 < \alpha,\beta < 1 \), \(\mu,\sigma > 0\), \(\mu + \sigma = \lambda \). Define the following weight functions:
We find
For fixed y (≠0), setting \(u = (x - \alpha x)^{\delta} (|y| + \beta y) \) in the above first integral, we obtain that \(x = \frac{(|y| + \beta y)^{ - \frac{1}{\delta}}}{1 - \alpha} u^{\frac{1}{\delta}}\), \(dx = \frac{(|y| + \beta y)^{ - \frac{1}{\delta}}}{\delta (1 - \alpha )}u^{\frac{1}{\delta} - 1}\,du \), and
In the same way, setting \(u = (x + \alpha x)^{\delta} (|y| + \beta y) \) in the above second integral, it follows that
Hence, we have
By (8), we find
For fixed x (≠0), setting \(u = (|x| + \alpha x)^{\delta} (1 - \beta )y \) in the above first integral, we obtain \(y = \frac{u}{(1 - \beta )(|x| + \alpha x)^{\delta}}\), and
In the same way, setting \(u = (|x| + \alpha x)^{\delta} (1 + \beta )y \) in the above second integral, we find
and then
Theorem 1
Suppose that \(p > 0\) (\(p \ne 1\)), \(\frac{1}{p} + \frac{1}{q} = 1\), \(\delta \in \{ - 1,1\}\), \(- 1 < \alpha,\beta < 1 \), \(\mu,\sigma > 0\), \(\mu + \sigma = \lambda \), and
If \(f(x) \ge 0\) (\(x \in \textbf{R} \)), satisfying \(0 < \int_{ - \infty}^{\infty} (|x| + \alpha x)^{p(1 - \delta \sigma ) - 1} f^{p}(x)\,dx < \infty \), then
-
(i)
for \(p > 1 \), we have the following inequality:
$$ \begin{aligned}[b] J&: = \biggl\{ \int_{ - \infty}^{\infty} \bigl( \vert y \vert + \beta y \bigr)^{p\sigma - 1}\biggl[ \int_{ - \infty}^{\infty} \frac{f(x)}{[1 + ( \vert x \vert + \alpha x)^{\delta} ( \vert y \vert + \beta y)]^{\lambda}} \,dx \biggr]^{p}\,dy\biggr\} ^{\frac{1}{p}} \\ &< \frac{2B(\mu,\sigma )}{(1 - \beta^{2})^{1/p}(1 - \alpha^{2})^{1/q}}\biggl[ \int_{ - \infty}^{\infty} \bigl( \vert x \vert + \alpha x \bigr)^{p(1 - \delta \sigma ) - 1} f^{p}(x)\,dx\biggr]^{\frac{1}{p}}; \end{aligned} $$(12) -
(ii)
for \(0 < p < 1 \), we have the reverse of (12).
Proof
(i) For \(p > 1 \), by Hölder’s inequality with weight [15] and (7), when \(y \ne 0 \), we find
We prove that (13) takes the form of strict inequality. Otherwise, there exists \(y \ne 0 \) such that (13) takes the form of equality. Then there exist constants A and B such that they are not all zero, and [15]
If \(A = 0 \), then \(B = 0 \), which is impossible. We suppose that \(A \ne 0 \), namely
which contradicts the fact that \(0 < \int_{ - \infty}^{\infty} (|x| + \alpha x)^{p(1 - \delta \sigma ) - 1} f^{p}(x)\,dx < \infty \).
Then by (11) and Fubini’s theorem [16], we find
In view of (10) and (11), we have (12).
(ii) For \(0 < p < 1 \), by the reverse Hölder’s inequality [15], (7), and (9), we have the reverses of (13) and (14). Then, by (10) and (11), we obtain the reverse of (12).
The theorem is proved. □
3 Main results
Theorem 2
Suppose that \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(\delta \in \{ - 1,1\} \), \(- 1 < \alpha,\beta < 1 \), \(\mu,\sigma > 0\), \(\mu + \sigma = \lambda \). If \(f(x), g(y) \ge 0 \), satisfying
then we have the following inequality equivalent to (12):
where the constant \(K(\sigma ) = \frac{2B(\mu,\sigma )}{(1 - \beta^{2})^{1/p}(1 - \alpha^{2})^{1/q}} \) in (15) and (12) is the best possible.
In particular, for \(\delta = 1 \), we have the following equivalent inequalities with the nonhomogeneous kernel and the best possible constant factor \(K(\sigma ) = \frac{2B(\mu,\sigma )}{(1 - \beta^{2})^{1/p}(1 - \alpha^{2})^{1/q}} \):
Proof
By Hölder’s inequality, we find
and then by (12) we have (15).
On the other hand, suppose that (15) is valid. We set
By (14) and the assumptions, we find \(J < \infty \). If \(J = 0 \), then (12) is trivially valid; if \(J > 0 \), then by (15) we obtain
Hence, we have (12), which is equivalent to (15).
For \(n \in \textbf{N} = \{ 1,2, \ldots \}\), \(n > \frac{1}{q\mu} \), we define the sets \(E_{\delta}: = \{ x \in \textbf{R};|x|^{\delta} \ge 1\} \),
and the following functions:
Then we obtain that
In the same way, we still find that
Hence, we obtain
If there exists a constant \(k \le K(\sigma ) \) such that (15) is valid when replacing \(K(\sigma ) \) by k, then in particular, we have
In view of the above results, it follows that
For \(n \to \infty \), we find
namely \(K(\sigma ) \le k \). Hence, \(k = K(\sigma ) \) is the best possible constant factor of (15).
The constant factor \(K(\sigma ) \) in (12) is also the best possible. Otherwise, we can conclude a contradiction by (18) that the constant factor in (17) is not the best possible.
The theorem is proved. □
Theorem 3
With regards to the assumptions of Theorem 2, replacing \(p > 1 \) by \(0 < p < 1 \), we have the equivalent reverses of (12) and (15) with the best possible constant factor \(K(\sigma ) \).
Proof
We only prove that the constant factor \(K(\sigma ) \) in the reverse of (15) is the best possible, and omit the others. If there exists a constant \(k \ge K(\sigma ) \) such that the reverse of (15) is valid when replacing \(K(\sigma ) \) by k, then in particular, for \(n \in \textbf{N} = \{ 1,2, \ldots \}\), \(n > \frac{1}{|q|\sigma} \), we have
For \(n \to \infty \), we obtain that \(k \le K(\sigma ) \). Hence, \(k = K(\sigma ) \) is the best possible constant factor of the reverse of (15).
The theorem is proved. □
4 Operator expressions and a remark
For \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(\delta \in \{ - 1,1\}\), \(- 1 < \alpha,\beta < 1\), \(\mu,\sigma > 0\), \(\mu + \sigma = \lambda \), we set the following functions: \(\varphi (x): = (|x| + \alpha x)^{p(1 - \delta \sigma ) - 1}\), \(\psi (y): = (|y| + \beta y)^{q(1 - \sigma ) - 1}\), where from
Define the following real normed linear spaces:
In view of Theorem 1, for any \(f \in L_{p,\varphi} (\textbf{R}) \), we set
By (12) we have
Definition 2
Define an extended Hilbert’s integral operator in the whole plane
as follows: For any \(f \in L_{p,\varphi} (\textbf{R}) \), there exists \(Tf = h \in L_{p,\psi^{1 - p}}(\textbf{R}) \).
In view of (20), the operator T is bounded with
Since by Theorem 2 the constant factor in (20) is the best possible, we have
If we define the normal inner product of Tf and g as follows:
then we can rewrite (15) and (12) as the following equivalent operator expressions:
Remark 1
(i) In Theorem 2, for \(\delta = - 1 \), replacing \((|x| + \alpha x)^{\lambda} f(x) \) by \(f(x) \), we have
and the following equivalent inequalities with the homogeneous kernel and the best possible constant factor \(K(\sigma ) = \frac{2B(\mu,\sigma )}{(1 - \beta^{2})^{1/p}(1 - \alpha^{2})^{1/q}} \):
(ii) For \(\alpha = \beta = 0 \), inequality (24) reduces to (6), and (17) reduces to
(iii) For \(p = q = 2\), \(\mu = \sigma = \frac{\lambda}{2}\), \(f( - x) = f(x)\), \(g( - y) = g(y) \) (\(x,y > 0 \)), inequality (6) reduces to (4). Hence, inequality (6) is an extended Hilbert’s integral inequality in the whole plane, and inequality (15) is a more general form of (6) with parameters.
5 Conclusions
In this paper, by introducing independent parameters and interval variables, applying the weight functions and the technique of real analysis, an extended Hilbert’s integral inequality in the whole plane with parameters and a best possible constant factor is provided in Theorem 2. The equivalent forms, the reverses, and the related homogeneous forms with particular parameters are considered. An extended Hilbert’s integral operator in the whole plane is defined, and the operator expressions for the equivalent inequalities are obtained. The method of weight functions is very important, which helps us to prove the equivalent inequalities with the best possible constant factor. The lemmas and theorems provide an extensive account of this type of inequalities.
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Funding
This work is supported by the National Natural Science Foundation (No. 61772140) and Science and Technology Planning Project of Guangzhou City (No. 201707010229). We are grateful for this help.
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BY carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. LH and YL participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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He, L., Li, Y. & Yang, B. An extended Hilbert’s integral inequality in the whole plane with parameters. J Inequal Appl 2018, 216 (2018). https://doi.org/10.1186/s13660-018-1810-z
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DOI: https://doi.org/10.1186/s13660-018-1810-z