Abstract
By introducing independent parameters, applying the weight coefficients, and Hermite-Hadamard’s inequality, we give a more accurate Mulholland-type inequality in the whole plane with a best possible constant factor. Furthermore, the equivalent forms, the reverses, a few particular cases, and the operator expressions are considered.
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1 Introduction
If \(p > 1 \), \(\frac{1}{p} + \frac{1}{q} = 1 \), \(a_{m},b_{n} \ge 0 \), \(0 < \sum_{m = 1}^{\infty} a_{m}^{p} < \infty \) and \(0 < \sum_{n = 1}^{\infty} b_{n}^{q} < \infty \), then we have the following Hardy-Hilbert’s inequality (cf. [1]):
where the constant factor \(\frac{\pi}{\sin (\pi /p)} \) is the best possible. In 1934, Hardy proved the following more accurate inequality of (1) with the same best possible constant factor (cf. Theorem 343 of [2]):
We still have the following Mulholland’s inequality with the same best possible constant factor \(\frac{\pi}{\sin (\pi /p)} \) (cf. Theorem 343 of [2], replacing \(\frac{a_{m}}{m} \), \(\frac{b_{n}}{n} \) by \(a_{m} \), \(b_{n} \)):
Inequalities (1)-(3) are important in analysis and its applications (cf. [2, 3]). In 2007, Yang [4] first gave a Hilbert-type integral inequality in the whole plane. Many extensions of this type inequalities and (1)-(3) were provided in [5–20].
In 2016, Yang and Chen [21] gave a more accurate Hardy-Hilbert’s inequality in the whole plane:
where the constant factor \(2B(\lambda_{1},\lambda_{2}) \) (\(0 < \lambda_{1},\lambda_{2} \le 1 \), \(\lambda_{1} + \lambda_{2} = \lambda \), \(\xi,\eta \in [0,\frac{1}{2}] \)) is the best possible.
In this paper, by introducing independent parameters and applying the weight coefficients and Hermite-Hadamard’s inequality, we give a new more accurate extension of (3) in the whole plane with a best possible constant factor similar to (4). Furthermore, we consider the equivalent forms, the reverses, a few particular cases, and the operator expressions.
2 Some lemmas
In the following, we agree that \(p \ne 0,1 \), \(\frac{1}{p} + \frac{1}{q} = 1 \), \(\lambda_{1},\lambda_{2} > 0 \), \(\lambda_{1} + \lambda_{2} = \lambda \),
\(\xi,\eta \in ( - \frac{3}{2},\frac{3}{2}) \), satisfying
and
Note 1
For \(\alpha,\beta = \frac{\pi}{2} \), we find \(\xi,\eta \in [ - \frac{1}{2},\frac{1}{2}] \). If \(\alpha,\beta \in [\arccos \sqrt{\frac{1}{3}},\pi - \arccos \sqrt{\frac{1}{3}} ] \), then \({\xi,\eta = 0} \) satisfy (5).
For \(|x|,|y| \ge \frac{3}{2} \), we set \(A_{\xi,\alpha} (x): = |x - \xi | + (x - \xi )\cos \alpha \),
and
Definition 1
Define two weight coefficients as follows:
where \(\sum_{|j| = 2}^{\infty} \cdots = \sum_{j = - 2}^{ - \infty} \cdots \sum_{j = 2}^{\infty} \cdots \) (\(j = m,n \)).
Lemma 1
For \(\lambda_{2} \le 1 \), we have the following inequalities:
where
Proof
For \(|m| \in \mathbb{N}\setminus \{ 1\} \), we set
wherefrom
We find
For fixed \(|m| \in \mathbb{N}\setminus \{ 1\} \), since \(\lambda > 0 \), \(0 < \lambda_{2} \le 1 \),we find that, for \(y > \frac{3}{2} \),
and it follows that
are strict decreasing and strictly convex in \((\frac{3}{2},\infty )\). By Hermite-Hadamard’s inequality (see [22]) and (12) we find
In view of (5), it follows that \((\frac{3}{2} \pm \eta )(1 \mp \cos \beta ) \ge 1 \). Setting \(u = \frac{\ln [(y + \eta )(1 - \cos \beta )]}{\ln A_{\xi,\alpha} (m)} \) (\(u = \frac{\ln [(y - \eta )(1 + \cos \beta )]}{\ln A_{\xi,\alpha} (m)} \)) in the first (second) integral, by simplifications we obtain
By monotonicity and (12) we still have
where \(\theta (\lambda_{2},m) \) is indicated by (11). It follows that \(\theta (\lambda_{2},m) < 1 \) and
Hence, (10) and (11) are valid. □
In the same way, we still have the following:
Lemma 2
For \(\lambda_{1} \le 1 \), we have the following inequalities:
where
Lemma 3
If \(\rho > 0 \), \(\gamma \in [\arccos \sqrt{\frac{1}{3}},\pi - \arccos \sqrt{\frac{1}{3}} ] \) (\(\gamma = \alpha,\beta \)), and
then for \((\varsigma,\gamma ) = (\xi,\alpha ) \) (or \((\eta,\beta ) \)), we have
Proof
By Hermite-Hadamard’s inequality we find
We still can find that
Hence, we have (15). □
3 Main results and the reverses
We also set
Theorem 1
Suppose that \(p > 1 \), \(\lambda_{1},\lambda_{2} \le 1 \), \(a_{m},b_{n} \ge 0 \) (\(|m|,|n| \in \mathbb{N}\setminus \{ 1\} \)), and
We have the following equivalent inequalities:
In particular, (i) for \(\xi,\eta \in [ - \frac{1}{2},\frac{1}{2}]\) (\(\alpha = \beta = \frac{\pi}{2}\)), we have the following equivalent inequalities:
(ii) For \(\alpha,\beta \in [\arccos \frac{1}{3},\pi - \arccos \frac{1}{3}]\) (\(\xi = \eta = 0\)), we have the following equivalent inequalities:
Proof
By Hölder’s inequality with weight (cf. [22]) and (9) we find
By (13) it follows that
By (10) and (16) we have (18).
Using Hölder’s inequality again, we have
and then by (18) we have (17).
On the other hand, assuming that (17) is valid, we set
and find
By (23) it follows that \(J < \infty \). If \(J = 0 \), then (18) is trivially valid; if \(J > 0 \), then we have
Hence (18) is valid, which is equivalent to (17). □
Theorem 2
With regards to the assumptions of Theorem 1, the constant factor \(K(\lambda_{1}) \) in (17) and (18) is the best possible.
Proof
For \(0 < \varepsilon < q\lambda_{2} \), we set \(\tilde{\lambda}_{1} = \lambda_{1} + \frac{\varepsilon}{q}\) (>0), \(\tilde{\lambda}_{2} = \lambda_{2} - \frac{\varepsilon}{q}\) (\(\in (0,1)\)), and
If there exists a positive number \(k \le K(\lambda_{1}) \), such that (17) is still valid when replacing \(K(\lambda_{1}) \) by k, then in particular we have
We obtain from the above results that
and then
namely, \(K(\lambda_{1}) = 2B(\lambda_{1},\lambda_{2})\csc^{\frac{2}{p}}\beta \csc^{\frac{2}{q}}\alpha \le k \). Hence, \(k = K(\lambda_{1}) \) is the best value of (17).
The constant factor \(K(\lambda_{1}) \) in (18) is still the best possible. Otherwise, we would reach a contradiction by (24) that the constant factor in (17) is not the best possible. □
Theorem 3
Suppose that \(0 < p < 1\), \(\lambda_{1},\lambda_{2} \le 1\), \(a_{m},b_{n} \ge 0\) (\(|m|,|n| \in \mathbb{N}\setminus \{ 1\} \)), and
We have the following equivalent inequalities:
where the constant factor \(K(\lambda_{1}) \) in (25), (26), and (27) is the best possible.
Proof
By the reverse Hölder inequality with weight (cf. [22]), and (9), we find
Since \(p - 1 < 0 \), by (13) it follows that
By (10) and (16) we have (26).
Using the reverse Hölder’s inequality again, we have
and then by using (26) we have (25).
On the other hand, assuming that (25) is valid, we set
and find
By (28) it follows that \(J > 0 \). If \(J = \infty \), then (26) is trivially valid; if \(0 < J < \infty \), then we have
Hence (26) is valid, which is equivalent to (25).
By the reverse Hölder inequality with weight (cf. [22]), and (9) we find
Since \(q < 0 \), by (10) it follows that
By (13) and (16) we have (27).
In the same way, we find
and then we can prove that (27) and (25) are equivalent.
For \(0 < \varepsilon < \min \{ p\lambda_{1},|q|\lambda_{1}\} \), we set \(\tilde{\lambda}_{1} = \lambda_{1} - \frac{\varepsilon}{p}\) (\(\in (0,1)\)), \(\tilde{\lambda}_{2} = \lambda_{2} + \frac{\varepsilon}{p}\) (>0), and
If there exists a positive number \(k \ge K(\lambda_{1}) \), such that (25) is still valid when replacing \(K(\lambda_{1}) \) by k, then in particular we have
We obtain from the above results that
and then
namely, \(K(\lambda_{1}) = 2B(\lambda_{1},\lambda_{2})\csc^{\frac{2}{p}}\beta \csc^{\frac{2}{q}}\alpha \ge k \). Hence, \(k = K(\lambda_{1}) \) is the best value of (25).
The constant factor \(K(\lambda_{1}) \) in (26) is still the best possible. Otherwise, we would reach a contradiction by (30) that the constant factor in (25) is not the best possible.
In the same way, by (30) we can proved that the constant factor \(K(\lambda_{1}) \) in (27) is still the best possible. □
4 Operator expressions
Setting \(\varphi (m): = \frac{\ln^{p(1 - \lambda_{1}) - 1}A_{\xi,\alpha} (m)}{(A_{\xi,\alpha} (m))^{1 - p}}\) (\(|m| \in \mathbb{N}\setminus \{ 1\} \)), and \(\psi (n): = \frac{\ln^{q(1 - \lambda_{2}) - 1}B_{\eta,\beta} (n)}{(B_{\eta,\beta} (n))^{1 - q}} \), wherefrom \(\psi^{1 - p}(n) = \frac{\ln^{p\lambda_{2} - 1}B_{\eta,\beta} (n)}{B_{\eta,\beta} (n)}\) (\(|n| \in \mathbb{N}\setminus \{ 1\} \)), we define the real weighted normed function spaces as follows:
For \(a = \{ a_{m}\}_{|m| = 2}^{\infty} \in l_{p,\varphi} \), putting \(c_{n} = \sum_{|m| = 2}^{\infty} k(m,n)a_{m} \) and \(c = \{ c_{n}\}_{|n| = 2}^{\infty} \), it follows by (18) that \(\|c\|_{p,\psi^{1 - p}} < K(\lambda_{1})\|a\|_{p,\varphi} \), namely \(c \in l_{p,\psi^{1 - p}} \).
Definition 2
Define the Mulholland-type operator \(T:l_{p,\varphi} \to l_{p,\psi^{1 - p}} \)as follows: For \(a_{m} \ge 0\), \(a = \{ a_{m}\}_{|m| = 2}^{\infty} \in l_{p,\varphi} \), there exists a unique representation \(Ta = c \in l_{p,\psi^{1 - p}} \). We also define the following formal inner product of Ta and \(b = \{ b_{n}\}_{|n| = 2}^{\infty} \in l_{q,\psi}\) (\(b_{n} \ge 0\)):
Hence, we may rewrite (17) and (18) in the following operator expressions:
It follows that the operator T is bounded by
Since the constant factor \(K(\lambda_{1}) \) in (18) is the best possible, we have
Remark 1
-
(i)
For \(\xi = \eta = 0 \) in (19), we have the following new inequality:
$$\begin{aligned}[b] &\sum_{|n| = 2}^{\infty} \sum _{|m| = 2}^{\infty} \frac{a_{m}b_{n}}{\ln^{\lambda} |mn|}\\ &\quad < 2B( \lambda_{1},\lambda_{2}) \Biggl[ \sum _{|m| = 2}^{\infty} \frac{\ln^{p(1 - \lambda_{1}) - 1}|m|}{|m|^{1 - p}}a_{m}^{p} \Biggr]^{\frac{1}{p}} \Biggl[ \sum_{|n| = 2}^{\infty} \frac{\ln^{q(1 - \lambda_{2}) - 1}|n|}{|n|^{1 - q}}b_{n}^{q} \Biggr]^{\frac{1}{q}}.\end{aligned} $$(36)It follows that (19) is a more accurate inequalitythan (36); so is (17).
-
(ii)
If \(a_{ - m} = a_{m}\), \(b_{ - n} = b_{n}\) (\(m,n \in \mathbb{N}\setminus \{ 1\}\)), \(\xi,\eta \in [0,\frac{1}{2}] \), then (19) reduces to the following inequality:
$$\begin{aligned}[b] &\sum_{n = 2}^{\infty} \sum _{m = 2}^{\infty} \biggl[ \frac{1}{\ln^{\lambda} [(m - \xi )(n - \eta )]} + \frac{1}{\ln^{\lambda} [(m - \xi )(n + \eta )]} \\ &\qquad+ \frac{1}{\ln^{\lambda} [(m + \xi )(n - \eta )]} + \frac{1}{\ln^{\lambda} [(m + \xi )(n + \eta )]} \biggr]a_{m}b_{n} \\ &\quad< 2B(\lambda_{1},\lambda_{2}) \Biggl\{ \sum _{m = 2}^{\infty} \biggl[ \frac{\ln^{p(1 - \lambda_{1}) - 1}(m - \xi )}{(m - \xi )^{1 - p}} + \frac{\ln^{p(1 - \lambda_{1}) - 1}(m + \xi )}{(m + \xi )^{1 - p}} \biggr]a_{m}^{p} \Biggr\} ^{\frac{1}{p}} \\&\qquad\times \Biggl\{ \sum_{n = 2}^{\infty} \biggl[ \frac{\ln^{q(1 - \lambda_{2}) - 1}(n - \eta )}{(n - \eta )^{1 - q}} + \frac{\ln^{q(1 - \lambda_{2}) - 1}(n + \eta )}{(n + \eta )^{1 - q}} \biggr]b_{n}^{q} \Biggr\} ^{\frac{1}{q}}.\end{aligned} $$(37) -
(iii)
If \(\lambda = 1\), \(\lambda_{1} = \frac{1}{q}\), \(\lambda_{2} = \frac{1}{p}\), \(\xi = \eta \in [0,\frac{1}{2}] \), then (37) reduces to
$$\begin{aligned}[b] &\sum_{n = 2}^{\infty} \sum _{m = 2}^{\infty} \biggl[ \frac{1}{\ln (m - \xi )(n - \xi )} + \frac{1}{\ln (m - \xi )(n + \xi )}\\ &\quad\quad + \frac{1}{\ln (m + \xi )(n - \xi )} + \frac{1}{\ln (m + \xi )(n + \xi )} \biggr]a_{m}b_{n} \\&\quad< \frac{2\pi}{\sin (\frac{\pi}{p})} \Biggl\{ \sum_{m = 2}^{\infty} \biggl[ \frac{1}{(m - \xi )^{1 - p}} + \frac{1}{(m + \xi )^{1 - p}} \biggr]a_{m}^{p} \Biggr\} ^{\frac{1}{p}} \\ &\qquad\times\Biggl\{ \sum_{n = 2}^{\infty} \biggl[ \frac{1}{(n - \xi )^{1 - q}} + \frac{1}{(n + \xi )^{1 - q}} \biggr]b_{n}^{q} \Biggr\} ^{\frac{1}{q}}.\end{aligned} $$(38)For \(\xi = 0 \), (38) reduces to (3). Hence, (17) is a more accurate extension of (3).
5 Conclusions
In this paper, by introducing independent parameters and applying the weight coefficients and Hermite-Hadamard’s inequality we give a more accurate Mulholland-type inequality in the whole plane with a best possible constant factor in Theorems 1-2. Furthermore, the equivalent forms, the reverses in Theorem 3, a few particular cases, and the operator expressions are considered. The method of real analysis is very important, which is the key to prove the equivalent inequalities with the best possible constant factor. The lemmas and theorems provide an extensive account of this type inequalities.
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Acknowledgements
This work is supported by the National Natural Science Foundation (Nos. 61370186, 61640222, and 61562016) and Science and Technology Planning Project Item 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. YZ and QC participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Zhong, Y., Yang, B. & Chen, Q. A more accurate Mulholland-type inequality in the whole plane. J Inequal Appl 2017, 315 (2017). https://doi.org/10.1186/s13660-017-1589-3
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DOI: https://doi.org/10.1186/s13660-017-1589-3