Abstract
A new discrete Mulholland-type inequality in the whole plane with a best possible constant factor is presented by introducing multi-parameters, applying weight coefficients, and using Hermite–Hadamard’s inequality. Moreover, the equivalent forms, some particular cases, and the operator expressions are considered.
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1 Introduction
Assume that \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\), \(a_{m},b_{n} \ge0\), \(0 < \sum_{m = 1}^{\infty} a_{m}^{p} < \infty\), and \(0 < \sum_{n = 1}^{\infty} b_{n}^{q} < \infty\), Hardy–Hilbert’s inequality is provided as follows (cf. [1]):
where \(\frac{\pi}{\sin(\pi/p)}\) is the best possible constant factor. By Theorem 343 in [1] (replacing \(\frac{a_{m}}{m} \) and \(\frac {b_{n}}{n} \) by \(a _{m}\) and \(b _{n}\), respectively), it yields the following Mulholland’s inequality:
Equations (1) and (2) are important inequalities in analysis and its applications (cf. [1, 2]).
In 2007, Yang [3] firstly provided the following Hilbert-type integral inequality in the whole plane:
where \(B(\frac{\lambda}{2},\frac{\lambda}{2})\) (\(\lambda> 0\)) is the best possible constant factor. Various extensions of (1)–(3) have been presented since then (cf. [4–15]).
Recently, Yang and Chen [16] presented an extension of (1) in the whole plane as follows:
where \(2B(\lambda_{1},\lambda_{2})\) (\(0 < \lambda_{1},\lambda_{2} \le 1\), \(\lambda_{1} + \lambda_{2} = \lambda\), \(\xi,\eta\in[0,\frac{1}{2}]\)) is the best possible constant factor. In addition, Yang et al. [17, 18] also carried out a few similar works.
In this paper, we present a new discrete Mulholland-type inequality in the whole plane with a best possible constant factor that is similar to that in (4) via introducing multi-parameters, applying weight coefficients, and using Hermite–Hadamard’s inequality. Moreover, the equivalent forms, some particular cases, and the operator expressions are considered.
2 An example and two lemmas
In what follows, we assume that \(0 < \lambda_{1},\lambda_{2} < 1\), \(\lambda_{1} + \lambda_{2} = \lambda\le1\), \(\xi,\eta\in[0,\frac{1}{2}]\), \(\alpha,\beta\in[\arccos\frac{1}{3},\frac{\pi}{2}] \), and
Remark 1
In view of the assumptions that \(\xi,\eta\in [0,\frac{1}{2}]\), \(\alpha,\beta\in[\arccos\frac{1}{3},\frac{\pi}{2}] \), it follows that
Example 1
For \(u > 0 \), we set \(g(u): = \frac{\ln u}{u - 1}\) (\(u > 0\)), \(g(1): = \lim_{u \to1}g(u) = 1 \). Then we have \(g(u) > 0\), \(g'(u) < 0\), \(g''(u) > 0\) (\(u > 0\)). In fact, we find
and then \(g^{(k)}(1) = \frac{( - 1)^{k}k!}{k + 1}\) (\(k = 0,1,2, \ldots\)). Hence, \(g^{(0)}(1) = g(1)\), \(g'(1) = - \frac{1}{2}\), \(g''(1) = \frac{2}{3} \). It is evident that \(g(u) > 0 \). We obtain \(g'(u) = \frac{h(u)}{u(u - 1)^{2}}\), \(h(u): = u - 1 - u\ln u \). Since
it follows that \(h_{\max} = h(1) = 0 \) and \(h(u) < 0\) (\(u \ne1\)). Then we have \(g'(u) < 0\) (\(u \ne1\)). In view of \(g'(1) = - \frac{1}{2} < 0 \), it follows that \(g'(u) < 0\) (\(u > 0\)). We find
\(J'(u) = - 4(u - 1) + 4u\ln u \), and
It follows that \(J'_{\min} = J'(1) = 0 \), \(J'(u) > 0\) (\(u \ne1\)) and \(J(u)\) is strictly increasing. In view of \(J(1) = 0 \), we have
and \(g''(u) > 0\) (\(u \ne1\)). Since \(g''(1) = \frac{2}{3} > 0 \), we find \(g''(u) > 0\) (\(u > 0\)).
For \(0 < \lambda\le1\), \(0 < \lambda_{2} < 1 \), setting \(G(u): = g(u^{\lambda} )u^{\lambda_{2} - 1}\) (\(u > 0\)), we still have \(G(u) > 0 \), \(G'(u) = \lambda g'(u^{\lambda} )u^{\lambda+ \lambda_{2} - 2} + (\lambda_{2} - 1)g(u^{\lambda} )u^{\lambda_{2} - 2} < 0 \), and
We set \(F(x,y): = \frac{\ln(x/y)}{x^{\lambda} - y^{\lambda}} (\frac{y}{x})^{\lambda_{2} - 1}\) (\(x,y > 0\)). Since \(F(x,y) = \frac{1}{x^{\lambda}} G(\frac{y}{x})\), we have
Hence, for \(x,y > 1 \), we still have
Lemma 1
(cf. [19])
If \(f(u) > 0\), \(f'(u) < 0\), \(f''(u) > 0\) (\(u > \frac{3}{2}\)) and \(\int_{\frac{3}{2}}^{\infty} f(u)\,du < \infty \), then we have the following Hermite–Hadamard’s inequality:
and then
For \(|x|,|y| \ge\frac{3}{2} \), let the functions
\(A_{\eta,\beta} (y) = |y - \eta| + (y - \eta)\cos\beta \), and
We 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 2
The inequalities
are valid, where
Proof
For \(|m| \in\mathbf{N}\setminus\{ 1\} \), let
Then we have
yields
In virtue of \(0 < \lambda\le1\), \(0 < \lambda_{2} < 1 \), and Example 1, we find that for \(y > \frac{3}{2} \),
it follows that
are strictly decreasing and convex in \(( \frac{3}{2},\infty )\). Then, by (5), (12) yields
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 above first (second) integral, in view of Remark 1, we obtain
by simplifications. Similarly, by (5), (12) also yields
where \(\theta(\lambda_{2},m)\) (<1) is indicated by (11). Since
there exists a positive constant C such that \(\frac{\ln u}{u^{\lambda} - 1}u^{\lambda_{2}/2} \le C\) (\(0 < u \le1\)), and then for \(A_{\xi,\alpha} (m) \ge(2 + \eta)(1 + \cos\beta)\), we have
Hence, (10) and (11) are valid. □
Similarly, we have the following.
Lemma 3
For \(0 < \lambda\le1\), \(0 < \lambda_{1} < 1 \), the inequalities
are valid, where
Lemma 4
If \((\varsigma,\gamma) = (\xi,\alpha )\) (or \((\eta,\beta )\)), \(\rho> 0 \), then we have
Proof
According to (5), we obtain
and
Therefore, (16) is valid. □
3 Main results
Theorem 1
Suppose that \(p > 1\), \(\frac{1}{p} + \frac{1}{q} = 1 \), we set
If \(a_{m},b_{n} \ge0\) (\(|m|,|n| \in\mathbf{N}\setminus\{ 1\} \)) satisfy
then we obtain the following equivalent inequalities:
Particularly, (i) for \(\alpha= \beta= \frac{\pi}{2}\), \(\xi,\eta\in [0,\frac{1}{2}] \), we have the following equivalent inequalities:
(ii) For \(\xi= \eta= 0\), \(\alpha,\beta\in[\arccos\frac{1}{3},\frac{\pi}{ 2}] \), we have the following equivalent inequalities:
Proof
According to Hölder’s inequality with weight (cf. [20]) and (9), we find
Then, by (14), it yields
Combining (10) and (17), we obtain (19).
Using Hölder’s inequality again, we obtain
Then, according to (19), we obtain (18).
On the other hand, assuming that (18) is valid, we let
According to (24), it follows that \(J < \infty \). If \(J = 0 \), then (20) is trivially valid; if \(J > 0 \), then we have
Thus (19) is valid, which is equivalent to (18). □
Theorem 2
With regards to the assumptions in Theorem 1, \(k(\lambda _{1})\) is the best possible constant factor in (18) and (19).
Proof
For \(0 < \varepsilon< \min\{ q(1 - \lambda_{1}),q\lambda_{2}\} \), we let \(\tilde{\lambda}_{1} = \lambda_{1} + \frac{\varepsilon}{q}\) (\(\in(0,1)\)), \(\tilde{\lambda}_{2} = \lambda_{2} - \frac{\varepsilon}{q}\) (\(\in(0,1)\)), and
If there exists a positive number \(K \le k(\lambda_{1})\) such that (18) is still valid when replacing \(k(\lambda_{1})\) by K, then we obtain
Hence, in view of the above results, it follows that
and then
namely
Hence, \(K = k(\lambda_{1})\) is the best possible constant factor in (18).
\(k(\lambda_{1})\) in (19) is still the best possible constant factor. Otherwise we would reach a contradiction by (25) that \(k(\lambda_{1})\) in (18) is not the best possible constant factor. □
4 Operator expressions and a remark
Let \(\varphi(m): = \frac{\ln^{p(1 - \lambda_{1}) - 1}A_{\xi,\alpha} (m)}{(A_{\xi,\alpha} (m))^{1 - p}}\) (\(|m| \in\mathbf{N}\setminus\{ 1\} \)), and \(\psi(n): = \frac{\ln^{q(1 - \lambda_{2}) - 1}A_{\eta,\beta} (n)}{(A_{\eta,\beta} (n))^{1 - q}} \), wherefrom
We define the real weighted normed function spaces as follows:
For \(a = \{ a_{m}\}_{|m| = 2}^{\infty} \in l_{p,\varphi} \), we let \(c_{n} = \sum_{|m| = 2}^{\infty} k(m,n)a_{m} \) and \(c = \{ c_{n}\}_{|n| = 2}^{\infty} \), it follows by (19) that \(\|c\|_{p,\psi^{1 - p}} < k(\lambda_{1})\|a\|_{p,\varphi} \), namely \(c \in l_{p,\psi^{1 - p}} \).
Further, we define a Mulholland-type operator \(T:l_{p,\varphi} \to l_{p,\psi^{1 - p}} \) as follows: For \(a_{m} \ge0\), \(a = \{ a_{m}\}_{|m| = 2}^{\infty} \in l_{p,\varphi} \), 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 can respectively rewrite (18) and (19) as the following operator expressions:
It follows that the operator T is bounded with
Since \(k(\lambda_{1})\) in (19) is the best possible constant factor, we obtain
Remark 2
(i) For \(\xi= \eta= 0 \) in (20), we have the following new inequality:
It follows that (20) is an extension of (31). In particular, for \(\lambda= 1\), \(\lambda_{1} = \frac{1}{q}\), \(\lambda_{2} = \frac{1}{p} \), we have the following simple Mulholland-type inequality in the whole plane with the best possible constant factor \(\frac{2\pi^{2}}{\sin^{2}(\frac{\pi}{p})} \):
(ii) If \(a_{ - m} = a_{m}\), \(b_{ - n} = b_{n}\) (\(m,n \in\mathbf{N}\setminus \{ 1\} \)), then (20) reduces to
In particular, for \(\lambda= 1\), \(\lambda_{1} = \frac{1}{q}\), \(\lambda_{2} = \frac{1}{p}\), \(\xi= \eta\in[0,\frac{1}{2}] \), we obtain
For \(\xi= 0 \), (34) reduces to the following simple Mulholland-type inequality with the best possible constant factor \(\frac{\pi^{2}}{\sin^{2}(\frac{\pi}{p})} \):
5 Conclusions
In this paper, we present a new discrete Mulholland-type inequality in the whole plane with a best possible constant factor that is similar to that in (4) via introducing multi-parameters, applying weight coefficients, and using Hermite–Hadamard’s inequality in Theorem 1 and Theorem 2. Moreover, the equivalent forms, some particular cases, and the operator expressions are considered. The lemmas and theorems provide an extensive account of this type of inequalities.
References
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Mitrinović, D.S., Pecarić, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)
Yang, B.: A new Hilbert’s type integral inequality. Soochow J. Math. 33(4), 849–859 (2007)
Hong, Y.: All-sided generalization about Hardy–Hilbert integral inequalities. Acta Math. Sin. 44(4), 619–626 (2001)
Milovanović, G.V., Rassias, M.Th. (eds.): Analytic Number Theory, Approximation Theory and Special Functions. Springer, Berlin (2014)
Rassias, M.Th., Yang, B.: On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 242, 800–813 (2014)
Rassias, M.Th., Yang, B.: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput. 225, 263–277 (2013)
Krnić, M., Pečarić, J.E.: General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 8(1), 29–51 (2005)
Perić, I., Vuković, P.: Multiple Hilbert’s type inequalities with a homogeneous kernel. Banach J. Math. Anal. 5(2), 33–43 (2011)
Agarwal, R.P., O’Regan, D., Saker, S.H.: Some Hardy-type inequalities with weighted functions via Opial type inequalities. Adv. Dyn. Syst. Appl. 10, 1–9 (2015)
Adiyasuren, V., Tserendorj, B., Krnić, M.: Multiple Hilbert-type inequalities involving some differential operators. Banach J. Math. Anal. 10(2), 320–337 (2016)
Li, Y., He, B.: On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 76(1), 1–13 (2007)
Krnić, M., Vuković, P.: On a multidimensional version of the Hilbert type inequality. Anal. Math. 38(4), 291–303 (2012)
Huang, Q., Yang, B.: A more accurate half-discrete Hilbert inequality with a nonhomogeneous kernel. J. Funct. Spaces Appl. 2013, Article ID 628250 (2013)
He, B., Wang, Q.: A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor. J. Math. Anal. Appl. 431(2), 889–902 (2015)
Yang, B., Chen, Q.: A new extension of Hardy–Hilbert’s inequality in the whole plane. J. Funct. Spaces 2016, Article ID 9197476 (2016)
Xin, D., Yang, B., Chen, Q.: A discrete Hilbert-type inequality in the whole plane. J. Inequal. Appl. 2016, Article ID 133 (2016)
Zhong, Y., Yang, B., Chen, Q.: A more accurate Mulholland-type inequality in the whole plane. J. Inequal. Appl. 2017, Article ID 315 (2017)
Yang, B.: A more accurate multidimensional Hardy–Hilbert’s inequality. J. Appl. Anal. Comput. 8(2), 559–573 (2018)
Kuang, J.: Applied Inequalities. Shangdong Science Technic Press, Jinan (2010) (in Chinese)
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).
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BY carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. 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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Yang, B., Chen, Q. On a new discrete Mulholland-type inequality in the whole plane. J Inequal Appl 2018, 184 (2018). https://doi.org/10.1186/s13660-018-1777-9
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DOI: https://doi.org/10.1186/s13660-018-1777-9