Abstract
By using weight coefficients, technique of real analysis, and Hermite-Hadamard’s inequality, we give a more accurate Hardy-Mulholland-type inequality with multiparameters and a best possible constant factor related to the beta function. The equivalent forms, the reverses, the operator expressions, and some particular cases are also considered.
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1 Introduction
Assuming that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(a_{m},b_{n}\geq 0\), \(a=\{a_{m}\}_{m=1}^{\infty}\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty }\in l^{q}\), \(\|a\|_{p}=(\sum_{m=1}^{\infty}a_{m}^{p})^{\frac{1}{p}}>0\), and \(\|b\|_{q}>0\), we have the following Hardy-Hilbert inequality with the best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) (see [1], Theorem 315):
The more accurate and extended inequality of (1) is given as follows (see [1], Theorem 323 and [2]):
where the constant factor \(\frac{\pi}{\sin(\pi/p)}\) is the best possible. Also, we have the following Mulholland inequality similar to (1) with the same best possible constant factor \(\frac{\pi}{\sin (\pi/p)}\) (see [3] or [1], Theorem 343, replacing \(\frac {a_{m}}{n}\), \(\frac{b_{n}}{n}\) by \(a_{m}\), \(b_{n}\)):
Inequalities (1)-(3) are important in analysis and its applications (see [1, 2, 4–20]).
Suppose that \(\mu_{i},\upsilon_{j}>0\) (\(i,j\in\mathbf{N}=\{1,2,\ldots \}\)) and
Then we have the following Hardy-Hilbert-type inequality ([1], Theorem 321):
For \(\mu_{i}=\upsilon_{j}=1\) (\(i,j\in\mathbf{N}\)), inequality (5) reduces to (1). Replacing \(\mu_{m}^{1/q}a_{m}\) and \(\upsilon _{n}^{1/p}b_{n}\) by \(a_{m}\) and \(b_{n}\) in (5), respectively, we obtain the equivalent form of (5) as follows:
In 2015, Yang [21] gave the following extension of (6). For \(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda_{2}=\lambda \), decreasing sequences \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon_{n}\}_{n=1}^{\infty}\), and \(U_{\infty}=V_{\infty}=\infty\), we have the following inequality with the best possible constant factor \(B(\lambda _{1},\lambda _{2})\):
where \(B(u,v)\) is the beta function (see [22]):
In this paper, by using weight coefficients, technique of real analysis, and the Hermite-Hadamard inequality, we give a Hardy-Mulholland-type inequality with a best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) as follows.
For \(\mu_{1}=\upsilon_{1}=\)1, decreasing sequences \(\{\mu_{m}\} _{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\), and \(U_{\infty}=V_{\infty}=\infty\), we have
which is an extension of (3). So, we have obtained a more accurate and extended inequality of (9) with multiparameters and a best possible constant factor \(B(\lambda _{1},\lambda_{2})\). We also consider the equivalent forms, the reverses, the operator expressions, and some particular cases.
2 Some lemmas and an example
In the following, we make appointment that \(p\neq0,1\), \(\frac{1}{p}+\frac {1}{q}=1\), \(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda _{2}=\lambda\), \(\mu_{i},\upsilon_{j}>0\) (\(i,j\in\mathbf{N}\)), with \(\mu _{1}=\upsilon _{1}=1\), \(U_{m}\) and \(V_{n}\) are defined by (4), \(\frac {1}{1+\frac{\mu_{2}}{2}}\leq\alpha\leq1\), \(\frac{1}{1+\frac{\upsilon _{2}}{2}}\leq \beta\leq1\), \(a_{m},b_{n}\geq0\), \(\|a\|_{p,\Phi_{\lambda }}:=(\sum_{m=2}^{\infty}\Phi_{\lambda}(m)a_{m}^{p})^{\frac{1}{p}}\), and \(\|b\|_{q,\Psi_{\lambda}}:=(\sum_{n=2}^{\infty}\Psi_{\lambda }(n)b_{n}^{q})^{\frac{1}{q}}\), where
Lemma 1
If \(a\in\mathbf{R}\), \(f(x)\) is continuous in \([a-\frac{1}{2},a+\frac{1}{2}]\), and \(f^{\prime}(x)\) is strictly increasing in the intervals \((a-\frac{1}{2},a)\) and \((a,a+\frac{1}{2})\) and satisfying
then we have the following Hermite-Hadamard inequality (cf. [23]):
Proof
Since \(f^{\prime}(a-0)\) (\(\leq f^{\prime}(a+0)\)) is finite, we define the linear function \(g(x)\) as follows:
Since \(f^{\prime}(x)\) is strictly increasing in \((a-\frac{1}{2},a)\), we have that, for \(x\in(a-\frac{1}{2},a)\),
Since \(f(a)-g(a)=0\), it follows that \(f(x)-g(x)>0\), \(x\in(a-\frac{1}{2},a)\). In the same way, we obtain \(f(x)-g(x)>0\), \(x\in(a,a+\frac{1}{2})\). Hence, we find
that is, (11) follows. □
Example 1
If \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are decreasing, then we define the functions \(\mu(t):=\mu_{m}\), \(t\in(m-1,m]\) (\(m\in\mathbf{N}\)); \(\upsilon(t):=\upsilon_{n}\), \(t\in (n-1,n]\) (\(n\in\mathbf{N}\)), and
Then it follows that \(U(m)=U_{m}\), \(V(n)=V_{n}\), \(U(\infty)=U_{\infty }\), \(V(\infty)=V_{\infty}\), and
For fixed \(m,n\in\mathbf{N}\backslash\{1\}\), we also define the function
Then \(f(x)\) is continuous in \([n-\frac{1}{2},n+\frac{1}{2}]\). For \(x\in (n-\frac{1}{2},n)\) (\(n\in\mathbf{N}\backslash\{1\}\)), we find
Since \(1-\lambda_{2}\geq0\), it follows that \(f^{\prime}(x)\) (<0) is strictly increasing in \((n-\frac{1}{2},n)\) and
In the same way, for \(x\in(n,n+\frac{1}{2})\) (\(n\in\mathbf {N}\backslash\{1\}\)), we find
so that \(f^{\prime}(x)\) (<0) is strict increasing in \((n,n+\frac {1}{2})\). In view of \(\upsilon_{n+1}\leq\upsilon_{n}\), it follows that
Then by (11), for \(m,n\in\mathbf{N}\backslash\{1\}\), we have
Definition 1
Define the following weight coefficients:
Lemma 2
If \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are decreasing and \(U_{\infty}=V_{\infty }=\infty\), then for \(m,n\in\mathbf{N}\backslash\{1\}\), we have the following inequalities:
Proof
For \(x\in(n-\frac{1}{2},n+\frac{1}{2})\backslash \{n\}\), \(\upsilon_{n+1}\leq V^{\prime}(x)\), by (13) we find
Setting \(t=\frac{\ln\beta V(x)}{\ln\alpha U_{m}}\), since \(\beta V(\frac{3}{2})=\beta(1+\frac{\upsilon_{2}}{2})\geq1\) and \(\frac{V^{\prime }(x)}{V(x)}\,dx=(\ln\alpha U_{m})\,dt\), we find
Hence, we obtain (16). In the same way, we obtain (17). □
Note
For example, \(\mu_{n},\upsilon_{n}=\frac{1}{n^{\sigma}}\) (\(0\leq\sigma\leq1\)) satisfy the conditions of Lemma 2.
Lemma 3
With the assumptions of Lemma 2, (i) for \(m,n\in\mathbf{N}\backslash\{1\}\), we have
where
(ii) for any \(c>0\), we have
Proof
In view of \(\beta\leq1\) and \(\beta\geq\frac{1}{ 1+\upsilon_{2}/2}>\frac{1}{1+\upsilon_{2}}\), it follows that \(1\leq \frac{1-\beta}{\beta\upsilon_{2}}+1<2\). Since, by Example 1, \(f(x)\) is strictly decreasing in \([n,n+1]\), for \(m\in\mathbf{N}\backslash\{1\}\), we find
Setting \(t=\frac{\ln\beta V(x)}{\ln\alpha U_{m}}\), we have \(\ln\beta V(\frac{1-\beta}{\beta\upsilon_{2}}+1)=\ln\beta(1+\frac{1-\beta }{\beta\upsilon_{2}}\upsilon_{2})=0\) and
where
There exists \(\theta(m)\in(\frac{1-\beta}{\beta\upsilon_{2}},1)\) such that
Since we find
namely, \(\theta(\lambda_{2},m)=O(\frac{1}{\ln^{\lambda_{2}}\alpha U_{m}})\), we obtain (18) and (20). In the same way, we obtain (19) and (21).
For any \(c>0\), we find
3 Main results and operator expressions
In the following, we also set
Theorem 1
(i) For \(p>1\), we have the following equivalent inequalities:
(ii) for \(0< p<1\) (or \(p<0\)), we have the equivalent reverses of (25) and (26).
Proof
(i) By Hölder’s inequality with weight (see [23]) and (15) we have
Then by (14) we find
and then (26) follows.
By Hölder’s inequality we have
On the other hand, assuming that (25) is valid, we set
Then we find \(J^{p}=\|b\|_{q,\widetilde{\Psi}_{\lambda}}^{q}\). If \(J=0\), then (26) is trivially valid; if \(J=\infty\), then by (28), (26) takes the form of equality. Suppose that \(0< J<\infty\). By (25) it follows that
and then (26) follows, which is equivalent to (25).
(ii) For \(0< p<1\) (or \(p<0\)), by the reverse Hölder inequality with weight and (15), we obtain the reverse of (27) (or (27)), then we have the reverse of (28), and then the reverse of (26) follows. By Hölder’s inequality we have the reverse of (29), and then by the reverse of (26) the reverse of (25) follows.
On the other hand, assuming that the reverse of (25) is valid, we set \(b_{n}\) as in (30). Then we find \(J^{p}=\|b\|_{q,\widetilde{\Psi }_{\lambda}}^{q}\). If \(J=\infty\), then the reverse of (26) is trivially valid; if \(J=0\), then by the reverse of (28), (26) takes the form of equality (=0). Suppose that \(0< J<\infty\). By the reverse of (25) it follows that the reverses of (31) and (32) are valid, and then the reverse of (26) follows, which is equivalent to the reverse of (25). □
Theorem 2
If \(p>1\), \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are decreasing, \(U_{\infty}=V_{\infty}=\infty \), \(\|a\|_{p,\Phi_{\lambda}}\in\mathbf{R}_{+}\), and \(\|b\|_{q,\Psi _{\lambda}}\in\mathbf{R}_{+}\), then we have the following equivalent inequalities:
where the constant factor \(B(\lambda_{1},\lambda_{2})\) is the best possible.
Proof
Using (16) and (17) in (25) and (26), we obtain equivalent inequalities (33) and (34).
For \(\varepsilon\in(0,p\lambda_{1})\), we set \(\tilde{\lambda}_{1}=\lambda_{1}-\frac{\varepsilon}{p}\) (\(\in(0,1)\)), \(\tilde{\lambda} _{2}=\lambda_{2}+\frac{\varepsilon}{p}\) (>0), and
Then by (22), (23), and (19) we have
If there exists a positive constant \(K\leq B(\lambda_{1},\lambda_{2})\) such that (33) is valid when replacing \(B(\lambda_{1},\lambda_{2})\) by K, then, in particular, we have \(\varepsilon\tilde {I}<\varepsilon K\|\tilde{a}\|_{p,\Phi_{\lambda}}\|\tilde{b}\|_{q,\Psi _{\lambda}}\), namely
It follows that \(B(\lambda_{1},\lambda_{2})\leq K\) (\(\varepsilon \rightarrow 0^{+}\)). Hence, \(K=B(\lambda_{1},\lambda_{2})\) is the best possible constant factor of (33).
Similarly to (29), we still can find the following inequality:
Hence, we can prove that the constant factor \(B(\lambda_{1},\lambda_{2})\) in (34) is the best possible. Otherwise, we would reach a contradiction by (36) that the constant factor in (33) is not the best possible. □
Remark 1
(i) For \(\alpha=\beta=1\) in (33) and (34), setting
we have the following equivalent Mulholland-type inequalities:
which are extensions of (9), and the following inequality:
(ii) For \(\mu_{i}=\upsilon_{j}=1\) (\(i,j\in\mathbf{N}\)), \(\lambda =1\), \(\lambda_{1}=\frac{1}{q}\), \(\lambda_{2}=\frac{1}{p}\), (33) reduces to the following more accurate and extended Mulholland’s inequality:
where \(\frac{2}{3}\leq\alpha,\beta\leq1\).
For \(p>1\), \(\Psi_{\lambda}^{1-p}(n)=\frac{\upsilon_{n+1}}{V_{n}}(\ln \beta V_{n})^{p\lambda_{2}-1}\), we define the following normed spaces:
Assuming that \(a=\{a_{m}\}_{m=2}^{\infty}\in l_{p,\Phi_{\lambda}}\) and setting
we can rewrite (34) as follows:
that is, \(c\in l_{p,\Psi_{\lambda}^{1-p}}\).
Definition 2
Define the Mulholland-type operator \(T:l_{p,\Phi _{\lambda}}\rightarrow l_{p,\Psi_{\lambda}^{1-p}}\) as follows: For any \(a=\{a_{m}\}_{m=2}^{\infty}\in l_{p,\Phi_{\lambda}}\), there exists a unique representation \(Ta=c\in l_{p,\Psi_{\lambda}^{1-p}}\). Define the formal inner product of Ta and \(b=\{b_{n}\}_{n=2}^{\infty}\in l_{q,\Psi_{\lambda}}\) as follows:
Then we can rewrite (33) and (34) as follows:
Define the norm of the operator T as follows:
Then by (43) we find \(\|T\|\leq B(\lambda_{1},\lambda_{2})\). Since the constant factor in (43) is the best possible, we have
4 Some reverses
In the following, we also set
For \(0< p<1\) or \(p<0\), we still use the formal symbols \(\|a\|_{p,\Phi _{\lambda}}\), \(\|b\|_{q,\Psi_{\lambda}}\), \(\|a\|_{p,\widetilde {\Omega}_{\lambda}}\), and \(\|b\|_{q,\widetilde{\digamma}_{\lambda}}\), and so on.
Theorem 3
If \(0< p<1\), \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are decreasing, \(U_{\infty}=V_{\infty}=\infty \), \(\|a\|_{p,\Phi_{\lambda}}\in\mathbf{R}_{+}\), and \(\|b\|_{q,\Psi _{\lambda}}\in\mathbf{R}_{+}\), then we have the following equivalent inequalities with the best possible constant factor \(B(\lambda_{1},\lambda_{2})\):
Proof
Using (18) and (17) in the reverses of (25) and (26), since
and
we obtain equivalent inequalities (46) and (47).
For \(\varepsilon\in(0,p\lambda_{1})\), we set \(\tilde{\lambda }_{1}\), \(\tilde{\lambda}_{2}\), \(\tilde{a}_{m}\), and \(\tilde{b}_{n}\) as in (35). Then by (22), (23), and (17) we find
If there exists a positive constant \(K\geq B(\lambda_{1},\lambda_{2})\) such that (46) is valid when replacing \(B(\lambda_{1},\lambda_{2})\) by K, then, in particular, we have \(\varepsilon\tilde {I}>\varepsilon K\|\tilde{a}\|_{p,\widetilde{\Omega}_{\lambda}}\|\tilde{b}\|_{q,\Psi_{\lambda}}\), namely
It follows that \(B(\lambda_{1},\lambda_{2})\geq K\) (\(\varepsilon \rightarrow 0^{+}\)). Hence, \(K=B(\lambda_{1},\lambda_{2})\) is the best possible constant factor of (46).
The constant factor \(B(\lambda_{1},\lambda_{2})\) in (47) is still the best possible. Otherwise, we would reach a contradiction by the reverse of (36) that the constant factor in (46) is not the best possible. □
Remark 2
For \(\alpha=\beta=1\), setting
it is evident that (46) and (47) are extensions of the following equivalent inequalities:
where the constant factor \(B(\lambda_{1},\lambda_{2})\) is still the best possible.
Theorem 4
If \(p<0\), \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are decreasing, \(U_{\infty}=V_{\infty}=\infty \), \(\|a\|_{p,\Phi_{\lambda}}\in\mathbf{R}_{+}\), and \(\|b\|_{q,\Psi _{\lambda}}\in\mathbf{R}_{+}\), then we have the following equivalent inequalities with the best possible constant factor \(B(\lambda_{1},\lambda_{2})\):
Proof
Using (16) and (19) in the reverses of (25) and (26), since
and
we obtain equivalent inequalities (50) and (51).
For \(\varepsilon\in(0,q\lambda_{2})\), we set \(\tilde{\lambda}_{1}=\lambda_{1}+\frac{\varepsilon}{q}\) (>0), \(\tilde{\lambda}_{2}=\lambda_{2}-\frac{\varepsilon}{q}\) (\(\in(0,1)\)), and
Then by (22), (23), and (16) we have
If there exists a positive constant \(K\geq B(\lambda_{1},\lambda_{2})\) such that (50) is valid when replacing \(B(\lambda_{1},\lambda_{2})\) by K, then, in particular, we have \(\varepsilon\tilde {I}>\varepsilon K\|\tilde{a}\|_{p,\Phi_{\lambda}}\|\tilde{b}\|_{q,\widetilde{\digamma}_{\lambda}}\), namely
It follows that \(B(\lambda_{1},\lambda_{2})\geq K\) (\(\varepsilon \rightarrow 0^{+}\)). Hence, \(K=B(\lambda_{1},\lambda_{2})\) is the best possible constant factor of (50).
Similarly to the reverse of (29), we still can find that
Hence, the constant factor \(B(\lambda_{1},\lambda_{2})\) in (51) is still the best possible. Otherwise, we would reach a contradiction by (52) that the constant factor in (50) is not the best possible. □
Remark 3
For \(\alpha=\beta=1\), setting
it is evident that (50) and (51) are extensions of the following equivalent inequalities:
where the constant factor \(B(\lambda_{1},\lambda_{2})\) is still the best possible.
References
Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934)
Yang, BC: Discrete Hilbert-Type Inequalities. Bentham Science Publishers, Sharjah (2011)
Mulholland, HP: Some theorems on Dirichlet series with positive coefficients and related integrals. Proc. Lond. Math. Soc. 29(2), 281-292 (1929)
Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)
Yang, BC: Hilbert-Type Integral Inequalities. Bentham Science Publishers, Sharjah (2009)
Yang, BC: On Hilbert’s integral inequality. J. Math. Anal. Appl. 220, 778-785 (1998)
Yang, BC: An extension of Mulholand’s inequality. Jordan J. Math. Stat. 3(3), 151-157 (2010)
Yang, BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)
Rassias, MT, Yang, BC: On half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75-93 (2013)
Rassias, MT, Yang, BC: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput. 225, 263-277 (2013)
Huang, QL, Yang, BC: A more accurate half-discrete Hilbert inequality with a non-homogeneous kernel. J. Funct. Spaces Appl. 2013, Article ID 628250 (2013)
Huang, QL, Wang, AZ, Yang, BC: A more accurate half-discrete Hilbert-type inequality with a general non-homogeneous kernel and operator expressions. Math. Inequal. Appl. 17(1), 367-388 (2014)
Liu, T, Yang, BC, He, L: On a half-discrete reverse Mulholland-type inequality and extension. J. Inequal. Appl. 2014, 103 (2014)
Rassias, MT, Yang, BC: On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 242, 800-813 (2014)
Huang, QL, Wu, SH, Yang, BC: Parameterized Hilbert-type integral inequalities in the whole plane. Sci. World J. 2014, Article ID 169061 (2014)
Chen, Q, Yang, BC: On a more accurate multidimensional Mulholland-type inequality. J. Inequal. Appl. 2014, 322 (2014)
Rassias, MT, Yang, BC: On a multidimensional Hilbert-type integral inequality associated to the gamma function. Appl. Math. Comput. 249, 408-418 (2014)
Rassias, MT, Yang, BC: A Hilbert-type integral inequality in the whole plane related to the hyper geometric function and the beta function. J. Math. Anal. Appl. 428(2), 1286-1308 (2015)
Gao, MZ, Yang, BC: On the extended Hilbert’s inequality. Proc. Am. Math. Soc. 126(3), 751-759 (1998)
Chen, Q, Yang, BC: A survey on the study of Hilbert-type inequalities. J. Inequal. Appl. 2015, 302 (2015)
Yang, BC: An extension of a Hardy-Hilbert-type inequality. J. Guangdong Univ. Educ. 35(3), 1-7 (2015)
Wang, DX, Guo, DR: Introduction to Spectral Functions. Science Press, Beijing (1979)
Kuang, JC: Applied Inequalities. Shangdong Science Technic Press, Jinan (2004)
Acknowledgements
This work is supported by the National Natural Science Foundation (No. 61370186), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). We are grateful for their help.
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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. Both authors read and approved the final manuscript.
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Yang, B., Chen, Q. On a more accurate Hardy-Mulholland-type inequality. J Inequal Appl 2016, 82 (2016). https://doi.org/10.1186/s13660-016-1026-z
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DOI: https://doi.org/10.1186/s13660-016-1026-z