Abstract
By means of the way of weight coefficients, technique of real analysis, and Hermite-Hadamard’s inequality, a strengthened version of the Mulholland-type inequality with the best possible constant factor and multi-parameters is given. The equivalent forms, the reverses, the operator expressions and a few particular cases are considered.
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1 Introduction
If \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(a_{m},b_{n}\geq 0\), \(a=\{a_{m}\}_{m=1}^{\infty}\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\|a\|_{p}=(\sum_{m=1}^{\infty}a_{m}^{p})^{\frac{1}{p}}>0\), \(\|b\|_{q}>0\), then we have the following well-known Hardy-Hilbert’s inequality with the best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) (cf. [1], Theorem 315):
Also we have the following Mulholland’s inequality similar to (1) with the same best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) (cf. [2] or [1], Theorem 343, replacing \(\frac {a_{m}}{n}\), \(\frac{b_{n}}{n}\) by \(a_{m}\), \(b_{n}\)):
Inequalities (1) and (2) are important in analysis and its applications (cf. [1, 3–8]).
In 1998, Gao and Yang [9] gave a strengthened version of (1) as follows:
where \(1-\gamma=0.42278433^{+}\) (γ is Euler constant).
Suppose that \(\mu_{i},\upsilon_{j}>0\) (\(i,j\in\mathbf{N}=\{1,2,\ldots \}\)),
we have the following Hardy-Hilbert-type inequality (cf. [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 [10] gave an extension of (6) as follows: For \(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda_{2}=\lambda\), we have
where \(B(u,v)\) is the beta function indicated by (cf. [11])
In this paper, by using the way of weight coefficients, the technique of real analysis, and Hermite-Hadamard’s inequality, a Mulholland-type inequality with the best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) is given as follows: For \(\mu_{1}=\upsilon_{1}=1\), \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon_{n}\}_{n=1}^{\infty}\) are decreasing, and \(U_{\infty }=V_{\infty}=\infty\), we have
which is an extension of (2) (Note: the series on the right-hand side of (9) are positive). Moreover, a strengthened version of (9) and some extended Mulholland-type inequalities with multi-parameters are obtained. The equivalent forms, the reverses, the operator expressions and a few particular cases are considered.
2 Some lemmas
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), \(a_{m},b_{n}\geq 0\), \(\|a\|_{p,\Phi_{\lambda}}:=(\sum_{m=2}^{\infty}\Phi_{\lambda }(m)a_{m}^{p})^{\frac{1}{p}}\) 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 \([a-\frac{1}{2},a+\frac{1}{2}]\), \(f^{\prime}(x)\) is strictly increasing in \((a-\frac {1}{2},a)\) and \((a,a+\frac{1}{2})\), respectively, and
then we have the following Hermite-Hadamard’s inequality (cf. [12]):
Proof
Since \(f^{\prime}(a-0)\) (\(\leq f^{\prime}(a+0)\)) is finite, we set a function \(g(x)\) as follows:
In view of \(f^{\prime}(x)\) being strictly increasing in \((a-\frac{1}{2},a)\), then for \(x\in(a-\frac{1}{2},a)\), \((f(x)-g(x))^{\prime}=f^{\prime }(x)-f^{\prime}(a-0)<0\). 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 can obtain \(f(x)-g(x)>0\), \(x\in(a,a+\frac{1}{2})\). Hence, we find
namely (11) follows. □
Example 1
If \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are also decreasing, we set \(\mu(t):=\mu _{m}\), \(t\in(m-1,m]\) (\(m\in\mathbf{N}\)); \(\upsilon(t):=\upsilon_{n}\), \(t\in (n-1,n]\) (\(n\in\mathbf{N}\)),
Then it follows that \(U(m)=U_{m}\), \(V(n)=V_{n}\) (\(m,n\in\mathbf{N}\)), \(U(\infty )=U_{\infty}\), \(V(\infty)=V_{\infty}\) and
For fixed \(m,n\in\mathbf{N}\backslash\{1\}\), we also set a function \(f(x)\) as follows:
Then \(f(x)\) in 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 \(x\in(n,n+\frac{1}{2})\), we find
\(f^{\prime}(x)\) (<0) is strictly increasing in \((n,n+\frac{1}{2})\). In view of \(\upsilon_{n+1}\leq\upsilon_{n}\), it follows that \(\lim_{x\rightarrow n+}f^{\prime}(x)=f^{\prime}(n+0)\geq f^{\prime}(n-0)\). Then by (11) 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 we have the following inequalities:
where
Proof
Since 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 \(t=\frac{\ln V(x)}{\ln U_{m}}\), we obtain \(\frac{V^{\prime }(x)}{V(x)}\, dx=\ln U_{m}\, dt\) and
where
We find
Hence, by (20), we have (16) and (18). In the same way, we obtain (17) and (19). □
Note
For example, \(\mu_{n},\upsilon_{n}=\frac{1}{n^{\sigma }}\) (\(0\leq\sigma\leq1\)) are satisfied the assumptions 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 \(a>0\), we have
Proof
Since by Example 1, \(f(x)\) is strictly decreasing in \([n,n+1] \), then we find
where
There exists \(\theta(m)\in(0,1)\) such that
Since we obtain
namely \(\theta(\lambda_{2},m)=O(\frac{1}{\ln^{\lambda_{2}}U_{m}})\), we have (22). In the same way, we obtain (23).
For \(a>0\), we find
3 Main results and operator expressions
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 (29) and (30).
Proof
(i) By Hölder’s inequality with weight (cf. [12]) and (15), we have
Then by (14) we find
and then (30) follows.
By Hölder’s inequality (cf. [12]), we have
On the other hand, assuming that (29) is valid, we set
Then we find \(J^{p}=\|b\|_{q,\widetilde{\Psi}_{\lambda}}^{q}\). If \(J=0\), then (30) is trivially valid; if \(J=\infty\), then, by (32), (30) takes the form of equality. Suppose that \(0< J<\infty\). By (29), it follows that
and then (30) follows, which is equivalent to (29).
(ii) For \(0< p<1\) (or \(p<0\)), by the reverse Hölder’s inequality with weight (cf. [12]) and (15), we obtain the reverse of (31) (or (31)), then we have the reverse of (32), and then the reverse of (30) follows. By Hölder’s inequality (cf. [12]), we have the reverse of (33) and then by the reverse of (30), the reverse of (29) follows.
On the other hand, assuming that the reverse of (29) is valid, we set \(b_{n}\) as (34). Then we find \(J^{p}=\|b\|_{q,\widetilde{\Psi}_{\lambda}}^{q}\). If \(J=\infty\), then the reverse of (30) is trivially valid; if \(J=0\), then, by the reverse of (32), (30) takes the form of equality (=0). Suppose that \(0< J<\infty\). By the reverse of (29), it follows that the reverses of (35) and (36) are valid, and then the reverse of (30) follows, which is equivalent to the reverse of (29). □
Setting
we have the following.
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 (29) and (30), since
and
we obtain equivalent inequalities (38) and (39).
For \(\varepsilon\in(0,p\lambda_{1})\), we set \(\widetilde{\lambda}_{1}=\lambda_{1}-\frac{\varepsilon}{p}\) (\(\in(0,1)\)), \(\widetilde{\lambda} _{2}=\lambda_{2}+\frac{\varepsilon}{p}\) (>0), and
Then, by (26), (27) and (23), we have
If there exists a positive constant \(K\leq B(\lambda_{1},\lambda_{2})\) such that (38) is valid when replacing \(B(\lambda_{1},\lambda_{2})\) by K, then, in particular, we have \(\varepsilon\widetilde {I}<\varepsilon K\|\widetilde{a}\|_{p,\Omega_{\lambda}}\|\widetilde{b}\|_{q,\digamma _{\lambda}}\), 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 (38).
Similarly to (33), we still can find that
Hence, we can prove that the constant factor \(B(\lambda_{1},\lambda_{2})\) in (39) is the best possible. Otherwise, we would reach a contradiction by (41) that the constant factor in (38) is not the best possible. □
Remark 1
(i) It is evident that (38) and (39) are strengthened versions of the following equivalent Mulholland-type inequalities:
where the constant factor \(B(\lambda_{1},\lambda_{2})\) is still the best possible.
(ii) For \(\lambda=1\), \(\lambda_{1}=\frac{1}{q}\), \(\lambda _{2}=\frac{1}{p}\),
(38) reduces to the strengthened version of (9) as follows:
For \(\mu_{i}=\upsilon_{j}=1\) (\(i,j\in\mathbf{N}\)), (44) reduces to the following strengthened Mulholland’s inequality:
where \(\ln\sqrt{3/2}=0.20275^{+}\).
For \(p>1\), \(\Psi_{\lambda}^{1-p}(n)=\frac{\upsilon_{n+1}}{V_{n}}(\ln V_{n})^{p\lambda_{2}-1}\), we define the following normed spaces:
Assuming that \(a=\{a_{m}\}_{m=2}^{\infty}\in l_{p,\Phi_{\lambda}}\), setting
we can rewrite (43) as follows:
namely \(c\in l_{p,\Psi_{\lambda}^{1-p}}\).
Definition 2
Define a 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 (42) and (43) as follows:
Define the norm of operator T as follows:
Then by (43) we find \(\|T\|\leq B(\lambda_{1},\lambda_{2})\). Since the constant factor in (48) is the best possible, we have
4 Some strengthened versions of the reverses
In the following, we also set
For \(0< p<1\) or \(p<0\), we still use the formal symbols \(\|a\|_{p,\Phi _{\lambda}}\), \(\|b\|_{q,\Psi_{\lambda}}\), \(\|a\|_{p,\Omega_{\lambda }}\), \(\|b\|_{q,\digamma_{\lambda}}\), \(\|a\|_{p,\widetilde{\Omega}_{\lambda}}\) and \(\|b\|_{q,\widetilde{\digamma}_{\lambda}}\).
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 (22) and (17) in the reverses of (29) and (30), since
and
we obtain equivalent inequalities (51) and (52).
For \(\varepsilon\in(0,p\lambda_{1})\), we set \(\widetilde{\lambda }_{1}\),\(\widetilde{\lambda}_{2}\), \(\widetilde{a}_{m}\) and \(\widetilde{b}_{n}\) as (40). Then, by (26), (27) and (17), we find
If there exists a positive constant \(K\geq B(\lambda_{1},\lambda_{2})\) such that (51) is valid when replacing \(B(\lambda_{1},\lambda_{2})\) by K, then, in particular, we have \(\varepsilon\widetilde {I}>\varepsilon K\|\widetilde{a}\|_{p,\widetilde{\Omega}_{\lambda}}\|\widetilde{b}\|_{q,\digamma_{\lambda}}\), 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 (51).
The constant factor \(B(\lambda_{1},\lambda_{2})\) in (52) is still the best possible. Otherwise, we would reach a contradiction by the reverse of (41) that the constant factor in (51) is not the best possible. □
Remark 2
It is evident that (51) and (52) are strengthened versions 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 (23) in the reverses of (29) and (30), since
and
we obtain equivalent inequalities (55) and (56).
For \(\varepsilon\in(0,q\lambda_{2})\), we set \(\widetilde{\lambda}_{1}=\lambda_{1}+\frac{\varepsilon}{q}\) (>0), \(\widetilde{\lambda}_{2}=\lambda_{2}-\frac{\varepsilon}{q}\) (\(\in(0,1)\)), and
Then, by (26), (27) and (16), we have
If there exists a positive constant \(K\geq B(\lambda_{1},\lambda_{2})\) such that (55) is valid when replacing \(B(\lambda_{1},\lambda_{2})\) by K, then, in particular, we have \(\varepsilon\widetilde {I}>\varepsilon K\|\widetilde{a}\|_{p,\Omega_{\lambda}}\|\widetilde{b}\|_{q,\widetilde {\digamma}_{\lambda}}\), 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 (55).
Similarly to the reverse of (33), we still find that
Hence the constant factor \(B(\lambda_{1},\lambda_{2})\) in (56) is still the best possible. Otherwise, we would reach a contradiction by (57) that the constant factor in (55) is not the best possible. □
Remark 3
It is evident that (55) and (56) are strengthened versions of the following equivalent inequalities:
where the constant factor \(B(\lambda_{1},\lambda_{2})\) is still the best possible.
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Acknowledgements
The authors wish to express their thanks to the referees for their careful reading of the manuscript and for their valuable suggestions. This work is supported by the National Natural Science Foundation (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2013KJCX0140).
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BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. AW and QH participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Wang, A., Huang, Q. & Yang, B. A strengthened Mulholland-type inequality with parameters. J Inequal Appl 2015, 329 (2015). https://doi.org/10.1186/s13660-015-0852-8
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DOI: https://doi.org/10.1186/s13660-015-0852-8