Abstract
Using weight coefficients and applying the well-known Hermite-Hadamard inequality, a new Hardy-Mulholand-type inequality with a best possible constant factor is given. Furthermore, we also consider the more accurate equivalent forms, the operator expressions and some particular inequalities. The lemmas and theorems provide an extensive account of this type of inequalities.
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1 Introduction
Assuming that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(a_{m},b_{n}\geq 0\), \(a=\{a_{m}\}_{m=1}^{\infty}\in l^{p}\), \(b=\{b_{n}\}_{n=1}^{\infty}\in l^{q}\), \(\|a\|_{p}=(\sum_{m=1}^{\infty}a_{m}^{p})^{\frac{1}{p}}>0\), \(\|b\|_{q}>0\), we have the following Hardy-Hilbert inequality with the best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) (cf. [1], Theorem 315):
The more accurate inequality of (1) is given as follows (cf. [2] and Theorem 323 of [1]):
which is an extension of (1). We still have the following Mulholland inequality similar to (1) with the same best possible constant factor \(\frac{\pi}{\sin(\pi/p)}\) (cf. [3] or Theorem 343 of [1], replacing \(\frac{a_{m}}{n}\), \(\frac{b_{n}}{n}\) by \(a_{m}\), \(b_{n}\)):
Inequalities (1)-(3) are important in analysis and applications (cf. [2, 4–9]).
If \(\mu_{i},\upsilon_{j}>0 \) (\(i,j\in\mathbf{N}=\{1,2,\ldots\}\)),
then we have the following Hardy-Hilbert-type inequality (cf. Theorem 321 of [1], replacing \(\mu_{m}^{1/q}a_{m}\) and \(\upsilon_{n}^{1/p}b_{n}\) by \(a_{m}\) and \(b_{n}\)):
For \(\mu_{i}=\upsilon_{j}=1\) (\(i,j\in\mathbf{N}\)), (5) reduces to (1).
In 2015, Yang [10] gave an extension of (5) as follows: For \(0<\lambda_{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(\{\mu_{m}\}_{m=1}^{\infty}\), and \(\{\upsilon_{n}\}_{n=1}^{\infty}\) are decreasing, 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 defined by (cf. [11])
In a similar way, Huang and Yang [12] gave a more accurate inequality of (6) and Yang and Chen [13] obtained a Hardy-Hilbert-type inequality with another kernel and a best possible constant factor.
In this paper, using the way of weight coefficients and applying Hermite-Hadamard’s inequality, a Hardy-Mulholland-type inequality with a best possible constant factor similar to (6) is proved, which is an extension of (3). Furthermore, the more accurate Hardy-Mulholland-type inequality is built by introducing a few parameters. We also consider the equivalent forms, the operator expressions and some particular inequalities.
2 Some lemmas and an example
In the following of this paper, we assume that \(p>1\), \(\frac{1}{p}+\frac {1}{q}=1\), \(\mu_{i},\upsilon_{j}>0\) (\(i,j\in\mathbf{N}\)), with \(\mu_{1}=\upsilon _{1}=1\), \(U_{m}\) and \(V_{n}\) are indicated by (4), \(\alpha\leq \frac{\mu_{2}}{2}\), \(\beta\leq\frac{\upsilon_{2}}{2}\), \(0<\lambda _{1},\lambda_{2}\leq1\), \(\lambda_{1}+\lambda_{2}=\lambda\), \(a_{m},b_{n}\geq0\), \(\|a\|_{p,\Phi_{\lambda}}:=(\sum_{m=2}^{\infty }\Phi _{\lambda}(m)a_{m}^{p})^{\frac{1}{p}}\), and \(\|b\|_{q,\Psi_{\lambda }}:=(\sum_{n=2}^{\infty}\Psi_{\lambda}(n)b_{n}^{q})^{\frac{1}{q}}\), where
Lemma 1
Suppose that \(a\in\mathbf{R}\), \(f(x)\) in continuous in \([a-\frac{1}{2},a+\frac{1}{2}]\), \(f^{\prime}(x)\) is strictly increasing in \((a-\frac {1}{2},a)\) and \((a,a+\frac{1}{2})\), and \(f^{\prime}(a-0)\leq f^{\prime }(a+0)\). We have the following Hermite-Hadamard inequality (cf. Lemma 1 of [14]):
Example 1
Assuming that \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are 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 we have \(U(m)=U_{m}\), \(V(n)=V_{n}\), \(U(\infty)=U_{\infty }\), \(V(\infty)=V_{\infty}\) and
For fixed \(m,n\in\mathbf{N}\backslash\{1\}\), we define the function \(h(x)\) as follows:
Then \(h(x)\) in continuous in \([n-\frac{1}{2},n+\frac{1}{2}]\), and, for \(x\in(n-\frac{1}{2},n)\) (\(n\in\mathbf{N}\backslash\{1\}\)),
In view of \(1-\lambda_{2}\geq0\), \(h^{\prime}(x)\) (<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
\(h^{\prime}(x)\) (<0) is strictly increasing in \((n,n+\frac{1}{2})\) and
Since \(\upsilon_{n+1}\leq\upsilon_{n}\), we have \(h^{\prime}(n-0)\leq h^{\prime}(n+0)\). Then by (9), for \(m,n\in\mathbf{N}\backslash\{1\}\), it follows that
Lemma 2
For \(m,n\in \mathbf{N}\backslash\{1\}\), we define the following weight coefficients:
If \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are decreasing, and \(U(\infty)=V(\infty )=\infty \), then
Proof
For \(x\in(n-\frac{1}{2},n+\frac{1}{2})\backslash \{n\}\), \(\upsilon_{n+1}\leq V^{\prime}(x)\), by (11), we obtain
Setting \(t=\frac{\ln(V(x)-\beta)}{\ln(U_{m}-\alpha)}\), since \(V(\frac{3}{2})-\beta=1+\frac{\upsilon_{2}}{2}-\beta\geq1\) and \(\frac{V^{\prime }(x)}{V(x)-\beta}\,dx=\ln(U_{m}-\alpha)\,dt\), we find
Hence, we obtain (14). In the same way, we obtain (15). □
Lemma 3
Suppose that \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are decreasing, and \(U(\infty)=V(\infty )=\infty \). (i) For \(m,n\in\mathbf{N}\backslash\{1\}\), we have
where
(ii) for any \(c>0\), we have
Proof
In view of \(0\leq\beta\leq\frac{\upsilon_{2}}{2}<\upsilon_{2}\), it follows that \(\frac{\beta}{\upsilon_{2}}+1\geq1\) and \(\frac{\beta}{\upsilon_{2}}+1<2\). By Example 1, \(h(x)\) is strictly decreasing in \([n,n+1]\), then for \(m\in\mathbf{N}\backslash\{1\}\), we obtain
Setting \(t=\frac{\ln(V(x)-\beta)}{\ln(U_{m}-\alpha)}\), since
we find
where
In view of the integral mid value theorem, there exists a \(\theta (m)\in(\frac{\beta}{\upsilon_{2}},1)\), satisfying
Since we find
namely, \(\theta(\lambda_{2},m)=O(\frac{1}{\ln^{\lambda _{2}}(U_{m}-\alpha )})\), we have (16) and (18). In the same way, we obtain (17) and (19).
For any \(c>0\), it follows that
3 Main results
We define the following functions:
Theorem 1
We have the following equivalent inequalities:
Proof
By Hölder’s inequality (cf. [15]) and (13), we find
Then by (12) we obtain
namely, (24) follows. By Hölder’s inequality (cf. [15]), we find
On the other hand, suppose that (23) is valid. We set
Then we have \(J^{p}=\|b\|_{q,\widetilde{\Psi}_{\lambda}}^{q}\). If \(J=0\), then (24) is trivially valid; if \(J=\infty\), then in view of (26), (24) takes the form of an equality. Suppose that \(0< J<\infty\). By (23), we obtain
namely, (24) follows, which is equivalent to (23). □
Theorem 2
Assuming that \(\{\mu_{m}\}_{m=1}^{\infty}\) and \(\{\upsilon _{n}\}_{n=1}^{\infty}\) are decreasing, \(U(\infty)=V(\infty)=\infty\), \(0<\|a\|_{p,\Phi_{\lambda}},\|b\|_{q,\Psi_{\lambda }}<\infty\), we have the following equivalent inequalities:
where the constant factor \(B(\lambda_{1},\lambda_{2})\) is the best possible.
Proof
Applying (14) and (15) in (23) and (24), we have the equivalent inequalities (31) and (32).
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 (20), (21), and (17), we obtain
If there exists a positive constant \(K\leq B(\lambda_{1},\lambda_{2})\), such that (31) is valid when replacing \(B(\lambda_{1},\lambda_{2})\) by K, then in particular, we have \(\varepsilon\widetilde {I}<\varepsilon K\|\widetilde{a}\|_{p,\Phi_{\lambda}}\|\widetilde{b}\|_{q,\Psi _{\lambda }}\), 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 (31).
Similarly, we can obtain
Hence, we can prove that the constant factor \(B(\lambda_{1},\lambda_{2})\) in (32) is the best possible. Otherwise, we would reach a contradiction by (34) that the constant factor in (31) is not the best possible. □
We find \(\Psi_{\lambda}^{1-p}(n)=\frac{\upsilon_{n+1}}{V_{n}-\beta }\ln ^{p\lambda_{2}-1}(V_{n}-\beta)\), and we define the following weighted normed spaces:
Assuming that \(a=\{a_{m}\}_{m=2}^{\infty}\in l_{p,\Phi_{\lambda}}\), setting
we can rewrite (32) as follows:
namely, \(c\in l_{p,\Psi_{\lambda}^{1-p}}\).
Definition 1
Define a Hardy-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 \(Ta=c\in l_{p,\Psi_{\lambda}^{1-p}}\). We set the formal inner product of Ta and \(b=\{b_{n}\}_{n=2}^{\infty}\in l_{q,\Psi _{\lambda}}\) as follows:
Then we can rewrite (31) and (32) as follows:
We set the norm of operator T as follows:
By (37), we find \(\|T\|\leq B(\lambda_{1},\lambda_{2})\). Since the constant factor in (37) is the best possible, it follows that \(\|T\|=B(\lambda_{1},\lambda_{2})\).
Remark 1
(i) For \(\alpha=\beta=0\) in (31) and (32), setting
we have the following equivalent Hardy-Mulholland-type inequalities:
For \(\lambda=1\), \(\lambda_{1}=\frac{1}{q}\), \(\lambda_{2}=\frac{1}{p}\) in (38) and (39), we have the following equivalent inequality:
Hence, (38) is an extension of (40), and (31) is a more accurate inequality of (38) (for \(0<\alpha\leq\frac {\mu_{2}}{2}\), \(0<\beta\leq\frac{\upsilon_{2}}{2}\)).
(ii) For \(\mu_{i}=\upsilon_{j}=1 \) (\(i,j\in\mathbf{N}\)), \(\lambda =1\), \(\lambda _{1}=\frac{1}{q}\), \(\lambda_{2}=\frac{1}{p}\) in (31), we reduce our case to the following inequality: For \(\alpha,\beta\leq\frac{1}{2}\),
Hence, (42) is a more accurate inequality of (3) (for \(0<\alpha,\beta\leq\frac{1}{2}\)).
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Acknowledgements
This work is supported by Hunan Province Natural Science Foundation (No. 2015JJ4041), and the National Natural Science Foundation of China (No. 61370186). Thanks for their help.
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BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. AL and LH participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
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Li, A., Yang, B. & He, L. On a new Hardy-Mulholland-type inequality and its more accurate form. J Inequal Appl 2016, 69 (2016). https://doi.org/10.1186/s13660-016-1012-5
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DOI: https://doi.org/10.1186/s13660-016-1012-5