Abstract
By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.
Similar content being viewed by others
1 Introduction
Suppose that \(f(x)\) and \(\mu (x)\) (>0) are two measurable functions defined on a measurable set Ω, and \(p>1\). Define
Specially, for \(\mu (x)=1\), we have the abbreviated forms: \(\|f\|_{p,\mu }:=\|f\|_{p}\) and \(L^{p}_{\mu }(\Omega ):= L^{p}(\Omega )\).
Consider two real-valued functions \(f, g\geq 0\) and \(f, g\in L^{p}(\mathbb{R}_{+})\). Suppose that \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). Then we have the following two classical Hilbert-type inequalities [1]:
where the constant factors \(\frac{\pi }{\sin \frac{\pi }{p}}\) and pq in (1.2) and (1.3) are the best possible.
In the past 100 years, especially after the 1990s, by the introduction of several parameters and special functions such as β-function and Γ-function, some classical Hilbert-type integral inequalities like (1.2) and (1.3) as well as their discrete forms were extended to more general forms (see [2–12]). The inequality below is a typical extension of (1.2) which was established by Yang [13] in 2004:
where \(\rho >0\), \(\mu (x)=x^{p(1-\frac{\beta }{r})-1}\), \(\nu (x)=x^{q(1-\frac{\beta }{s})-1}\), \(\frac{1}{r}+\frac{1}{s}=1\), and the constant factor is the best possible.
In recent years, by constructing new kernel functions and studying their discrete form, half-discrete form, reverse form, multi-dimensional extension and coefficient refinement, researchers have established a large number of new Hilbert-type inequalities (see [14–25]). It should be noted that the establishment of these inequalities fully demonstrates the techniques of modern analysis and proves to be critical in the development of modern analysis [26].
In these numerous publications related to the Hilbert inequality, we will present some results with the kernels involving exponent function, and the motivation of this work is precisely from these results. The first result presented below was established by Yang [27] in 2012, that is,
where \(a>0\), \(\beta >0\), \(\mu (x)=x^{-2\beta +1}\) and \(\nu (y)=y^{2\beta +1}\).
In addition, Liu [28] established an inequality with the kernel involving hyperbolic secant function in 2013, and Yang [29] established an inequality with the kernel involving hyperbolic cosecant function in 2014. The two inequalities can be written as follows:
where \(\operatorname{sech} (t)=\frac{2}{e^{t}+e^{-t}}\), \(\operatorname{csch} (t)=\frac{2}{e^{t}-e^{-t}}\), \(\mu (x)=x^{-3}\) and \(c_{0}=\sum_{k=0}^{\infty }\frac{(-1)^{k}}{(2k+1)^{2}}=0.91596559^{+}\), which is the Catalan constant.
In this work, the integral interval of inequality (1.6) and (1.7) will be extended to the whole plane, and the following new inequalities will be established:
where \(\mu (x)=| x| ^{-4n-1}\), \(\nu (x)=| x| ^{-4n+1}\), \(E_{n}\) is an Euler number [30, 31] and \(B_{n}\) is a Bernoulli number [30, 31].
Furthermore, we also present some interesting inequalities involving other hyperbolic functions in this paper. More generally, a new kernel function including both the homogeneous case and the non-homogeneous case is constructed, and a Hilbert-type inequality involving this new kernel is established. It will be shown that the newly obtained inequality is a unified extension of (1.8), (1.9) and some other special Hilbert-type inequalities.
2 Some lemmas
Lemma 2.1
Let \(r,s>0\), \(r+s=1\), \(\varphi (x)=\cot {x}\). Then
Proof
Consider the rational fraction expansion of \(\varphi (x)=\cot x\) [30]:
and find the \((2n-1)\)th derivative of \(\varphi (x)\), then we obtain
Set \(x=r\pi \) in (2.3). Since \(r+s=1\), it follows that
Therefore, (2.1) is proved, and similar computation yields (2.2). □
Furthermore, by considering the following rational fraction expansion of \(\psi (x)=\csc x\) [30]:
we can obtain Lemma 2.2.
Lemma 2.2
Let \(r,s>0\), \(r+s=1\) and \(\psi (x)=\csc {x}\). Then
Remark 2.3
Let \(r=\frac{1}{2}\) in (2.1). For \(n\in {\mathbb{N}}^{+}\), we have
By using the equality [30, 31] \(\sum_{k=1}^{\infty }\frac{1}{k^{2n}}=\frac{(2\pi )^{2n}}{2(2n)!}B_{n} \), where \(B_{n}\) is a Bernoulli number, we have
Applying (2.7) to (2.6), we obtain
In addition, letting \(r=\frac{1}{4}\) in (2.1), we can also obtain
Furthermore, let \(r=\frac{1}{4}\) in (2.2) and \(r=\frac{1}{2}\) in (2.5). In view of [30] \(\sum_{k=0}^{\infty }\frac{(-1)^{k}}{(2k+1)^{2n+1}}= \frac{\pi ^{2n+1}}{2^{2n+2}(2n)!}E_{n} \), where \(E_{n}\) is an Euler number, we obtain
Lemma 2.4
Let \(\eta _{1},\eta _{2}, \delta \in \{1,-1\}\), \(a>c\geq d>b>0 \) and \(c\neq d\) for \(\eta _{2}=-1\). Let β be such that \(\beta \geq 1 \), and \(\beta \neq 1\) for \(\eta _{1}=-1\), \(\eta _{2}=1\). Define
and
Then
where \(\Gamma (\beta )=\int _{\mathbb{R}_{+}}x^{\beta -1}e^{-x}\,\mathrm{d} x\) (\(\beta >0\)) is the second type of Euler integral (Γ-function) [30, 31], and \(\Gamma (\beta )=(\beta -1)!\) for \(\beta \in {\mathbb{N}}^{+}\).
Proof
Observing that \(a>b>0 \) and \(\eta _{1}\in \{1,-1\}\), we obtain
Hence
Setting \(t=\frac{u}{k\ln \frac{a}{b}+\ln \frac{a}{c}}\), we have
Similarly, we can obtain
Applying (2.17) and (2.18) to (2.16), we obtain
Similarly, we can obtain
Plugging (2.19) and (2.20) into (2.14), and using (2.12), we have (2.13). □
Lemma 2.5
Let \(\eta _{1},\eta _{2}, \delta \in \{1,-1\}\), \(a>c\geq d>b>0\) and \(c\neq d\) for \(\eta _{2}=-1\). Let β be such that \(\beta \geq 1 \), and \(\beta \neq 1\) for \(\eta _{1}=-1\), \(\eta _{2}=1\). \(C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta ) \) is defined by Lemma 2.4. Suppose \(D_{\delta }=\{y: | y| ^{\delta }<1\}\), and, for arbitrary natural number n which is large enough, set
Then
Proof
Setting \(D_{\delta }^{+}=\{y: y>0, y\in D_{\delta }\}\), \(D_{\delta }^{-}=\{y: y<0, y\in D_{\delta }\}\), we get
Setting \(xy^{\delta }=t\), and using Fubini’s theorem, where δ is equal to 1 or −1, we can get
Similarly, setting \(xy^{\delta }=-t\), we can also obtain
Applying (2.23) and (2.24) to (2.22), we have
Letting \(n\to \infty \) in (2.25), and using (2.13), we get
The proof of Lemma 2.5 is completed. □
3 Main results
Theorem 3.1
Let \(\eta _{1},\eta _{2}, \delta \in \{1,-1\}\), \(a>c\geq d >b>0\) and \(c\neq d\) for \(\eta _{2}=-1\). Let β be such that \(\beta \geq 1 \), and \(\beta \neq 1\) for \(\eta _{1}=-1\), \(\eta _{2}=1\). Suppose that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\mu (x)=| x| ^{p(1-\beta )-1}\) and \(\nu (y)=| y| ^{q(1-\delta \beta )-1}\). Let \(f(x)\), \(g(y)\geq 0\) with \(f(x)\in L_{\mu }^{p}(\mathbb{R})\) and \(g(y)\in L_{\nu }^{q}(\mathbb{R})\). \(K(x,y)\) and \(C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta )\) are defined via Lemma 2.4. Then
where the constant factor \(\Gamma (\beta )C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta )\) is the best possible.
Proof
By Hölder’s inequality, we obtain
where \(\omega (x)=\int _{\mathbb{R}} K(x, y)| y| ^{\delta \beta -1} \,\mathrm{d} y\), and \(\varpi (y)=\int _{\mathbb{R}} K(x, y)| x| ^{\beta -1}{ \,\mathrm{d} x} \).
Setting \(xy^{\eta }=t\), and using (2.13), we have
and
Plugging (3.3) and (3.4) back to (3.2), we have
If (3.5) takes the form of an equality, then there must be two constants \(A_{1}\) and \(A_{2}\) that are not both equal to zero, such that
a.e. in \(\mathbb{R}\times \mathbb{R}\), that is,
a.e. in \(\mathbb{R}\times \mathbb{R}\). Therefore, there exists a constant A such that \(A_{1}| x| ^{p(1-\beta )}f^{p}(x)=A\), a.e. in \(\mathbb{R}\), and \(A_{2}| y| ^{q(1-\delta \beta )}g^{q}(y)=A\), a.e. in \(\mathbb{R}\). Without loss of generality, we assume that \(A_{1}\neq 0\), then we can obtain \(x^{p(1-\beta )-1}f^{p}(x)=\frac{A}{A_{1}x}\) a.e. in \(\mathbb{R}\), which contradicts the fact \(f(x)\in L_{\mu }^{p}(\mathbb{R})\). Hence, (3.5) keeps the form of a strict inequality, and (3.1) is obtained.
What we need to prove next is that the constant factor \(\Gamma (\beta )C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta )\) in inequality (3.1) is the best possible. Suppose that there exists a positive constant \(k<\Gamma (\beta )C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta )\), such that (3.1) still holds if \(C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta )\) is replaced by k. That is,
Replacing f and g in (3.6) with \(f_{n}\) and \(g_{n}\) defined in Lemma 2.5, respectively, we obtain
Combining (3.7) and (2.21), we have \(C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta )+o(1)< k \). Let \(n\rightarrow \infty \), then we have \(C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta )\leq k\). This contradicts the hypothesis obviously. Therefore, the constant factor in (3.1) is the best possible. □
Theorem 3.2
Let \(\eta _{1},\eta _{2}, \delta \in \{1,-1\}\), \(a>c\geq d> b>0\) and \(c\neq d\) for \(\eta _{2}=-1\). Let β be such that \(\beta \geq 1 \), and \(\beta \neq 1\) for \(\eta _{1}=-1\), \(\eta _{2}=1\). Suppose that \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), \(\mu (x)=| x| ^{p(1-\beta )-1}\) and \(\nu (y)=| y | ^{q(1-\delta \beta )-1}\). Let \(f(x)\geq 0\) with \(f(x)\in L_{\mu }^{p}(0, \infty )\). \(K(x,y)\) and \(C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta )\) be defined via Lemma 2.4. Then
where the constant factor \((\Gamma (\beta )C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta ) )^{p}\) is the best possible, and (3.8) is equivalent to (3.1).
Proof
Consider \(g(y):=| y| ^{p\delta \beta -1} (\int _{\mathbb{R}}K(x, y)f(x){ \,\mathrm{d} x} )^{p-1}\). By Theorem 3.1, we can get
Therefore
Since \(f(x)\in L_{\mu }^{p}(\mathbb{R})\), by using (3.10), we obtain \(g(y)\in L_{\nu }^{q}(\mathbb{R})\). By using Theorem 3.1 again, we can deduce that both (3.9) and (3.1) take the form of a strict inequality, and (3.8) is proved. On the other hand, if (3.8) is valid, by Hölder’s inequality, we obtain
Combining (3.8) and (3.11), we can get (3.1). Therefore, (3.1) is equivalent to (3.8). From the equivalence of (3.1) and (3.8), we can easily show that the constant factor \((\Gamma (\beta )C_{\eta _{1},\eta _{2}}(a, b, c, d, \beta ) )^{p}\) in (3.8) is the best possible. Theorem 3.2 is proved. □
4 Applications
Setting \(\eta _{1}=-1\), \(\eta _{2}=1\), \(c=d=1\) and \(\beta =2n\) (\(n\in {\mathbb{N}}^{+}\)) in Theorem 3.1, and using (2.1), we can obtain
Therefore, the following corollary holds.
Corollary 4.1
Let \(\delta \in \{1,-1\}\), \(a>1> b>0\). Let \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). Suppose that \(\varphi (x)=\cot x\), \(\mu (x)=| x| ^{p(1-2n)-1}\) and \(\nu (y)=| y | ^{q(1-2\delta n)-1}\), where \(n\in {\mathbb{N}}^{+}\). Let \(f(x)\), \(g(y)\geq 0\) with \(f(x)\in L_{\mu }^{p}(\mathbb{R})\) and \(g(y)\in L_{\nu }^{q}(\mathbb{R})\). Then
Particularly, let \(a=b^{-1}=e\) in (4.1), by virtue of (2.8), then (4.1) is transformed to
Let \(p=q=2 \), \(\delta =1\) in (4.2), then we have (1.9).
Setting \(\eta _{1}=1\), \(\eta _{2}=1\), \(c=d=1\) and \(\beta =2n+1\) (\(n\in {\mathbb{N}}\)) in Theorem 3.1, and using (2.5), we have
Therefore, we obtain the following corollary.
Corollary 4.2
Let \(\delta \in \{1,-1\}\) and \(a>1> b>0 \). Let \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). Suppose that \(\psi (x)=\csc x\), \(\mu (x)=| x| ^{-2np-1}\) and \(\nu (y)=| y | ^{-2\delta nq-1}\), where \(n\in {\mathbb{N}}\). Let \(f(x)\), \(g(y)\geq 0\) with \(f(x)\in L_{\mu }^{p}(\mathbb{R})\) and \(g(y)\in L_{\nu }^{q}(\mathbb{R})\). Then
Particularly, let \(a=b^{-1}=e\) in (4.3), by virtue of (2.10), then (4.3) is transformed to
Let \(p=q=2\) and \(\delta =1\) in (4.4), then we get (1.8).
Setting \(\eta _{1}=-1\), \(\eta _{2}=1\), \(ab=cd\) and \(\beta =2n\) (\(n\in {\mathbb{N}}^{+}\)) in Theorem 3.1, and using (2.1), we obtain
Hence, we can obtain another corollary as follows.
Corollary 4.3
Let \(\delta \in \{1,-1\}\), \(a>c\geq d >b>0\) and \(ab=cd\). Let \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). Suppose that \(\varphi (x)=\cot x\), \(\mu (x)=| x| ^{p(1-2n)-1}\) and \(\nu (y)=| y | ^{q(1-2\delta n)-1}\), where \(n\in {\mathbb{N}}^{+}\). Let \(f(x)\), \(g(y)\geq 0\) with \(f(x)\in L_{\mu }^{p}(\mathbb{R})\) and \(g(y)\in L_{\nu }^{q}(\mathbb{R})\). Then
Let \(a=e^{\lambda _{1}}\), \(b=e^{-\lambda _{1}}\), \(c=e^{\lambda _{2}}\) and \(d=e^{-\lambda _{2}}\) in (4.5), where \(\lambda _{1}>\lambda _{2}>0\), then we have
Letting \(\lambda _{1}=2\) and \(\lambda _{2}=1\) in (4.6), in view of (2.9), we can also obtain (4.2).
Letting \(\lambda _{1}=4\) and \(\lambda _{2}=1\) in (4.6), we have
Setting \(\eta _{1}=-1\), \(\eta _{2}=-1\), \(ab=cd\) and \(\beta =2n+1\) (\(n\in {\mathbb{N}}\)) in Theorem 3.1, and using (2.2), we have
Therefore, the following corollary holds obviously.
Corollary 4.4
Let \(\delta \in \{1,-1\}\), \(a>c\geq d >b>0\) and \(ab=cd\). Let \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\). Suppose that \(\varphi (x)=\cot x\), \(\mu (x)=| x| ^{-2np-1}\) and \(\nu (y)=| y | ^{-2\delta nq-1}\), where \(n\in {\mathbb{N}}\). Let \(f(x)\), \(g(y)\geq 0\) with \(f(x)\in L_{\mu }^{p}(\mathbb{R})\) and \(g(y)\in L_{\nu }^{q}(\mathbb{R})\). Then
Let \(a=e^{\lambda _{1}}\), \(b=e^{-\lambda _{1}}\), \(c=e^{\lambda _{2}}\) and \(d=e^{-\lambda _{2}}\) in (4.8), where \(\lambda _{1}>\lambda _{2}>0\), then we have
Let \(\lambda _{1}=2\) and \(\lambda _{2}=1\) in (4.9), then it follows from (2.10) that we also get (4.4).
Let \(\lambda _{1}=4\) and \(\lambda _{2}=1\) in (4.9), then we get
Setting \(\eta _{1}=1\), \(\eta _{2}=-1\), \(ab=cd\) and \(\beta =2n\) (\(n\in {\mathbb{N}}^{+}\)) in Theorem 3.1, and using (2.4), we have
Therefore, we obtain the following corollary.
Corollary 4.5
Let \(\delta \in \{1,-1\}\), \(a>c\geq d >b>0\) and \(ab=cd\). Let \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). Suppose that \(\psi (x)=\csc x\), \(\mu (x)=| x| ^{p(1-2n)-1}\) and \(\nu (y)=| y | ^{q(1-2\delta n)-1}\), where \(n\in {\mathbb{N}}^{+}\). Let \(f(x)\), \(g(y)\geq 0\) with \(f(x)\in L_{\mu }^{p}(\mathbb{R})\) and \(g(y)\in L_{\nu }^{q}(\mathbb{R})\). Then
Let \(a=e^{\lambda _{1}}\), \(b=e^{-\lambda _{1}}\), \(c=e^{\lambda _{2}}\) and \(d=e^{-\lambda _{2}}\) in (4.11), where \(\lambda _{1}>\lambda _{2}>0\), then we have
Let \(\lambda _{1}=2\) and \(\lambda _{2}=1\) in (4.12), then we can have
At last, setting \(\eta _{1}=1\), \(\eta _{2}=1\), \(ab=cd\) and \(\beta =2n+1\) (\(n\in {\mathbb{N}}\)) in Theorem 3.1, by virtue of (2.5), then the following corollary holds.
Corollary 4.6
Let \(\delta \in \{1,-1\}\), \(a>c\geq d >b>0\) and \(ab=cd\). Let \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). Suppose that \(\psi (x)=\csc x\), \(\mu (x)=| x| ^{-2np-1}\) and \(\nu (y)=| y | ^{-2\delta nq-1}\), where \(n\in {\mathbb{N}}\). Let \(f(x)\), \(g(y)\geq 0\) with \(f(x)\in L_{\mu }^{p}(\mathbb{R})\) and \(g(y)\in L_{\nu }^{q}(\mathbb{R})\). Then
Let \(a=e^{\lambda _{1}}\), \(b=e^{-\lambda _{1}}\), \(c=e^{\lambda _{2}}\) and \(d=e^{-\lambda _{2}}\) in (4.14), where \(\lambda _{1}>\lambda _{2}>0\), then we have
Letting \(\lambda _{1}=2\), \(\lambda _{2}=1\) in (4.15), we have
Availability of data and materials
Not applicable.
References
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, London (1952)
Rassias, M.Th., Yang, B.C.: On a Hilbert-type integral inequality in the whole plane related to the extended Riemann zeta function. Complex Anal. Oper. Theory 13(4), 1765–1782 (2019)
Rassias, M.Th., Yang, B.C.: On a Hilbert-type integral inequality related to the extended Hurwitz zeta function in the whole plane. Acta Appl. Math. 160(1), 67–80 (2019)
Rassias, M.Th., Yang, B.C.: A Hilbert-type integral inequality in the whole plane related to the hypergeometric function and the beta function. J. Math. Anal. Appl. 428, 1286–1308 (2015)
Yang, B.C.: A note on Hilbert’s integral inequalities. Chin. Q. J. Math. 13, 83–85 (1998)
Yang, B.C.: On Hilbert’s integral inequality. J. Math. Anal. Appl. 220, 778–785 (1998)
Krnić, M., Pečarić, J.: General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 8, 29–51 (2005)
Krnić, M., Pečarić, J., Perić, I., et al.: Advances in Hilbert-Type Inequalities. Element Press, Zagreb (2012)
Krnić, M., Pečarić, J., Vuković, P.: Discrete Hilbert-type inequalities with general homogeneous kernels. Rend. Circ. Mat. Palermo 60, 161–171 (2011)
Krnić, M., Pečarić, J.: Extension of Hilbert’s inequality. J. Math. Anal. Appl. 324, 150–160 (2006)
Kuang, J.C., Debnath, L.: On new generalizations of Hilbert’s inequality and their applications. J. Math. Anal. Appl. 245, 248–265 (2000)
You, M.H.: On a new discrete Hilbert-type inequality and its application. Math. Inequal. Appl. 18, 1575–1587 (2015)
Yang, B.C.: On an extension of Hilbert’s integral inequality with some parameters. Aust. J. Anal. Appl. 1(1), 1–8 (2004)
Rassias, M.Th., Yang, B.C.: On an equivalent property of a reverse Hilbert-type integral inequality related to the extended Hurwitz-zeta function. J. Math. Inequal. 13(2), 315–334 (2019)
Rassias, M.Th., Yang, B.C., Raigorodskii, A.: Two kinds of the reverse Hardy-type integral inequalities with the equivalent forms related to the extended Riemann zeta function. Appl. Anal. Discrete Math. 12, 273–296 (2018)
Rassias, M.Th., Yang, B.C., Raigorodskii, A.: On a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic secant function related to the Hurwitz zeta function. In: Trigonometric Sums and Their Applications, pp. 229–259. Springer, Berlin (2020)
Rassias, M.Th., Yang, B.C.: On half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75–93 (2013)
Yang, B.C., Wu, S.H., Wang, A.Z.: On a reverse half-discrete Hardy–Hilbert’s inequality with parameters. Mathematics (2019). https://doi.org/10.3390/math7111054
Yang, B.C.: An extended multi-dimensional Hardy-Hilbert-type inequality with a general homogeneous kernel. Int. J. Nonlinear Anal. Appl. 9, 131–143 (2018)
Yang, B.C., Debnath, L.: Half-Discrete Hilbert-Type Inequalities. World Scientific, Singapore (2014)
Yang, B.C.: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)
Krnić, M., Pečarić, J., Vuković, P.: A unified treatment of half-discrete Hilbert-type inequalities with a homogeneous kernel. Mediterr. J. Math. 10, 1697–1716 (2013)
Gao, X., Gao, M.Z.: A new Hilbert-type integral inequality with parameters. J. Math. Res. Expo. 31, 467–473 (2011)
Hong, Y., He, B., Yang, B.C.: Necessary and sufficient conditions for the validity of Hilbert-type inequalities with a class of quasi-homogeneous kernels ans its applications in operator theory. J. Math. Inequal. 12, 777–788 (2018)
He, B., Yang, B.C., Chen, Q.: A new multiple half-discrete Hilbert-type inequality with parameters and a best possible constant factor. Mediterr. J. Math. (2014). https://doi.org/10.1007/s00009-014-0468-0
Mintrinovic, D.S., Pečarić, J., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)
Yang, B.C.: A new Hilbert-type integral inequality with the homogeneous kernel of degree 0. J. Zhejiang Univ. Sci. Ed. 39, 390–392 (2012)
Liu, Q., Long, S.C.: A Hilbert-type integral inequality with the kernel of hyperbolic secant function. J. Zhejiang Univ. Sci. Ed. 40, 255–259 (2013)
Yang, B.C., Chen, Q.: A Hilbert-type integral inequality related to Riemann zeta function. J. Jilin Univ. Sci. Ed. 52, 869–872 (2014)
Wang, Z.X., Guo, D.R.: Introduction to Special Functions. Higher Education Press, Beijing (2012)
Richard, C.F.J.: Introduction to Calculus and Analysis. Springer, New York (1989)
Acknowledgements
The author is indebted to the anonymous referees for their valuable suggestions and comments that helped improve the paper significantly.
Funding
No funding.
Author information
Authors and Affiliations
Contributions
The author carried out the results, and read and approved the current version of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
You, M. On a class of Hilbert-type inequalities in the whole plane related to exponent function. J Inequal Appl 2021, 33 (2021). https://doi.org/10.1186/s13660-021-02563-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-021-02563-5