Abstract
Using methods of weight functions, techniques of real analysis as well as the Hermite-Hadamard inequality, a half-discrete Hardy-Hilbert-type inequality with multi-parameters and a best possible constant factor related to the Hurwitz zeta function and the Riemann zeta function is obtained. Equivalent forms, normed operator expressions, their reverses and some particular cases are also considered.
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1 Introduction
If \(p> 1, \frac{1} {p} + \frac{1} {q} = 1,f(x),g(y) \geq 0,f \in L^{p}(\mathbf{R}_{ +}),g \in L^{q}(\mathbf{R}_{ +}),\)
| | g | | q > 0, then we have the following Hardy-Hilbert’s integral inequality (cf. [3]):
where the constant factor \(\frac{\pi }{\sin (\pi /p)}\) is the best possible. Assuming that
we have the following Hardy-Hilbert’s inequality with the same best constant \(\frac{\pi }{\sin (\pi /p)}\) (cf. [3]):
Inequalities (1) and (2) are important in Analysis and its applications (cf. [3, 11, 19, 20, 22]).
If μ i , v j > 0(i, j ∈ N ={1, 2, ⋯}),
then we have the following inequality (cf. [3], Theorem 321):
Replacing μ m 1∕q a m and v n 1∕p b n by a m and b n in (4), respectively, we obtain the following equivalent form of (4):
For μ i = v j = 1(i, j ∈ N), both (4) and (5) reduce to (2). We call (4) and (5) as Hardy-Hilbert-type inequalities.
Note.
The authors did not prove that (4) is valid with the best possible constant factor in [3].
In 1998, by introducing an independent parameter λ ∈ (0, 1], Yang [17] gave an extension of (1) with the kernel 1∕(x + y)λ for p = q = 2. Optimizing the method used in [17], Yang [20] provided some extensions of (1) and (2) as follows:
If λ 1, λ 2 ∈ R, λ 1 +λ 2 = λ, k λ (x, y) is a non-negative homogeneous function of degree −λ, with
\(\phi (x) = x^{p(1-\lambda _{1})-1},\psi (x) = x^{q(1-\lambda _{2})-1},f(x),g(y) \geq 0,\)
g ∈ L q, ψ (R +), | | f | | p, ϕ , | | g | | q, ψ > 0, then we have
where the constant factor k(λ 1) is the best possible.
Moreover, if k λ (x, y) remains finite and \(k_{\lambda }(x,y)x^{\lambda _{1}-1}(k_{\lambda }(x,y)y^{\lambda _{2}-1})\) is decreasing with respect to x > 0 (y > 0), then for a m, b n ≥ 0,
b = { b n } n = 1 ∞ ∈ l q, ψ , | | a | | p, ϕ , | | b | | q, ψ > 0, we have
where the constant factor k(λ 1) is still the best possible.
Clearly, for
inequality (6) reduces to (1), while (7) reduces to (2). For
we set
Then by (7), we have
where the constant B(λ 1, λ 2) is the best possible, and
is the beta function.
In 2015, subject to further conditions, Yang [26] proved an extension of (8) and (5) as follows:
where the constant B(λ 1, λ 2) is still the best possible.
Further results including some multidimensional Hilbert-type inequalities can be found in [18, 21, 23–25, 27, 33].
On the topic of half-discrete Hilbert-type inequalities with non-homogeneous kernels, Hardy et al. provided a few results in Theorem 351 of [3]. But, they did not prove that the constant factors are the best possible. However, Yang [18] presented a result with the kernel 1∕(1 + nx)λ by introducing a variable and proved that the constant factor is the best possible. In 2011, Yang [21] gave the following half-discrete Hardy-Hilbert’s inequality with the best possible constant factor \(B\left (\lambda _{1},\lambda _{2}\right )\):
where λ 1 > 0, 0 < λ 2 ≤ 1, λ 1 +λ 2 = λ. Zhong et al. [36, 37, 39–41] investigated several half-discrete Hilbert-type inequalities with particular kernels.
Using methods of weight functions and techniques of discrete and integral Hilbert-type inequalities with some additional conditions on the kernel, a half-discrete Hilbert-type inequality with a general homogeneous kernel of degree −λ ∈ R and a best constant factor \(k\left (\lambda _{1}\right )\) is obtained as follows:
which is an extension of (11) (cf. Yang and Chen [28]). Additionally, a half-discrete Hilbert-type inequality with a general non-homogeneous kernel and a best constant factor is given by Yang [24]. The reader is referred to the three books of Yang [23, 25] and Yang and Debnath [29], where half-discrete Hilbert-type inequalities and their operator expressions are extensively treated. The interested reader will find a vast literature on both old and new results on half-discrete Hardy-Hilbert-type inequality with emphasis to the study of best constants in references [1–42].
In this chapter, using methods of weight functions, techniques of real analysis as well as the Hermite-Hadamard inequality, a half-discrete Hardy-Hilbert-type inequality with multi-parameters and a best possible constant factor related to the Hurwitz zeta function and the Riemann zeta function is studied, which is an extension of (12) for λ = 0 in a particular kernel. Equivalent forms, normed operator expressions, their reverses and some particular cases are also considered.
2 An Example and Some Lemmas
In the following, we assume that μ i , ν j > 0 (i, j ∈ N), U m and V n are defined by (3),
μ(t) is a positive continuous function in R + = (0, ∞),
\(\nu (t):=\nu _{n},t \in (n -\frac{1} {2},n + \frac{1} {2}](n \in \mathbf{N}),\) and
\(p\neq 0,1,\ \frac{1} {p} + \frac{1} {q} = 1,\delta \in \{-1,1\},\ f(x),a_{n} \geq 0(x \in \mathbf{R}_{+},n \in \mathbf{N}),\)
\(\vert \vert a\vert \vert _{q,\tilde{\varPsi }} = (\sum _{n=1}^{\infty }\tilde{\varPsi }(n)b_{n}^{q})^{\frac{1} {q} },\) where,
Example 1.
For 0 < γ < σ, 0 ≤ α ≤ ρ (ρ > 0),
is the hyperbolic cosecant function (cf. [34]). We set
-
(i)
Setting u = ρ t γ, we find
$$\displaystyle\begin{array}{rcl} k(\sigma )&:=& \int _{0}^{\infty }\frac{\mbox{csc } h(\rho t^{\gamma })} {e^{\alpha t^{\gamma }}} t^{\sigma -1}dt {}\\ & =& \frac{1} {\gamma \rho ^{\sigma /\gamma }}\int _{0}^{\infty }\frac{\mbox{csc } h(u)} {e^{\frac{\alpha }{\rho }u }} u^{\frac{\sigma }{\gamma }-1 }du {}\\ & =& \frac{2} {\gamma \rho ^{\sigma /\gamma }}\int _{0}^{\infty }\frac{e^{-\frac{\alpha }{\rho }u}u^{\frac{\sigma }{\gamma }-1}} {e^{u} - e^{-u}}du {}\\ & =& \frac{2} {\gamma \rho ^{\sigma /\gamma }}\int _{0}^{\infty }\frac{e^{-(\frac{\alpha }{\rho }+1)u}u^{\frac{\sigma }{\gamma }-1}} {1 - e^{-2u}} du {}\\ & =& \frac{2} {\gamma \rho ^{\sigma /\gamma }}\int _{0}^{\infty }\sum _{ k=0}^{\infty }e^{-(2k+\frac{\alpha }{\rho }+1)u}u^{\frac{\sigma }{\gamma }-1}du. {}\\ \end{array}$$By the Lebesgue term by term integration theorem (cf. [34]), setting \(v = \left (2k + \frac{\alpha }{\rho } + 1\right )u\), we have
$$\displaystyle\begin{array}{rcl} k(\sigma )& =& \int _{0}^{\infty }\frac{\mbox{csc } h(\rho t^{\gamma })} {e^{\alpha t^{\gamma }}} t^{\sigma -1}dt \\ & =& \frac{2} {\gamma \rho ^{\sigma /\gamma }}\sum _{k=0}^{\infty }\int _{ 0}^{\infty }e^{-(2k+\frac{\alpha }{\rho }+1)u}u^{\frac{\sigma }{\gamma }-1}du \\ & =& \frac{2} {\gamma \rho ^{\sigma /\gamma }}\sum _{k=0}^{\infty } \frac{1} {(2k + \frac{\alpha }{\rho } + 1)^{\sigma /\gamma }}\int _{0}^{\infty }e^{-v}v^{\frac{\sigma }{\gamma }-1}dv \\ & =& \frac{2\varGamma (\frac{\sigma }{\gamma })} {\gamma (2\rho )^{\sigma /\gamma }}\sum _{k=0}^{\infty } \frac{1} {(k + \frac{\alpha +\rho } {2\rho } )^{\sigma /\gamma }} \\ & =& \frac{2\varGamma (\frac{\sigma }{\gamma })} {\gamma (2\rho )^{\sigma /\gamma }}\zeta (\frac{\sigma } {\gamma }, \frac{\alpha +\rho } {2\rho } ) \in \mathbf{R}_{+}, {}\end{array}$$(13)where
$$\displaystyle{\zeta (s,a):=\sum _{ k=0}^{\infty } \frac{1} {(k + a)^{s}}\ (s> 1;0 <a \leq 1)}$$is the Hurwitz zeta function, ζ(s) = ζ(s, 1) is the Riemann zeta function, and
$$\displaystyle{\varGamma (y):=\int _{ 0}^{\infty }e^{-v}v^{y-1}dv^{}\ (y> 0)}$$is the Gamma function (cf. [16]).
In particular, (1) for α = ρ > 0, we have \(h(t) = \frac{\mbox{csc } h(\rho t^{\gamma })} {e^{\rho t^{\gamma }}}\) and \(k(\sigma ) = \frac{2\varGamma (\frac{\sigma }{\gamma })\zeta (\frac{\sigma }{\gamma })} {\gamma (2\rho )^{\sigma /\gamma }}.\) In this case, for \(\gamma = \frac{\sigma } {2},\) we have \(h(t) = \frac{\mbox{csc } h(\rho \sqrt{t^{\sigma }})} {e^{\rho \sqrt{t^{\sigma }}}}\) and \(k(\sigma ) = \frac{\pi ^{2}} {6\sigma \rho ^{2}};\) (2) for α = 0, we have h(t) = csch(ρ t γ) and \(\frac{2\varGamma (\frac{\sigma }{\gamma })} {\gamma (2\rho )^{\sigma /\gamma }}\zeta (\frac{\sigma }{\gamma }, \frac{1} {2}).\) In this case, for \(\gamma = \frac{\sigma } {2},\) we find \(h(t) =\mbox{csc } h(\rho \sqrt{t^{\sigma }})\) and \(k(\sigma ) = \frac{\pi ^{2}} {2\sigma \rho ^{2}}.\)
-
(ii)
We obtain for \(u> 0, \frac{1} {e^{u}-e^{-u}}> 0,\)
$$\displaystyle\begin{array}{rcl} \frac{d} {du}( \frac{1} {e^{u} - e^{-u}})& =& - \frac{e^{u} + e^{-u}} {(e^{u} - e^{-u})^{2}} <0, {}\\ \frac{d^{2}} {du^{2}}( \frac{1} {e^{u} - e^{-u}})& =& \frac{2(e^{u} + e^{-u})^{2} - (e^{u} - e^{-u})^{2}} {(e^{u} - e^{-u})^{3}}> 0. {}\\ \end{array}$$If g(u) > 0, g ′(u) < 0, g ′ ′(u) > 0, then for 0 < γ ≤ 1,
$$\displaystyle\begin{array}{rcl} g(\rho t^{\gamma })&>& 0, \frac{d} {dt}g(\rho t^{\gamma }) =\rho \gamma t^{\gamma -1}g^{{\prime}}(\rho t^{\gamma }) <0, {}\\ \frac{d^{2}} {dt^{2}}g(\rho t^{\gamma })& =& \rho \gamma (\gamma -1)t^{\gamma -2}g^{{\prime}}(\rho t^{\gamma }) +\rho ^{2}\gamma ^{2}t^{2\gamma -2}g^{{\prime\prime}}(\rho t^{\gamma })> 0; {}\\ \end{array}$$for \(y \in (n -\frac{1} {2},n + \frac{1} {2}),g(V (y))> 0,\)
$$\displaystyle\begin{array}{rcl} \frac{d} {dy}g(V (y))& =& g^{{\prime}}(V (y))\nu _{ n} <0, {}\\ \frac{d^{2}} {dy^{2}}g(V (y))& =& g^{{\prime\prime}}(V (y))\nu _{ n}^{2}> 0(n \in \mathbf{N}). {}\\ \end{array}$$If g i (u) > 0, g i ′(u) < 0, g i ′ ′(u) > 0(i = 1, 2), then
$$\displaystyle\begin{array}{rcl} g_{1}(u)g_{2}(u)&>& 0, {}\\ (g_{1}(u)g_{2}(u))^{{\prime}}& =& g_{ 1}^{{\prime}}(u)g_{ 2}(u) + g_{1}(u)g_{2}^{{\prime}}(u) <0, {}\\ (g_{1}(u)g_{2}(u))^{{\prime\prime}}& =& g_{ 1}^{{\prime\prime}}(u)g_{ 2}(u) + 2g_{1}^{{\prime}}(u)g_{ 2}^{{\prime}}(u) + g_{ 1}(u)g_{2}^{{\prime\prime}}(u)> 0(u> 0). {}\\ \end{array}$$ -
(iii)
Therefore, for 0 < γ < σ ≤ 1, 0 ≤ α ≤ ρ(ρ > 0), we have k(σ) ∈ R +, with h(t) > 0, h ′(t) < 0, h ′ ′(t) > 0, and then for \(c> 0,y \in (n -\frac{1} {2},n + \frac{1} {2})(n \in \mathbf{N}),\) it follows that
$$\displaystyle\begin{array}{rcl} & & h(cV (y))V ^{\sigma -1}(y)> 0, {}\\ & & \frac{d} {dy}h(cV (y))V ^{\sigma -1}(y) <0, {}\\ & & \frac{d^{2}} {dy^{2}}h(cV (y))V ^{\sigma -1}(y)> 0. {}\\ \end{array}$$
Lemma 1.
If g(t)(> 0) is decreasing in R + and strictly decreasing in [n 0 ,∞) where n 0 ∈ N , satisfying ∫ 0 ∞ g(t)dt ∈ R + , then we have
Proof.
Since we have
then it follows that
Similarly, we still have
Hence, (14) follows and therefore the lemma is proved.
Lemma 2.
If 0 ≤α ≤ρ(ρ > 0),0 < γ < σ ≤ 1, define the following weight coefficients:
Then, we have the following inequalities:
where k(σ) is given by ( 13 ).
Proof.
Since we find
and for \(t \in (n -\frac{1} {2},n + \frac{1} {2}],V ^{{\prime}}(t) =\nu _{ n},\) hence by Example 1(iii) and Hermite-Hadamard’s inequality (cf. [8]), we have
Setting u = U δ(x)V (t), by (13), we find
Hence, (17) follows.
Setting \(u =\tilde{ V }_{n}U^{\delta }(x)\) in (16), we find \(du =\delta \tilde{ V }_{n}U^{\delta -1}(x)\mu (x)dx\) and
If δ = 1, then
If δ = −1, then
Then by (13), we have (18). The lemma is proved.
Remark 1.
We do not need the constraint σ ≤ 1 to obtain (18). If U(∞) = ∞, then we have
For example, if we set \(\mu (t) = \frac{1} {(1+t)^{\beta }}(t> 0;0 \leq \beta \leq 1),\) then for x ≥ 0, we find
and
Lemma 3.
If 0 ≤α ≤ρ (ρ > 0), 0 < γ < σ ≤ 1, there exists n 0 ∈ N , such that ν n ≥ν n+1 (n ∈ { n 0 ,n 0 + 1,⋯ }), and V (∞) = ∞, then
-
(i)
for x ∈ R + , we have
$$\displaystyle{ k(\sigma )(1 -\theta _{\delta }(\sigma,x)) <\omega _{\delta }(\sigma,x), }$$(20)where, θ δ (σ,x) = O((U(x)) δ(σ−γ) ) ∈ (0,1);
-
(ii)
for any b > 0, we have
$$\displaystyle{ \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1+b}} = \frac{1} {b}\left ( \frac{1} {V _{n_{0}}^{b}} + bO(1)\right ). }$$(21)
Proof.
Since v n ≥ v n+1(n ≥ n 0), and
by Example 1(iii), we have
Setting u = U δ(x)V (t), in view of V (∞) = ∞, by (13), we find
Since \(F(u) = \frac{u^{\gamma }\mbox{csc } h(\rho u^{\gamma })} {e^{\alpha u^{\gamma }}}\) is continuous in (0, ∞), satisfying
there exists a constant L > 0, such that F(u) ≤ L, namely,
Hence we find
and then (20) follows.
For b > 0, we find
Hence we have (21). The lemma is proved.
Note.
For example, \(\nu _{n} = \frac{1} {(n-\tau )^{\beta }}(n \in \mathbf{N};0 \leq \beta \leq 1,0 \leq \tau <1)\) satisfies the conditions of Lemma 3 (for n 0 ≥ 1).
3 Equivalent Inequalities and Operator Expressions
Theorem 1.
If 0 ≤α ≤ρ(ρ > 0),0 < γ < σ ≤ 1,k(σ) is given by ( 13 ), then for \(p> 1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\tilde{\varPsi }} <\infty,\) we have the following equivalent inequalities:
Proof.
By Hölder’s inequality with weight (cf. [8]), we have
In view of (18) and the Lebesgue term by term integration theorem (cf. [9]), we find
By Hölder’s inequality (cf. [8]), we have
On the other hand, assuming that (22) is valid, we set
Then, we find \(J_{1}^{p} = \vert \vert a\vert \vert _{q,\tilde{\varPsi }}^{q}.\)
If J 1 = 0, then (23) is trivially valid.
If J 1 = ∞, then (23) keeps impossible.
Suppose that 0 < J 1 < ∞. By (22), it follows that
and then (23) follows, which is equivalent to (22).
By Hölder’s inequality with weight (cf. [8]), we obtain
Then by (17) and Lebesgue term by term integration theorem (cf. [9]), it follows that
By Hölder’s inequality (cf. [8]), we have
On the other hand, assuming that (24) is valid, we set
Then we find \(J_{2}^{q} = \vert \vert f\vert \vert _{p,\varPhi _{\delta }}^{p}.\)
If J 2 = 0, then (24) is trivially valid.
If J 2 = ∞, then (24) keeps impossible.
Suppose that 0 < J 2 < ∞. By (22), it follows that
and then (24) follows, which is equivalent to (22).
Therefore, (22), (23) and (24) are equivalent. The theorem is proved.
Theorem 2.
With the assumptions of Theorem 1 , if there exists n 0 ∈ N , such that v n ≥ v n+1 (n ∈ { n 0 ,n 0 + 1,⋯ }), and U(∞) = V (∞) = ∞, then the constant factor k(σ) in ( 22 ), ( 23 ) and ( 24 ) is the best possible.
Proof.
For ɛ ∈ (0, q(σ −γ)), we set \(\tilde{\sigma }=\sigma -\frac{\varepsilon }{q}(\in (\gamma,1)),\) and \(\tilde{f }= \tilde{f } (x),x \in \mathbf{R}_{+},\tilde{a} =\{\tilde{ a}_{n}\}_{n=1}^{\infty },\)
Then for δ = ±1, since U(∞) = ∞, we find
By (21), (33) and (20), we obtain
If there exists a positive constant K ≤ k(σ), such that (22) is valid when replacing k(σ) to K, then in particular, by Lebesgue term by term integration theorem, we have \(\varepsilon \tilde{I} <\varepsilon K\vert \vert \tilde{f } \vert \vert _{p,\varPhi _{\delta }}\vert \vert \tilde{a}\vert \vert _{q,\tilde{\varPsi }},\) namely,
It follows that k(σ) ≤ K(ɛ → 0+). Hence, K = k(σ) is the best possible constant factor of (22).
The constant factor k(σ) in (23) (respectively, (24)) is still the best possible. Otherwise, we would reach a contradiction by (27) (respectively, (30)) that the constant factor in (22) is not the best possible. The theorem is proved.
For p > 1, we find
and define the following real normed spaces:
Assuming that \(f \in L_{p,\varPhi _{\delta }}(\mathbf{R}_{+}),\) setting
we can rewrite (23) as follows:
namely, \(c \in l_{p,\tilde{\varPsi }^{1-p}}.\)
Definition 1.
Define a half-discrete Hardy-Hilbert-type operator
as follows:
For any \(f \in L_{p,\varPhi _{\delta }}(\mathbf{R}_{+}),\) there exists a unique representation \(T_{1}f = c \in l_{p,\tilde{\varPsi }^{1-p}}.\) Define the formal inner product of T 1 f and \(a =\{ a_{n}\}_{n=1}^{\infty }\in l_{q,\tilde{\varPsi }}\) as follows:
Then we can rewrite (22) and (23) as:
Define the norm of operator T 1 as follows:
Then by (37), it is evident that | | T 1 | | ≤ k(σ). Since by Theorem 2, the constant factor in (37) is the best possible, we have
Assuming that \(a =\{ a_{n}\}_{n=1}^{\infty }\in l_{q,\tilde{\varPsi }},\) setting
we can rewrite (24) as follows:
namely, \(h \in L_{q,\varPhi _{\delta }^{1-q}}(\mathbf{R}_{+}).\)
Definition 2.
Define a half-discrete Hardy-Hilbert-type operator
as follows:
For any \(a =\{ a_{n}\}_{n=1}^{\infty }\in l_{q,\tilde{\varPsi }},\) there exists a unique representation \(T_{2}a = h \in L_{q,\varPhi _{\delta }^{1-q}}(\mathbf{R}_{+}).\) Define the formal inner product of T 2 a and \(f \in L_{p,\varPhi _{\delta }}(\mathbf{R}_{+})\) by:
Then we can rewrite (22) and (24) as follows:
Define the norm of operator T 2 by:
Then by (41), we find | | T 2 | | ≤ k(σ). Since by Theorem 2, the constant factor in (41) is the best possible, we have
4 Some Equivalent Reverses
In the following, we also set
For 0 < p < 1 or p < 0, we still use the formal symbols \(\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\), \(\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{\delta }}\) and \(\vert \vert a\vert \vert _{q,\tilde{\varPsi }}.\)
Theorem 3.
If 0 ≤α ≤ρ(ρ > 0),0 < γ < σ ≤ 1,k(σ) is given by ( 13 ), there exists n 0 ∈ N , such that v n ≥ v n+1 (n ∈ { n 0 ,n 0 + 1,⋯ }), and U(∞) = V (∞) = ∞, then for \(p <0,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\tilde{\varPsi }} <\infty,\) we have the following equivalent inequalities with the best possible constant factor k(σ):
Proof.
By the reverse Hölder’s inequality with weight (cf. [8]), since p < 0, similarly to the way we obtained (25) and (26), we have
and then by (19) and Lebesgue term by term integration theorem, it follows that
By the reverse Hölder’s inequality (cf. [8]), we have
On the other hand, assuming that (43) is valid, we set a n as in Theorem 1. Then we find \(J_{1}^{p} = \vert \vert a\vert \vert _{q,\tilde{\varPsi }}^{q}.\)
If J 1 = ∞, then (44) is trivially valid.
If J 1 = 0, then (44) is impossible.
Suppose that 0 < J 1 < ∞. By (43), it follows that
and then (44) follows, which is equivalent to (43).
By the reverse of Hölder’s inequality with weight (cf. [8]), since 0 < q < 1, similarly to the way we obtained (28) and (29), we have
and then by (17) and Lebesgue term by term integration theorem, it follows that
By the reverse Hölder’s inequality (cf. [8]), we get
On the other hand, assuming that (45) is valid, we set f(x) as in Theorem 1. Then we find \(J_{2}^{q} = \vert \vert f\vert \vert _{p,\varPhi _{\delta }}^{p}.\)
If J 2 = ∞, then (45) is trivially valid.
If J 2 = 0, then (45) keeps impossible.
Suppose that 0 < J 2 < ∞. By (43), it follows that
and then (45) follows, which is equivalent to (43).
Therefore, inequalities (43), (44) and (45) are equivalent.
For ɛ ∈ (0, q(σ −γ)), we set \(\tilde{\sigma }=\sigma -\frac{\varepsilon }{q}(\in (\gamma,1)),\) and \(\tilde{f }= \tilde{f } (x),x \in \mathbf{R}_{+}\), \(\tilde{a} =\{\tilde{ a}_{n}\}_{n=1}^{\infty },\)
By (21), (33) and (17), we obtain
If there exists a positive constant K ≥ k(σ), such that (43) is valid when replacing k(σ) by K, then in particular, we have \(\varepsilon \tilde{I}>\varepsilon K\vert \vert \tilde{f } \vert \vert _{p,\varPhi _{\delta }}\vert \vert \tilde{a}\vert \vert _{q,\tilde{\varPsi }},\) namely,
It follows that k(σ) ≥ K(ɛ → 0+). Hence, K = k(σ) is the best possible constant factor of (43).
The constant factor k(σ) in (44) (respectively, (45)) is still the best possible. Otherwise, we would reach a contradiction by (46) (respectively, (47)) that the constant factor in (43) is not the best possible. The theorem is proved.
Theorem 4.
With the assumptions of Theorem 3 , if
then we have the following equivalent inequalities with the best possible constant factor k(σ):
Proof.
By the reverse Hölder’s inequality with weight (cf. [8]), since 0 < p < 1, similarly to the way we obtained (25) and (26), we have
and then in view of (19) and Lebesgue term by term integration theorem, we find
By the reverse Hölder’s inequality (cf. [8]), we have
On the other hand, assuming that (49) is valid, we set a n as in Theorem 1. Then we find \(J_{1}^{p} = \vert \vert a\vert \vert _{q,\tilde{\varPsi }}^{q}.\)
If J 1 = ∞, then (50) is trivially valid.
If J 1 = 0, then (50) remains impossible.
Suppose that 0 < J 1 < ∞. By (49), it follows that
and then (50) follows, which is equivalent to (49).
By the reverse Hölder’s inequality with weight (cf. [8]), since q < 0, we have
and then by (20) and Lebesgue term by term integration theorem, it follows that
By the reverse Hölder’s inequality (cf. [8]), we have
On the other hand, assuming that (49) is valid, we set f(x) as in Theorem 1. Then we find \(J^{q} = \vert \vert f\vert \vert _{p,\tilde{\varPhi }_{\delta }}^{p}.\)
If J = ∞, then (51) is trivially valid.
If J = 0, then (51) remains impossible.
Suppose that 0 < J < ∞. By (49), it follows that
and then (51) follows, which is equivalent to (49).
Therefore, inequalities (49), (50) and (51) are equivalent.
For ɛ ∈ (0, p(σ −γ)), we set \(\tilde{\sigma }=\sigma + \frac{\varepsilon }{p}(>\gamma ),\) and \(\tilde{f }= \tilde{f } (x),x \in \mathbf{R}_{+},\tilde{a} =\{\tilde{ a}_{n}\}_{n=1}^{\infty },\)
By (20), (21) and (33), we obtain
If there exists a positive constant K ≥ k(σ), such that (43) is valid when replacing k(σ) by K, then in particular, we have \(\varepsilon \tilde{I}>\varepsilon K\vert \vert \tilde{f } \vert \vert _{p,\tilde{\varPhi }_{\delta }}\vert \vert \tilde{a}\vert \vert _{q,\tilde{\varPsi }},\) namely,
It follows that k(σ) ≥ K(ɛ → 0+). Hence, K = k(σ) is the best possible constant factor of (49).
The constant factor k(σ) in (50) (respectively, (51)) is still the best possible. Otherwise, we would reach a contradiction by (52) (respectively, (53)) that the constant factor in (49) is not the best possible. The theorem is proved.
5 Some Particular Inequalities
For \(\tilde{\nu }_{n} = 0,\tilde{V }_{n} = V _{n},\) we set
In view of Theorems 2–4, we have
Corollary 1.
If 0 ≤α ≤ρ(ρ > 0),0 < γ < σ ≤ 1,k(σ) is given by (13), there exists n 0 ∈ N , such that v n ≥ v n+1 (n ∈ { n 0 ,n 0 + 1,⋯ }), and U(∞) = V (∞) = ∞, then
-
(i)
for \(p> 1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} a_{n}f(x)dx <k(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(54)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} f(x)dx\right ]^{p} <k(\sigma )\vert \vert f\vert \vert _{ p,\varPhi _{\delta }}, {}\end{array}$$(55)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\delta \sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } <k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }; }$$(56) -
(ii)
for \(p <0,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} a_{n}f(x)dx> k(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(57)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} f(x)dx\right ]^{p}> k(\sigma )\vert \vert f\vert \vert _{ p,\varPhi _{\delta }}, {}\end{array}$$(58)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\delta \sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} }> k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }; }$$(59) -
(iii)
for \(0 <p <1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} a_{n}f(x)dx> k(\sigma )\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{\delta }}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(60)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} f(x)dx\right ]^{p}> k(\sigma )\vert \vert f\vert \vert _{ p,\tilde{\varPhi }_{\delta }}, {}\end{array}$$(61)$$\displaystyle\begin{array}{rcl} & & \left \{\int _{0}^{\infty }\frac{(1 -\theta _{\delta }(\sigma,x))^{1-q}\mu (x)} {U^{1-q\delta \sigma }(x)} \left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)V _{ n})^{\gamma })} {e^{\alpha (U^{\delta }(x)V _{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } \\ &>& k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }. {}\end{array}$$(62)
The above inequalities have the best possible constant factor k(σ).
In particular, for δ = 1, we have the following inequalities with the non-homogeneous kernel:
Corollary 2.
If 0 ≤α ≤ρ(ρ > 0),0 < γ < σ ≤ 1,k(σ) is given by ( 13 ), there exists n 0 ∈ N , such that v n ≥ v n+1 (n ∈ { n 0 ,n 0 + 1,⋯ }), and U(∞) = V (∞) = ∞, then
-
(i)
for \(p> 1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{1}},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} a_{n}f(x)dx <k(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{1}}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(63)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} f(x)dx\right ]^{p} <k(\sigma )\vert \vert f\vert \vert _{ p,\varPhi _{1}}, {}\end{array}$$(64)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } <k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }; }$$(65) -
(ii)
for \(p <0,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{1}},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} a_{n}f(x)dx> k(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{1}}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(66)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} f(x)dx\right ]^{p}> k(\sigma )\vert \vert f\vert \vert _{ p,\varPhi _{1}}, {}\end{array}$$(67)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} }> k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }; }$$(68) -
(iii)
for \(0 <p <1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{1}},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} a_{n}f(x)dx> k(\sigma )\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{1}}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(69)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} f(x)dx\right ]^{p}> k(\sigma )\vert \vert f\vert \vert _{ p,\tilde{\varPhi }_{1}}, {}\end{array}$$(70)$$\displaystyle\begin{array}{rcl} & & \left \{\int _{0}^{\infty }\frac{(1 -\theta _{1}(\sigma,x))^{1-q}\mu (x)} {U^{1-q\sigma }(x)} \left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U(x)V _{n})^{\gamma })} {e^{\alpha (U(x)V _{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } \\ &>& k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }. {}\end{array}$$(71)
The above inequalities involve the best possible constant factor k(σ).
For δ = −1, we have the following inequalities with the homogeneous kernel of degree 0:
Corollary 3.
If 0 ≤α ≤ρ(ρ > 0),0 < γ < σ ≤ 1,k(σ) is given by ( 13 ), there exists n 0 ∈ N , such that v n ≥ v n+1 (n ∈ { n 0 ,n 0 + 1,⋯ }), and U(∞) = V (∞) = ∞, then
-
(i)
for \(p> 1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{-1}},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} a_{n}f(x)dx <k(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{-1}}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(72)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} f(x)dx\right ]^{p} <k(\sigma )\vert \vert f\vert \vert _{ p,\varPhi _{-1}}, {}\end{array}$$(73)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1+q\sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } <k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }; }$$(74) -
(ii)
for \(p <0,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{-1}},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} a_{n}f(x)dx> k(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{-1}}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(75)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} f(x)dx\right ]^{p}> k(\sigma )\vert \vert f\vert \vert _{ p,\varPhi _{-1}}, {}\end{array}$$(76)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1+q\sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} }> k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }; }$$(77) -
(iii)
for \(0 <p <1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{-1}},\vert \vert a\vert \vert _{q,\varPsi } <\infty,\) we have the following equivalent inequalities:
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} a_{n}f(x)dx> k(\sigma )\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{-1}}\vert \vert a\vert \vert _{q,\varPsi }, {}\end{array}$$(78)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {V _{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} f(x)dx\right ]^{p}> k(\sigma )\vert \vert f\vert \vert _{ p,\tilde{\varPhi }_{-1}}, {}\end{array}$$(79)$$\displaystyle\begin{array}{rcl} & & \left \{\int _{0}^{\infty }\frac{(1 -\theta _{-1}(\sigma,x))^{1-q}\mu (x)} {U^{1+q\sigma }(x)} \left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho ( \frac{V _{n}} {U(x)})^{\gamma })} {e^{\alpha ( \frac{V_{n}} {U(x)} )^{\gamma } }} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } \\ &>& k(\sigma )\vert \vert a\vert \vert _{q,\varPsi }. {}\end{array}$$(80)
The above inequalities involve the best possible constant factor k(σ).
For α = ρ in Theorems 2–4, we have
Corollary 4.
If ρ > 0,0 < γ < σ ≤ 1,
there exists n 0 ∈ N , such that v n ≥ v n+1 (n ∈ { n 0 ,n 0 + 1,⋯ }), and U(∞) = V (∞) = ∞, then
-
(i)
for \(p> 1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\tilde{\varPsi }} <\infty,\) we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} a_{n}f(x)dx <K(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(82)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} f(x)dx\right ]^{p} <K(\sigma )\vert \vert f\vert \vert _{ p,\varPhi _{\delta }}, {}\end{array}$$(83)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\delta \sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } <K(\sigma )\vert \vert a\vert \vert _{q,\tilde{\varPsi }}. }$$(84) -
(ii)
for \(p <0,0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\tilde{\varPsi }} <\infty,\) we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} a_{n}f(x)dx> K(\sigma )\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(85)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} f(x)dx\right ]^{p}> K(\sigma )\vert \vert f\vert \vert _{ p,\varPhi _{\delta }}, {}\end{array}$$(86)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\delta \sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} }> K(\sigma )\vert \vert a\vert \vert _{q,\tilde{\varPsi }}; }$$(87) -
(iii)
for \(0 <p <1,0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\tilde{\varPsi }} <\infty,\) we have the following equivalent inequalities with the best possible constant factor K(σ):
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} a_{n}f(x)dx> K(\sigma )\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(88)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} f(x)dx\right ]^{p}> K(\sigma )\vert \vert f\vert \vert _{ p,\tilde{\varPhi }_{\delta }}, {}\end{array}$$(89)$$\displaystyle\begin{array}{rcl} & & \left \{\int _{0}^{\infty }\frac{(1 -\theta _{\delta }(\sigma,x))^{1-q}\mu (x)} {U^{1-q\delta \sigma }(x)} \left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\gamma })} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\gamma }}} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } \\ &>& K(\sigma )\vert \vert a\vert \vert _{q,\tilde{\varPsi }}. {}\end{array}$$(90)
In particular, for \(\gamma = \frac{\sigma } {2},\theta _{\delta }(\sigma,x) = O((U(x))^{ \frac{\delta \sigma }{ 2} }),\)
-
(i)
for p > 1, we have the following equivalent inequalities with the best possible constant factor \(\frac{\pi ^{2}} {6\sigma \rho ^{2}}\):
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} a_{n}f(x)dx <\frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(91)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} f(x)dx\right ]^{p} <\frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\varPhi _{\delta }}, {}\end{array}$$(92)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\delta \sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } <\frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}; }$$(93) -
(ii)
for p < 0, we have the following equivalent inequalities with the best possible constant factor \(\frac{\pi ^{2}} {6\sigma \rho ^{2}}\):
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} a_{n}f(x)dx> \frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(94)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} f(x)dx\right ]^{p}> \frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\varPhi _{\delta }}, {}\end{array}$$(95)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\delta \sigma }(x)}\left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} }> \frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}; }$$(96) -
(iii)
for 0 < p < 1, we have the following equivalent inequalities with the best possible constant factor \(\frac{\pi ^{2}} {6\sigma \rho ^{2}}\):
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} a_{n}f(x)dx> \frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(97)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} f(x)dx\right ]^{p}> \frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{\delta }}, {}\end{array}$$(98)$$\displaystyle\begin{array}{rcl} & & \left \{\int _{0}^{\infty }\frac{(1 -\theta _{\delta }(\sigma,x))^{1-q}\mu (x)} {U^{1-q\delta \sigma }(x)} \left [\sum _{n=1}^{\infty }\frac{\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{\sigma /2})} {e^{\rho (U^{\delta }(x)\tilde{V }_{n})^{\sigma /2} }} a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } \\ &>& \frac{\pi ^{2}} {6\sigma \rho ^{2}}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}. {}\end{array}$$(99)
For \(\alpha = 0,\gamma = \frac{\sigma } {2},\theta _{\delta }(\sigma,x) = O((U(x))^{ \frac{\delta \sigma }{ 2} })\) in Theorems 2–4, we have
Corollary 5.
If ρ > 0,0 < σ ≤ 1, there exists n 0 ∈ N , such that v n ≥ v n+1 (n ∈ { n 0 ,n 0 + 1,⋯ }), and U(∞) = V (∞) = ∞, then
-
(i)
for \(p> 1,\ 0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\tilde{\varPsi }} <\infty,\) we have the following equivalent inequalities with the best possible constant factor \(\frac{\pi ^{2}} {2\sigma \rho ^{2}}\) :
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })a_{n}f(x)dx <\frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(100)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })f(x)dx\right ]^{p} <\frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\varPhi _{\delta }}, {}\end{array}$$(101)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\delta \sigma }(x)}\left [\sum _{n=1}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } <\frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}; }$$(102) -
(ii)
for \(p <0,0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\tilde{\varPsi }} <\infty,\) we have the following equivalent inequalities with the best possible constant factor \(\frac{\pi ^{2}} {2\sigma \rho ^{2}}\) :
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })a_{n}f(x)dx> \frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\varPhi _{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(103)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })f(x)dx\right ]^{p}> \frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\varPhi _{\delta }}, {}\end{array}$$(104)$$\displaystyle{ \left \{\int _{0}^{\infty } \frac{\mu (x)} {U^{1-q\delta \sigma }(x)}\left [\sum _{n=1}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} }> \frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}; }$$(105) -
(iii)
for \(0 <p <1,0 <\vert \vert f\vert \vert _{p,\varPhi _{\delta }},\vert \vert a\vert \vert _{q,\tilde{\varPsi }} <\infty,\) we have the following equivalent inequalities with the best possible constant factor \(\frac{\pi ^{2}} {2\sigma \rho ^{2}}\) :
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })a_{n}f(x)dx> \frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{\delta }}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}, {}\end{array}$$(106)$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty } \frac{\nu _{n}} {\tilde{V }_{n}^{1-p\sigma }}\left [\int _{0}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })f(x)dx\right ]^{p}> \frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert f\vert \vert _{p,\tilde{\varPhi }_{\delta }}, {}\end{array}$$(107)$$\displaystyle\begin{array}{rcl} & & \left \{\int _{0}^{\infty }\frac{(1 -\theta _{\delta }(\sigma,x))^{1-q}\mu (x)} {U^{1-q\delta \sigma }(x)} \left [\sum _{n=1}^{\infty }\mbox{csc } h(\rho (U^{\delta }(x)\tilde{V }_{ n})^{ \frac{\sigma }{ 2} })a_{n}\right ]^{q}dx\right \}^{\frac{1} {q} } \\ &>& \frac{\pi ^{2}} {2\sigma \rho ^{2}}\vert \vert a\vert \vert _{q,\tilde{\varPsi }}. {}\end{array}$$(108)
Remark 2.
-
(i)
For μ(x) = ν n = 1 in (54), we have the following inequality with the best possible constant factor k(σ):
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho (x^{\delta }n)^{\gamma })} {e^{\alpha (x^{\delta }n)^{\gamma }}} a_{n}f(x)dx {}\end{array}$$(109)$$\displaystyle\begin{array}{rcl} & <& k(\sigma )\left [\int _{0}^{\infty }x^{p(1-\delta \sigma )-1}f^{p}(x)dx\right ]^{\frac{1} {p} }\left [\sum _{n=1}^{\infty }n^{q(1-\sigma )-1}a_{n}^{q}\right ]^{\frac{1} {q} }. {}\end{array}$$(110)
In particular, for δ = 1, we have the following inequality with the non-homogeneous kernel:
for δ = −1, we have the following inequality with the homogeneous kernel:
-
(ii)
For \(\mu (x) =\nu _{n} = 1,\ \tilde{\nu }_{n} =\tau \in (0, \frac{1} {2}]\) in (22), we have the following more accurate inequality than (82) with the best possible constant factor k(σ):
$$\displaystyle\begin{array}{rcl} & & \sum _{n=1}^{\infty }\int _{ 0}^{\infty }\frac{\mbox{csc } h(\rho [x^{\delta }(n-\tau )]^{\gamma })} {e^{\alpha (x^{\delta }(n-\tau )]^{\gamma })^{\gamma }}} a_{n}f(x)dx {}\end{array}$$(115)$$\displaystyle\begin{array}{rcl} & <& k(\sigma )\left [\int _{0}^{\infty }x^{p(1-\delta \sigma )-1}f^{p}(x)dx\right ]^{\frac{1} {p} }\left [\sum _{n=1}^{\infty }(n-\tau )^{q(1-\sigma )-1}a_{n}^{q}\right ]^{\frac{1} {q} }. {}\end{array}$$(116)
In particular, for δ = 1, we have the following inequality with the non-homogeneous kernel:
for δ = −1, we have the following inequality with the homogeneous kernel:
We can still obtain a large number of other inequalities by using some special parameters in the above theorems and corollaries.
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Acknowledgements
B. Yang: This work is supported by the National Natural Science Foundation of China (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2013KJCX0140).
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Rassias, M.T., Yang, B. (2017). A Half-Discrete Hardy-Hilbert-Type Inequality with a Best Possible Constant Factor Related to the Hurwitz Zeta Function. In: Govil, N., Mohapatra, R., Qazi, M., Schmeisser, G. (eds) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-49242-1_10
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