A half-discrete Hardy-Hilbert-type inequality related to hyperbolic secant function

By applying weight functions and technique of real analysis, a half-discrete Hardy-Hilbert-type inequality related to the kernel of hyperbolic secant function and a best possible constant factor are given. The equivalent forms, the operator expressions with the norm, the reverses, and some particular cases are also considered.

Note The authors of [] did not prove that () is valid with the best possible constant factor.
Then by () we have ∞ m= ∞ n= a m b n (m + n) λ < B(λ  , λ  ) a p,φ b q,ψ , (  ) where the constant B(λ  , λ  ) is the best possible, and is the beta function. Clearly, for λ = , λ  =  q , λ  =  p , inequality () reduces to (). In , by adding some conditions, Yang [] gave an extension of () and () as follows: where the constant B(λ  , λ  ) is still the best possible. Some other results including multidimensional Hilbert-type inequalities are provided by [-].
About the topic of half-discrete Hilbert-type inequalities with inhomogeneous kernels, Hardy et al. provided a few results in Theorem  of [], but they did not prove that the constant factors are the best possible. However, Yang [] gave a result with the kernel  (+nx) λ by introducing a variable and proved that the constant factor is the best possible. In , Yang [] gave the following half-discrete Hardy-Hilbert's inequality with the best possible constant factor B(λ  , λ  ): where, λ  > ,  < λ  ≤ , λ  + λ  = λ. Zhong and Yang [, -] investigated several halfdiscrete Hilbert-type inequalities with particular kernels. Applying weight functions, a half-discrete Hilbert-type inequality with a general homogeneous kernel of degree -λ ∈ R with the best constant factor k(λ  ) is obtained as follows: which is an extension of () (cf. []). At the same time, a half-discrete Hilbert-type inequality with a general inhomogeneous kernel and the best constant factor is given by Yang  In this paper, by applying weight coefficients and technique of real analysis, a halfdiscrete Hardy-Hilbert-type inequality related to the kernel of hyperbolic secant function and the best possible constant factor is given, which is an extension of () for λ =  and a particular kernel. The equivalent forms, the operator expressions with the norm, the reverses, and some particular cases are also considered.
and then () follows.

Main results and operator expressions
, then for p > ,  < f p, δ , a q, < ∞, we have the following equivalent inequalities: In view of () and the Lebesgue term-by-term integration theorem (see []), we find Then by () we have (). By Hölder's inequality (see []) we have Then by () we have (). On the other hand, assuming that () is valid, we set Then we find J p  = a q q, . If J  = , then () is trivially valid; if J  = ∞, then () keeps impossible. Suppose that  < J  < ∞. By () we have and then () follows, which is equivalent to (). Again by Hölder's inequality with weight we have Then by () and the Lebesgue term-by-term integration theorem it follows that Then by () we have (). By Hölder's inequality we have Then by () we have (). On the other hand, assuming that () is valid, we set Then we find J q  = f p p, δ . If J  = , then () is trivially valid; if J  = ∞, then () keeps impossible. Suppose that  < J  < ∞. By () we have Proof By (), (), and () we obtain If there exists a positive constant K ≤ k(σ ) such that () is valid when replacing k(σ ) by K , then, in particular, by the Lebesgue term-by-term integration theorem we have εĨ < εK f p, δ ã q, , namely, It follows that k(σ ) ≤ K (ε →  + ). Hence, K = k(σ ) is the best possible constant factor of (). The constant factor k(σ ) in () (()) is still the best possible. Otherwise, we would reach a contradiction by () (()) that the constant factor in () is not the best possible.
Definition  Define a half-discrete Hardy-Hilbert-type operator T  : L p, δ (R + ) → l p, -p as follows: For any f ∈ L p, δ (R + ), the exists a unique representation T  f = c ∈ l p, -p . We define the formal inner product of T  f and a = {a n } ∞ n= ∈ l q, as follows: Then we can rewrite () and () as follows: Define the norm of operator T  as follows: Then by () it follows that T  ≤ k(σ ). Since by Theorem  the constant factor in () is the best possible, we have Assuming that a = {a n } ∞ n= ∈ l q, and setting Definition  Define a half-discrete Hardy-Hilbert-type operator T  : l q, → L q, -q δ (R + ) as follows: For any a = {a n } ∞ n= ∈ l q, , there exists a unique representation T  a = h ∈ L q, -q δ (R + ). We define the formal inner product of T  a and f ∈ L p, δ (R + ) as follows: Then we can rewrite () and () as follows: Define the norm of operator T  as follows: Then by () we find T  ≤ k(σ ). Since by Theorem  the constant factor in () is the best possible, we have

Some equivalent reverse inequalities
In the following, we also set For  < p <  or p < , we still use the formal symbols f p, δ , f p, δ , and a q, .
, there exists n  ∈ N such that υ n ≥ υ n+ (n ∈ {n  , n  + , . . .}), and U(∞) = V (∞) = ∞, then for p < ,  < f p, δ , a q, < ∞, we have the following equivalent inequalities with the best possible constant factor k(σ ): Proof By the reverse Hölder inequality with weight (see []), since p < , similarly as obtaining () and (), we have Then by () and the Lebesgue term-by-term integration theorem it follows that Then by () we have (). By the reverse Hölder inequality we have Then by () we have (). On the other hand, assuming that () is valid, we set a n as in Theorem . Then we find J p  = a q q, . If J  = ∞, then () is trivially valid; if J  = , then () keeps impossible. Suppose that  < J  < ∞. By () it follows that and then () follows, which is equivalent to ().
Still by the reverse Hölder inequality with weight, since  < q < , similarly as obtaining () and (), we have Then by () and the Lebesgue term-by-term integration theorem it follows that Then by () we have (). By the reverse Hölder inequality we have Then by () we have (). On the other hand, assuming that () is valid, we set f (x) as in Theorem . Then we find J q  = f p p, δ . If J  = ∞, then () is trivially valid; if J  = , then () keeps impossible. Suppose that  < J  < ∞. By () it follows that and then () follows, which is equivalent to (). Therefore, inequalities (), (), and () are equivalent.
By (), (), and () we obtain If there exists a positive constant K ≥ k(σ ) such that () is valid when replacing k(σ ) by K , then, in particular, we have εĨ > εK f p, δ ã q, , namely, It follows that k(σ ) ≥ K (ε →  + ). Hence, K = k(σ ) is the best possible constant factor of (). The constant factor k(σ ) in () (()) is still the best possible. Otherwise, we would reach a contradiction by () (()) that the constant factor in () is not the best possible.
Theorem  With the assumptions of Theorem , if  < p < ,  < f p, δ , and a q, < ∞, then we have the following equivalent inequalities with the best possible constant factor k(σ ): Proof By the reverse Hölder inequality with weight, since  < p < , similarly as obtaining () and (), we have In view of () and the Lebesgue term-by-term integration theorem, we find Then by () we have ().
By the reverse Hölder inequality we have Then by () we have (). On the other hand, assuming that () is valid, we set a n as in Theorem . Then we find J p  = a q q, . If J  = ∞, then () is trivially valid; if J  = , then () keeps impossible. Suppose that  < J  < ∞. By () it follows that and then () follows, which is equivalent to (). Again by the reverse Hölder inequality with weight, since q < , we have Then by () and the Lebesgue term-by-term integration theorem it follows that Then by () we have (). By the reverse Hölder inequality we have and then () follows, which is equivalent to ().

Some corollaries
For δ =  in Theorems -, we have the following inequalities with inhomogeneous kernel.
For δ = - in Theorems -, we have the following inequalities with the homogeneous kernel of degree .
For α = ρ and γ = σ in Theorems -, we have the following corollary.