A more accurate half-discrete Hardy-Hilbert-type inequality with the logarithmic function

By means of the weight functions, the technique of real analysis and Hermite-Hadamard’s inequality, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of logarithmic function and a best possible constant factor is given. Moreover, the equivalent forms, the operator expressions, the reverses and some particular cases are also considered.


Introduction
Assuming that p > ,  p +  q = , f (x), g(y) ≥ , f ∈ L p (R + ), g ∈ L q (R + ), f p = ( ∞  f p (x) dx)  p > , g q > , we have the following Hardy-Hilbert integral inequality (cf. []): x + y dx dy < π sin(π/p) f p g q , where the constant factor π sin(π /p) is the best possible. If a m , b n ≥ , a = {a m } ∞ m= ∈ l p , b = {b n } ∞ n= ∈ l q , a p = ( ∞ m= a p m )  p > , b q > , then we have the following Hardy-Hilbert inequality with the same best possible constant factor π sin(π /p) (cf. []): For μ i = ν j =  (i, j ∈ N), () reduces to (). In , by introducing an independent parameter λ ∈ (, ], Yang [] gave an extension of () with the kernel  (x+y) λ for p = q = . Recently, Yang [] gave some extensions of () and () as follows: If λ  , λ  ∈ R, λ  + λ  = λ, k λ (x, y) is a non-negative homogeneous function of degree -λ, with k(λ  ) = g ∈ L q,ψ (R + ), f p,φ , g q,ψ > , then we have where the constant factor k(λ  ) is the best possible. Moreover, if k λ (x, y) keeps a finite value and we have the following inequality with the same best possible constant factor k(λ  ): In , Yang [] gave an extension of () for the kernel k λ (m, n) =  (m+n) λ and () as follows: investigated a few half-discrete Hilbert-type inequalities with the particular kernels. Applying the weight functions, a half-discrete Hilbert-type inequality with a general homogeneous kernel of degree -λ ∈ R and a best constant factor k(λ  ) is proved as follows (cf. []): A half-discrete Hilbert-type inequality with a general non-homogeneous kernel and a best constant factor is given by Yang []. In this paper, by means of the weight functions, the technique of real analysis and Hermite-Hadamard's inequality, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of logarithmic function and a best possible constant factor is given, which is an extension of () in a particular kernel of degree  similar to (). The equivalent forms, the operator expressions, the equivalent reverses and some particular cases are also considered.

Some lemmas
In the following, we agree that ν j >  (j ∈ N), V n := n j= ν j , μ(t) is a positive continuous function in R + = (, ∞), (i) Setting u = ρt -γ , we find Proof For n  ∈ N\{}, by the assumptions and Hermite-Hadamard's inequality (cf. []), we have In the same way, we still have Hence, adding the above two inequalities, we have (). The lemma is proved.
Then by (), we have (). The lemma is proved.
Note If U(∞) = ∞, then () keeps the form of an equality. where .
Hence we find and then () follows.
(ii) For b > , by (), we find Hence we have (). The lemma is proved.

Main results and operator expressions
Theorem  If ρ > , k(σ ) is determined by (), then, for p > ,  < f p, δ , a q, < ∞, we have the following equivalent inequalities: Proof By Hölder's inequality with weight (cf. []), we have In view of () and Lebesgue term by term integration theorem (cf. []), we find Then by (), we have (). By Hölder's inequality (cf. []), we have In view of (), 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 () remains impossible. Suppose that  < J  < ∞. By (), we have and then () follows, which is equivalent to ().
Still by Hölder's inequality with weight (cf. []), we have Then by () and Lebesgue term by term integration theorem (cf. []), it follows that In view of (), we have (). By Hölder's inequality (cf. []), 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 and then () follows, which is equivalent to (). Therefore, inequalities (), () and () are equivalent. The theorem is proved.
Then, for δ = ±, since U(∞) = ∞, we obtain By (), () and (), we find If there exists a positive constant K ≤ k(σ ), such that () is valid when replacing k(σ ) to K , then in particular, by Lebesgue term by term integration theorem, we have ε I < εK f p, δ a 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. The theorem is proved.
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 + ), there exists a unique representation T  f = c ∈ l p, -p . 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, , 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 + ). 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 reverses
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, .
Still by the reverse Hölder inequality with weight (cf. []), since  < q < , in a similar way to obtaining () and (), we have Then by () and the Lebesgue term by term integration theorem (cf. []), it follows that In view of the note of Lemma , 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 () remains impossible. Suppose that  < J  < ∞. By (), it follows that and then () follows, which is equivalent to ().
We still can obtain some inequalities with the best possible constant factors in Theorems -, by using some particular parameters.

Conclusions
In this paper, by means of the weight functions, the technique of real analysis and Hermite-Hadamard's inequality, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of logarithmic function and a best possible constant factor is given by Theorems -. Moreover, the equivalent forms and the operator expressions are considered. We also obtain the reverses and some particular cases in Theorems -. The method of weight functions is very important, which is the key to help us proving the main inequalities with the best possible constant factor. The lemmas and theorems provide an extensive account of this type of inequalities.