New half-discrete Hilbert inequalities for three variables

In this paper, we obtain two new half-discrete Hilbert inequalities for three variables. The obtained inequalities are with the best constant factor. Moreover, we give their equivalent forms.


Introduction
Suppose that p > 1, 1 p + 1 q = 1, and f and g are nonnegative functions such that f ∈ L p (R + ) and g ∈ L q (R + ). Then we have the inequalities ∞ 0 ∞ 0 f (x)g(y) x + y dx dy < π sin(π/p) provided that the integrals on the right-hand side in both inequalities are positive. The constant factors π sin(π /p) and ( π sin(π /p) ) p are the best possible in (1) and (2), respectively. Moreover, (1) and (2) are equivalent. Inequality (1) was first studied by D. Hilbert at the end of the 19th century, and hence, in his honor, it is referred to as the Hilbert inequality.
The corresponding discrete forms of inequalities (1) and (2) for two nonnegative sequences of real numbers {a m } and {b n } are given as where {a m } ∈ p and {b n } ∈ q . The constant factors π sin(π /p) and ( π sin(π /p) ) p are also the best possible. Inequalities (3) and (4) are equivalent (see [1]).

Preliminaries and lemmas
As it is well known, the gamma function (θ ) and beta function B(μ, ν) are defined respectively by the improper integrals Using the definition of the gamma function, we may write To prove our main results, we need some lemmas. As we will see, their proofs are simple and based on the famous Hölder inequality for both integrals and sums and the fact that we can estimate the sum of a decreasing function by an integral. The first lemma (Lemma 2.1) is given in [19] and the second one (Lemma 2.2) is also given in [15]. For completeness, we give proofs of these two lemmas.
Proof By the Hölder inequality we obtain We compute the first integral on the right-hand side of (10) by using the substitutions y = ux and x = v t(u+1) : Substituting (11) into (10), we get (9).

Lemma 2.3
Let p > 1, 1 p + 1 q = 1, and a n,m > 0. Then, for t > 0 and 0 Proof The proof follows the lines of the proof of Lemma 2.2. Namely, applying Hölder inequality, we have The lemma is proved.
Similarly, we obtain the following lemma.

Main results
In this section, we give two new half-discrete versions of inequality (7). Both obtained inequalities are with the best constant factor expressed in terms of the beta function.
where the constant C = B( λ qγ , λ p + γ ) is the best possible. In particular: Proof Using (8) and applying the Hölder inequality, we have e -nt a n dt By Lemma 2.1 and Lemma 2.2 we obtain respectively n θq e -nt a q n .
Putting these two inequalities into (13), we get it follows that It remains to show that the constant factor C in (12)  q -γ -1 (n ≥ 1). Suppose that the constant C is not the best possible. Then there exist 0 < K < C such that On the other hand, estimating the left-hand side, we find (set u = z(x + y)) Obviously, as ε → 0 + , from (14) and (15) we obtain a contradiction. Therefore, the proof of the theorem is completed.
, and a n,m > 0. Suppose that f (x) is a nonnegative function defined on (0, ∞) and a double sequence a n,m > 0.
Proof Using (8) and applying the Hölder inequality, we have ∞ n=1 e -mt-nt a n,m t ∞ n=1 e -mt-nt a n,m

Equivalent forms
In this section, we give some equivalent forms of the inequalities obtained in Theorems 3.1 and 3.2. All the inequalities are with the same best constant factor.
Both inequalities (20) and (21) are equivalent to (12), and the constants C p and C q are the best possible.

we obtain inequality (20). On the other hand, by the Hölder inequality and (20), we find
Therefore, using (20), we obtain (12). Now, to prove the equivalence relation between (12) and (21), we set f (x, y) = (x + y) (q-1)λ+γ q-2 ∞ n=1 a n (x + y + n) λ q-1 . By inequality (12) Obviously, from the last inequality we get (21). On the other hand, using the Hölder inequality and (21) respectively, we find Thus, the equivalence relation between (21) and (12) is proved. Moreover, since the constant in (12) is the best possible, we deduce that the constants in both inequalities (20) and (21) are also the best possible. The theorem is proved.

Theorem 4.2
Under the assumptions of the Theorem 3.2, we have the following inequalities: Both inequalities (23) and (24) are equivalent to (16), and the constants C p and C q are the best possible.
Proof To prove (23), we set Applying inequality (16), we find Hence, we obtain inequality (23). On the other hand, by the Hölder inequality and (23) we find Therefore, using (23), we obtain (16). To prove the equivalence relation between (16) From the last inequality we obtain (24). On the other hand, using the Hölder inequality and (24), we have Thus, the equivalence relation between (24) and (16) is proved. Moreover, since the constant in (16) is the best possible, the constants in both inequalities (23) and (24) are also the best possible. The theorem is proved.

Conclusion
In the present study, we introduced two new half-discrete Hilbert inequalities for three variables. The equivalent forms are also considered. Moreover, we proved that the constants appearing on the right-hand sides of these inequalities are the best possible.