A Hilbert-Type Integral Inequality in the Whole Plane Related to the Arc Tangent Function

: In this work we establish a few equivalent statements of a Hilbert-type integral inequality in the whole plane related to the kernel of the arc tangent function. We prove that the constant factor, which is associated with the cosine function, is optimal. Some special cases as well as some operator expressions are also presented.

Most of them are constructed in the quarter plane of the first quadrant.
In this paper, we follow the idea of Hong's work in [23] and using techniques of real analysis as well as weight functions, we prove a few equivalent statements of a Hilbert-type integral inequality in the whole plane related to the kernel of the arc tangent function. The constant factor which is related to the cosine function is proved to be the best possible. Within this work, we also consider some particular cases of interest as well as operator expressions.

Some Lemmas
This completes the proof of the lemma.
In what follows, we assume that p > 1, 1 (3) and For n ∈ N = {1, 2, · · · }, E −1 = [−1, 1], x ∈ E δ , we define the following expressions: For fixed x ∈ E δ , setting u = x δ α y β , we obtain For n ∈ N, x ∈ F δ , we define the following expressions: Since for x ∈ E −δ , For fixed x ∈ E −δ , setting u = x δ α y β , we obtain In view of (8) and (10), we derive the lemma below: We have the following inequalities: Lemma 3. If there exists a constant M, such that for any nonnegative measurable functions f (x) and g(y) in R, the following inequality holds true, then we have σ 1 = σ.
Proof. If σ 1 > σ, then for n ≥ 1 σ 1 −σ (n ∈ N), we consider the functions and by (4) and (5), we obtain By (11) and (13) (for f = f n , g = g n ), we have Since for any n ≥ 1 In view of we derive that ∞ ≤ M J 1 < ∞, which is a contradiction. If σ > σ 1 , then for n ≥ 1 σ−σ 1 (n ∈ N), we consider the functions , and by (4) and (5), we obtain By (12) and (13) (for f = f n , g = g n ), we have This completes the proof of the lemma.
For σ 1 = σ, we also get the lemma below: If there exists a constant M, such that for any nonnegative measurable functions f (x) and g(y) in R, the following inequality holds true, then we have K Proof. For σ 1 = σ, by (8), we have In view of the presented results, for n > 1 q(γ−σ) , we obtain For γ > σ + d (d > 0), we have that (arctan ρ u γ )u σ+d is continuous in (0, ∞), and There exists a positive constant M 1 , such that By (4), it follows that and then by (15), it follows that Similarly, we have By (14) (for f = f n , g = g n ), we have For n → ∞, by Fatou's lemma (cf. [39]), (16) and (17), we obtain This completes the proof of the lemma.

Main Results and Some Particular Cases
Theorem 1. If M is a constant, then the following statements (i), (ii) and (iii) are equivalent: (i) For any f (x) ≥ 0, we have the following inequality: (ii) for any f (x), g(y) ≥ 0, we have the following inequality:
For σ 1 = σ, we deduce the theorem below: Theorem 2. If M is a constant, then the following statements (i), (ii) and (iii) are equivalent: we have the following inequality: and g(y) ≥ 0, satisfying we have the following inequality: (25) and (26) is the best possible.
In particular: (1) for δ = 1, we have the following equivalent inequalities with the nonhomogeneous kernel: where K (γ) α,β (σ) is the best possible constant factor; (2) for δ = −1, we have the following equivalent inequalities with the homogeneous kernel of degree 0: where K (γ) α,β (σ) is the best possible constant factor. Proof. For σ 1 = σ and the assumption of statement (i), if (24) assumes the form of equality for some y ∈ (−∞, 0) ∪ (0, ∞), then (see [40]) there exist constants A and B, such that they are not both zero, and We suppose that A = 0 (otherwise B = A = 0). Then it follows that it contradicts the fact that Hence, (24) takes the form of strict inequality, and so does (21). Hence, (25) and (26) are true.
In view of Theorem 1, we can establish the equivalency between the statements (i), (ii) and (iii) in Theorem 2.
In case the statement (iii) is valid, namely K  (26) is not optimal.
This completes the proof of the theorem.

Operator Expressions
We set the following functions: . Define the following real normed linear spaces: In view of Theorem 2, for f ∈ L p,ϕ (R), setting by (25), we have Definition 1. Define a Hilbert-type integral operator with the nonhomogeneous kernel T : L p,ϕ (R) → L p,ψ 1−p (R) as follows: For any f ∈ L p,ϕ (R), there exists a unique representation T f = h 1 ∈ L p,ψ 1−p (R), satisfying T f (y) = h 1 (y), for any y ∈ R.
In view of (31), it follows that and thus the operator T is bounded satisfying where ρ σ/γ π σ cos πσ 2γ is the optimal constant factor. If f (−x) = f (x), g(−y) = g(y) (x, y ∈ R + ), then we have the following equivalent inequalities: where ρ σ/γ π 2σ cos πσ 2γ is the best possible constant factor.

Conclusions
In this paper, making use of ideas of Hong [23], and by employing techniques of real analysis as well as weight functions, we obtain in Theorem 1 a few equivalent statements of a Hilbert-type integral inequality in the whole plane associated with the kernel of the arc tangent function. In Theorem 2, the constant factor associated with the cosine function is proved to be optimal. Furthermore, in Theorem 3 and Remark 1 we also consider some particular cases and operator expressions. The lemmas and theorems within this work provide an extensive account of this type of inequalities.
Author Contributions: All authors contributed equally during all stages of the preparation of the present work. All authors have read and agreed to the published version of the manuscript.