A more accurate half-discrete Hilbert-type inequality in the whole plane and the reverses

A more accurate half-discrete Hilbert-type inequality in the whole plane with multi-parameters is established by the use of Hermite–Hadamard’s inequality and weight functions. Furthermore, some equivalent forms and some special types of inequalities and operator representations as well as reverses are considered.


Introduction
For p > 1, 1 p + 1 q = 1, a m , b n > 0, the following discrete Hardy-Hilbert inequality (cf. [10], Theorem 315, and [4,11,36,40]) holds true: The constant factor sin( ∕p) is optimal. Let f(x), g(y) ≥ 0, such that Then the following Hardy-Hilbert integral inequality with the same best possible constant factor sin( ∕p) (cf. [11], Theorem 316) is valid: The following half-discrete Hardy-Hilbert inequality with the same best possible constant factor was recently formulated and proved (cf. [39]): Several inequalities with homogenous kernels of degree 0 as well as with nonhomogenous kernels have been proved in [7,11,19,37,42,45]. For a large variety of integral inequalities of Hilbert-type the interested reader is referred to [1-6, 8, 12, 15, 16, 20, 22-24, 26, 41, 46, 47]. The above inequalities are constructed in the quarter plane of the first quadrant. A Hilbert-type integral inequality in the whole plane was proved in [33] by Yang. Furthermore, a generalized form of a Hilbert-type integral inequality in the whole plane was considered in [34]: is optimal. Additionally, in [9,13,14,25,27,29,30,43,44] several integral and discrete Hilbert-type inequalities in the whole plane where formulated and proved. The goal of the present paper is to study the following half-discrete Hilbert-type inequality in the whole plane with parameters and a best possible constant factor, by applying the Hermite-Hadamard inequality and weight functions: where , > 0, + = ≤ 1 . Moreover, a more accurate half-discrete Hilbert-type inequality with multiparameters is proved. Some equivalent forms, a few special types of inequalities as well as operator representations and reverses are studied.

Some lemmas
In what follows, we assume that Setting in the above first (resp. second) integral, by simplifications, we deduce that Hence, (8) follows. We obtain By (11) and Hermite-Hadamard's inequality (cf. [17]), in view of ∈ [0, 1 2 ], we have that Setting in the above first (resp. second) integral, by simplifications, we deduce that By (11) and the decreasing property of series, we also have that Setting in the above first (resp. second) integral, by simplifications, we derive that For some ∈ (0, ), we obtain that and thus there exists a constant L > 0, such that 0 < u ln u u−1 ≤ L (u ∈ (0, ∞)) . Hence, we have and therefore (9) and (10) By (13) and the decreasing property of series, we derive that Hence, we obtain (12) in the above first (resp. second) integral, we obtain Hence, we get (14) and thus the lemma is proved. ◻
Therefore, inequalities (16), (17) and (18)  Proof For 0 < < q , we set ̃ = − q (∈ (0, )), and Then by (12) and (14), we obtain that By (9), we also have that If the constant factor K a,b ( ) in (16) is not the best possible, then there exists a positive number k, with K a,b ( ) > k , such that (16) is valid when we replace K a,b ( ) by k. Then in particular, we have Ĩ < k � I 1 , namely, It follows thatĨ This is a contradiction. Hence, the constant factor K a,b ( ) in ( 16) is the best possible.
The constant factor K a,b ( ) in (17) (resp. (18)) is still the best possible. Otherwise, we would reach a contradiction by (23) (resp. (25)) that the constant factor K a,b ( ) in (16) is not the best possible.
This completes the proof of the theorem. ◻

Operator expressions
Suppose that p > 1, 1 p + 1 q = 1. We set the following functions: wherefrom, Define the following real weight normed linear spaces: In view of (26), it follows that and then the operator T (1) is bounded satisfying Since the constant factor K a,b ( ) in (26) is the best possible, we have If we define the formal inner product of T (1) f and b (∈ l q,Ψ ) as follows: we can then rewrite (16) and (17) as follows: In view of Theorem 1, for b ∈ l q,Ψ , setting then by (18) we have namely H (2) ∈ L q,Ψ 1−q ( ). In view of (29), we have and then the operator T (2) is bounded satisfying Since the constant factor K a,b ( ) in (29) is the best possible, we have If we define the formal inner product of T (2) b and f (∈ L p,Φ ( )) as follows: then we can rewrite (16) and (18) in the following manner:  (5). If f (−x) = f (x) (x > 0), b −n = b n (n ∈ ), then (5) reduces to the following half-discrete Hilbert-type inequality (cf. [40]): (16) reduces to the following particular inequality with homogeneous kernel of degree − : (iii) For = −1, (16) reduces to the following particular inequality with nonhomogeneous kernel: The constant factors in the above inequalities are the best possible.
For 0 < < |p| , we set � = + p (> 0), and Then by (12) and (14), we obtain that By (9), we also have ||f ||   If the constant factor K a,b ( ) in (42) is not the best possible, then there exists a positive number k, with K a,b ( ) < k , such that (42) is satisfied when we replace K a,b ( ) by k. Then in particular, we have Ĩ > k � I 1 , namely It follows that that is This is a contradiction. Hence, the constant factor K a,b ( ) in ( 42) is the best possible.
The constant factor K a,b ( ) in (43) ((44)) is still the best possible. Otherwise, we would reach a contradiction by (46) ((48)) that the constant factor K a,b ( ) in (42) is not the best possible.
This completes the proof of the theorem. ◻ Funding Open Access funding provided by Universität Zürich.
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