Equivalent property of a half-discrete Hilbert’s inequality with parameters

By using the weight functions and the idea of introducing parameters, a half-discrete Hilbert inequality with a nonhomogeneous kernel and its equivalent form are given. The equivalent statements of the constant factor are best possible related to parameters, and some particular cases are considered. The cases of a homogeneous kernel are also deduced.

In 2016, Hong [20,21] also considered some equivalent statements of the extensions of (1) and (2) with a few parameters. For the following work we refer to [22][23][24].
In this paper, following [20], by the use of the weight functions and the idea of introducing parameters, a half-discrete Hilbert inequality with the nonhomogeneous kernel and its equivalent form are given. The equivalent statements of the constant factor are best possible related to parameters, and some particular cases are considered. The cases of a homogeneous kernel are also deduced.

Some lemmas
In what follows, we assume that p > 1, 1 Lemma 1 Define the following weight functions: We have the following equality and inequalities: Proof Setting u = xn, we have and then (7) follows. In view of the decreasing property, we find Hence, (8) follows.
Then we have For ε → 0 + , in view of the continuous property of the Beta function, we find Hence, M = B(σ , λσ ) is the best possible constant factor of (10). λ)), we may rewrite (9) as follows: The parameterσ in (11) also satisfies by Hölder's inequality, andσ ≤ 1 (11) is the best possible, then we have

Main results and some corollaries
Theorem 1 Inequality (9) is equivalent to the following inequalities: If the constant factor in (9) is the best possible, then so is the constant factor in (13) and (14).
If the constant factor in (9) is the best possible, then so is constant factor in (13) (or (14)). Otherwise, by (15) (or (16)), we would reach the contradiction that the constant factor in (9) is not the best possible.
Replacing x by 1 x , and then x λ-2 f ( 1 x ) by f (x) in Theorem 1, setting σ 1 = λμ, we have the following.

Corollary 1
The following inequalities with the homogeneous kernel are equivalent: If the constant factor in (20) is the best possible, then so is the constant factor in (21) and (22).