A FEW EQUIVALENT STATEMENTS OF A HILBERT-TYPE INTEGRAL INEQUALITY WITH THE RIEMANN-ZETA FUNCTION

By using the way of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality in the whole plane with the internal variables and the best possible constant factor related to the Riemann-zeta function is proved. The operator expression and some particular cases are considered.

In this paper, by using the way of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality in the whole plane with the internal variables and the best possible constant factor related to the Riemannzeta function is proved in Theorem 3.1 and Theorem 3.2. The operator expressions and some particular cases are considered in Theorem 4.1 and Remark 4.1. (1 − tanh(γu))u σ−1 du
The lemma is proved. For σ 1 = σ, we still have Lemma 2.4. 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 α,β (σ) ≤ M.
In particular, (1) for δ = 1, we have the following equivalent inequalities with the homogeneous kernel of degree 0 and the best possible constant factor K α,β (σ) : (2) for δ = −1, we have the following equivalent inequalities with the nonhomogeneous kernel and the best possible constant factor K α,β (σ) : (3.10) Proof. For σ 1 = σ and the assumption of (i), if (3.4) takes the form of equality for a y ∈ (−∞, 0) ∪ (0, ∞), then (see [11]), there exist constants A and B, such that they are not all zero, and We suppose that A = 0 (otherwise B = A = 0). Then it follows that Ax α a.e. in R. (3.4) takes the form of strict inequality; so does (3.1). Hence, (3.5) and (3.6) are valid.
In view of Theorem 3.1, we still can conclude that statements (i), (ii) and (iii) in Theorem 3.2 are equivalent.
The theorem is proved.