On a new extended half-discrete Hilbert’s inequality involving partial sums

By applying the weight functions, the idea of introducing parameters, and Euler–Maclaurin summation formula, a new extended half-discrete Hilbert’s inequality with the homogeneous kernel and the beta, gamma function is given. The equivalent statements of the best possible constant factor related to a few parameters are considered. As applications, a corollary about the case of the non-homogeneous kernel and some particular cases are obtained.

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In this paper, according to the way of [21,22], by the use of the weight functions, the idea of introducing parameters and the Euler-Maclaurin summation formula, a new extended half-discrete Hilbert's inequality with the homogeneous kernel 1 (x+n) λ (0 < λ ≤ 26) and the beta, gamma function is given. The equivalent statements of the best possible constant factor related to a few parameters are considered. As applications, a corollary about the case of non-homogeneous kernel and some particular cases are also obtained.

Some lemmas
In what follows, we assume that p > 1, 1 By the definition of the gamma function, for λ, x > 0, n ∈ N, the following equality holds: Proof Since {a n } ∞ n=1 ∈ l 1 , we find lim n→∞ A n = ∞ i=1 a i ∈ [0, ∞). Using Abel's summation by parts formula and the inequality 1e -t ≤ t, we have (cf. [21]) ∞ n=1 e -tn a n = lim e -tn A n , namely, inequality (5) follows. For f ∈ L 1 (R + ), and then expression (6) follows.
we can reduce (9) as follows:
In view of Fubini's theorem (cf. [28]), it follows that So we obtain .
For ε → 0 + in the above inequality, in view of the continuity of the beta function, we find is the best possible constant factor of (13).
Remark 3 If μ + σ = s,then inequality (9) reduces to We confirm that the constant factor B(μ, σ ) in (17) is the best possible. Otherwise, we would reach a contradiction by (14) that the constant factor in (13) is not the best possible.

A corollary and some particular cases
Replacing x by 1 x in (12), setting g(x) = x λ-2 f ( 1 x ), we define Then we obtain the following inequality with the non-homogeneous kernel: It is obvious that inequality (18) is equivalent to (12). In view of Theorem 3, we have the following.