On two kinds of the reverse half-discrete Mulholland-type inequalities involving higher-order derivative function

By means of the weight functions, Hermite–Hadamard’s inequality, and the techniques of real analysis, a new more accurate reverse half-discrete Mulholland-type inequality involving one higher-order derivative function is given. The equivalent statements of the best possible constant factor related to a few parameters, the equivalent forms, and several particular inequalities are provided. Another kind of the reverses is also considered.

In 2016, by the use of the techniques of real analysis, Hong et al. [21] gave an equivalent statement of the best possible constant factor related to several parameters in the general form of (1). The other similar results were provided by [22][23][24][25][26][27][28][29]. Recently, Yang et al. [30] gave a new result in a reverse half-discrete Hilbert-type inequality.
In this paper, following the way of [4,21], by means of the weight functions, Hermite-Hadamard's inequality, and the techniques of real analysis, a new more accurate reverse half-discrete Mulholland-type inequality with the kernel as 1 [x+ln α (n-ξ )] λ+m involving one higher-order derivative function is given. The equivalent statements of the best possible constant factor related to a few parameters, the equivalent forms, and several particular inequalities are provided. Another kind of the reverses is also considered.
Lemma 2 For p < 0 (0 < q < 1), we have the following reverse Mulholland-type inequality: Proof For α > 0, setting v = x/ ln α (nξ ), we can obtain another weight function: By reverse Hölder's inequality (cf. [31]), we have We show that (10) does not keep the form of equality. Otherwise (cf. [31]), there exist constants A and B such that both of them are not zero and Assuming that A = 0, there exists n ∈ N\{1} satisfying Then, by (6) and (9), we have (8).
The lemma is proved.

Lemma 3
For t > 0, we have the following expression: In view of it follows that lim x→∞ By substitution of k = 1, . . . , m in the above expression, (11) follows. The lemma is proved.

Main results
Theorem 1 For p < 0 (0 < q < 1), we have the following more accurate reverse half-discrete Mulholland-type inequality involving one higher-order derivative function: In particular, for λ 1 + λ 2 = λ, we have and the following inequality: Proof Since we have by the Lebesgue term by term integration theorem (cf. [32]) and (11), we find e -t ln α (n-ξ ) a n dt e -t ln α (n-ξ ) a n dt Then, by (8), we have (12). The theorem is proved.
Proof If λ 1 + λ 2 = λ, then by Theorem 2, the constant factor (12) is best possible. By (19), the constant factor in (17) is still best possible. Otherwise, we would reach a contradiction that the constant factor in (12) is not best possible.
On the other hand, if the constant factor in (17) is best possible, then, by the equivalency of (17) and (12), in view of J p = I (see the proof of Theorem 3), we still can show that the constant factor in (12) is best possible. By the assumption and Theorem 2, we have The theorem is proved.
In the same way of proving Theorem 4, we have the following.

Conclusions
In this paper, following the way of [4,21], by means of the weight functions, Hermite-Hadamard's inequality, and the techniques of real analysis, a new more accurate reverse half-discrete Mulholland-type inequality with the kernel as 1 [x+ln α (n-ξ )] λ+m involving one higher-order derivative function is given (for p < 0, 0 < q < 1) in Theorem 1. The equivalent statements of the best possible constant factor related to a few parameters, the equivalent forms, and several particular inequalities are provided in Theorems 2-4 and Remark 1. Another kind of the reverses is also considered (for 0 < p < 1, q < 0) in Theorems 5-6. The lemmas and theorems provide an extensive account of this type of inequalities.