On a reverse Mulholland-type inequality in the whole plane with general homogeneous kernel

By using the idea of introducing parameters and weight coefficients, a new reverse discrete Mulholland-type inequality in the whole plane with general homogeneous kernel is given, which is an extension of the reverse Mulholland inequality. The equivalent forms are obtained. The equivalent statements of the best possible constant factor related to several parameters and a few applied examples are presented.

In this paper, following [25], by means of the idea of introducing parameters and the weight coefficients, a reverse discrete Mulholland-type in the whole plane is given as follows: for r > 1, 1 r which is an extension of the reverse of (2). The general forms as well as the equivalent forms are obtained. The equivalent statements of the best possible constant factor related to several parameters are presented, and a few applied examples are considered.
Hence, we have (7). The lemma is proved.

Note
In the same way, we still have the following inequality:
and the following Mulholland-type inequality in the whole plane: In particular, for (13), we have the following Mulholland-type inequality in the whole plane: Proof For any 0 < δ < δ 0 , we have We find By Lebesgue dominated convergence theorem (cf. [48]), it follows that The lemma is proved.
The lemma is proved.
Proof If the constant factor (12) is the best possible, then, by (17) and (13) (for λ i =λ i (i = 1, 2)), we have the following inequality: , from which it follows that (17) keeps the form of equality.
We observe that (17)  Assuming that A = 0, it follows that u λ 2 +λ 1 -λ = B A a.e. in R + , and then λ 2 + λ 1λ = 0, namely, The lemma is proved. (12) is equivalent to the following reverse Mulholland-type inequalities in the whole plane:
The theorem is proved.
(iii) ⇒ (i). If λ 1 + λ 2 = λ, then we have Both k 1 p λ (λ 2 )k 1 q λ (λλ 1 ) and k λ ( λ 2 p + λ-λ 1 q ) are independent of p, q. Hence, we have (i) ⇔ (ii) ⇔ (iii). (iii) ⇔ (iv). By Lemma 5 and Lemma 6, we obtain the conclusions. (iv) ⇔ (v). If the constant factor in (12) is the best possible, then so is constant factor in (19). Otherwise, by (21), we would arrive at a contradiction that the constant factor in (12) is not the best possible. On the other hand, if the constant factor in (19) is the best possible, then so is constant factor in (12). Otherwise, by (22), we would reach a contradiction that the constant factor in (19) is not the best possible.
(iv) ⇔ (vi). If the constant factor in (12) is the best possible, then so is constant factor in (20). Otherwise, by (23), we would reach a contradiction that the constant factor in (12) is not the best possible. On the other hand, if the constant factor in (20) is the best possible, then so is constant factor in (12). Otherwise, by (24), we would reach a contradiction that the constant factor in (20) is not the best possible. Therefore, the statements (i), (ii), (iii), (iv), (v) and (vi) are equivalent. The theorem is proved.