A New Extension of Hardy-Hilbert’s Inequality Containing Kernel of Double Power Functions

In this paper, we provide a new extension of Hardy-Hilbert’s inequality with the kernel consisting of double power functions and derive its equivalent forms. The obtained inequalities are then further discussed regarding the equivalent statements of the best possible constant factor related to several parameters. The operator expressions of the extended Hardy-Hilbert’s inequality are also considered.

Recently, it has come to our attention that some results were provided by Hong and Wen in [23]: in the paper they studied the equivalent statements of the extended inequalities (1) and (2), and estimated the best possible constant factor for several parameters. Inspired by the ideas of Hong and Wen in [23], using the Euler-Maclaurin summation formula, Yang, Wu and Chen [24] presented an extension of Hardy-Littlewood-Polya's inequality involving the kernel 1 (max{ , }) m n  as follows Based on the obtained inequality, we derive its equivalent form and discuss the equivalent statements of the best possible constant factor related to several parameters. The operator expressions and some particular cases of the obtained inequality of Hardy-Hilbert type are also considered.

Some Lemmas
In what follows, we suppose that Then we have the following inequalities ). 0 ( : By using the Euler-Maclaurin summation formula (see [2,3]), for the Bernoulli function of Utilizing the Euler-Maclaurin summation formula (see [2,3]), we obtain On the other hand, we also have , by using the Euler-Maclaurin summation formula (see [2,3]), we obtain Proof. Following the way of the proof of Lemma 1, for  n N, we have the following inequality ).

Proof. For any
By using inequality (14) and setting In virtue of the above results, we have is the best possible constant factor in (15). The Lemma 3 is proved. □

Remark 2. Setting
and thus we can rewrite inequality (13) as If the constant factor q p k k (17) is the best possible, then in view of (15), we have q p k k 1 1 )) ( ( )) ( ( Applying Hӧlder's inequality (see [27]), we obtain , which implies that (18) keeps the form of equality.
We observe that (18) keeps the form of equality if and only if there exist constants A and B such that they are not both zero and (see [27]) . .
. This completes the proof of Lemma 4. □

Theorem 1. Inequality (13) is equivalent to the following
If the constant factor in (13) is the best possible, then so is the constant factor in (19).
By means of the above obtained result, we can conclude that if the constant factor in (13) is the best possible, then so is the constant factor in (19). Otherwise, if there exists a constant 1 1 then by (20), we would reach a contradiction that the constant factor (13) is not the best possible. The proof of Theorem 1 is complete. □ Theorem 2. The following statements (i), (ii), (iii) and (iv) are equivalent: (13) Proof. (i)  (ii). By (i) , in view of the continuity of the Beta function, we have , then (18) which are independent of , p q . Hence, it follows that (i)  (ii)  (iv).
Hence, we conclude that the statements (i), (ii), (iii) and (iv) are equivalent. This completes the proof of Theorem 2. □

Conclusions
We first provided a brief survey on the study of Hardy-Hilbert's inequality, and then we stated the main results, new extensions of Hardy-Hilbert's inequality, in Lemma 2 and Theorem 1, respectively. For further study on the obtained inequalities, the equivalent statements of the best possible constant factor related to several parameters are given in Theorem 2; the operator expressions of the extended Hardy-Hilbert's inequality are established in Theorem 3. It is worth noting that the extended Hardy-Hilbert's inequality (13) obtained in this paper differs from the inequality (9) that appeared in [26], since inequalities (9)  . We also note that the two kinds of kernels have a similar form; this prompts us to consider the meaningful problem of how to establish a unified extension of inequalities (9) and (13) in a subsequent study.
Author Contributions: B.Y. carried out the mathematical studies and drafted the manuscript. S.W. and Q.C. participated in the design of the study and performed the numerical analysis. All authors contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.
Funding: This work is supported by the National Natural Science Foundation (No. 61772140), and the Science and Technology Planning Project Item of Guangzhou City (No. 201707010229).