We apply Theorem 1.2 to obtain the following theorem.

Theorem.

Let with Let and be two nonnegative real sequences. If is homogeneous function of degree strictly decreasing in both parameters and then the following inequalities hold and are equivalent:

where and

Proof.

We use the inequalities (1.7), (1.8), and Theorem 1.2 with counting measure. First, we prove the inequality (2.1). Put and in the inequality (1.7). Then, we have

where and Since and the functions and are strictly decreasing, where we have

Using homogeneity of the functions and the substitution we get

In a similar manner we obtain

Now, the result follows from Theorem 1.2.

Remark.

Equality in the previous theorem is possible only if

for arbitrary constants and (see [5]). Condition (2.8) immediately gives that nontrivial case of equality in (2.1) and (2.2) leads to divergent series.

Now, we consider some special choice of the parameters and More precisely, let the parameters and satisfy constraint

Then, the constant from Theorem 2.1 becomes

Further, the inequalities (2.1) and (2.2) take form

In the following theorem we show, in a similar way as Yang did in [6], that if the parameters and satisfy condition (2.9), then one obtains the best possible constant. To prove this result we need the next lemma (see [6]).

Lemma.

If is decreasing in and strictly decreasing in a subinterval of and then

Theorem.

Let , and be defined as in Theorem 2.1. If the parameters and satisfy condition then the constants and in the inequalities (2.11) and (2.12) are the best possible.

Proof.

For this purpose, with , set and Now, let us suppose that there exists a smaller constant such that the inequality (2.11) is valid. Let denote the right-hand side of (2.11). Using Lemma 2.3, we have

Further, let denote the left-hand side of the inequality (2.11), for above choice of sequences and Applying, respectively, Lemma 2.3, Fubini's theorem, and substitution we have

From (2.11), (2.14), and (2.15) we get

By letting we obtain

Using symmetry of the function we have Now, from (2.17) we obtain a contradiction with assumption

Finally, equivalence of the inequalities (2.11) and (2.12) means that the constant is the best possible in the inequality (2.12). This completes the proof.

We proceed with some special homogeneous functions. Since the function is homogeneous of degree by using Theorem 2.4 we obtain the following.

Corollary.

Let Suppose that the parameters satisfy condition Then the following inequalities hold and are equivalent:

where the constant factors and are the best possible.

Remark.

If we put in Corollary 2.5, then the inequalities (2.18) become

where the constant factors , and are the best possible. For we obtain nonweighted case with the best possible constant Setting and in the inequalities (2.19) we obtain the inequalities (1.1) and (1.2) from Introduction.

Remark.

It is easy to see that Theorem 2.4 is the generalization of Theorem 1.1. Namely, let us define , and Note that the parameters satisfy condition Then, the best possible constant from Theorem 2.4 becomes from Theorem 1.1 (see also [4]).

Remark.

Similarly as in Corollary 2.5, for the homogeneous function of degree nonnegative real sequences and the parameters we have

where the constants and are the best possible.

Remark.

Let , and be defined as in Theorem 2.1. Take in the inequalities (2.11) and (2.12). By using Theorem 2.4 we get equivalent inequalities for general homogeneous kernel :

where the constant factors and are the best possible.

Setting in the inequalities (2.21) we obtain the result from [6]. Similarly, for above choice of the parameters , and we obtain Yang's result (1.3) from Introduction.