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.