Throughout this paper, let
denote the set of all real numbers, let
be the given subset of
, and
denote the set of nonnegative integers. For functions
, their first-order differences are defined by
,
, and
. We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. In what follows, we assume all functions which appear in the inequalities to be real-value,
and
are constants, and
.
Lemma 2.1.
Assume that
are nonnegative functions defined for
, and
is nonincreasing in each variable, if
then
Proof.
Define a function
by
The function
is nonincreasing in each variable, so is
, we have
Using Lemma 1.2, the desired inequality (2.2) is obtained from (2.1), (2.3), and (2.4). This completes the proof of Lemma 2.1.
Theorem 2.2.
Suppose that
and
are nonnegative functions defined for
,
satisfies the inequality (1.4). Then
where
Proof.
Define a function
by
From (1.4), we have
By applying Lemma 1.1, from (2.8), we obtain
It follows from (2.9) and (2.10) that
where we note the definitions of
and
in (2.6). From (2.6), we see
is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (3.3) is obtained from (2.9) and (2.11). This completes the proof of Theorem 2.2.
Theorem 2.3.
Suppose that
and
are nonnegative functions defined for
,
satisfies
where
, and
satisfies the inequality (1.5). Then
where
Proof.
Define a function
by
Then, as in the proof of Theorem 2.2, we have (2.8), (2.9), and (2.10). By (2.12),
It follows from (2.8), (2.9), (2.10), and (2.17) that
where we note the definitions of
and
in (2.14) and (2.15). From (2.14) we see
is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (2.19) is obtained from (2.9) and (2.18). This completes the proof of Theorem 2.3.
Theorem 2.4.
Suppose that
are the same as in Theorem 2.3,
satisfies the inequality (1.6). Then
where
Proof.
Define a function
by
Then, as in the proof of Theorem 2.2, we have (2.8), (2.9), and (2.10). By (2.12),
It follows from (2.8), (2.9), (2.10), and (2.23) that
where
and
are defined by (2.20) and (2.21), respectively. From (2.20), we see
is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (2.19) is obtained from (2.9) and (2.24). This completes the proof of Theorem 2.4.