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.