First of all, we monotonize some given functions , in the sums. Obviously, the sequence defined by in (2.1) consists of nondecreasing nonnegative functions and satisfies , for . Moreover,

as defined in [27] for comparison of monotonicity of functions , because every ratio is nondecreasing. By the definitions of functions , and , from (1.7) we get

Then, we discuss the case that for all . Because satisfies

it is positive and nondecreasing on . We consider the auxiliary inequality to (3.2), for all ,

where and are chosen arbitrarily, and claim that, for , a sublattice in ,

where is determined recursively by

, and is arbitrarily chosen on the boundary of the lattice

We note that , can be chosen appropriately such that

In fact, from the fact of being on the boundary of the lattice , we see that

Thus, it means that we can take . Moreover, , .

In the following, we will use mathematical induction to prove (3.5).

For , let . Then is nonnegative and nondecreasing in each variable on . From (3.4) we observe that

Moreover, we note that is nondecreasing and satisfies for and that . From (3.10) we have

On the other hand, by the Mean Value Theorem for integral and by the monotonicity of and , for arbitrarily given , there exists in the open interval such that

It follows from (3.11) and (3.12) that

Substituting with and summing both sides of (3.13) from to , we get, for all ,

We note from the definition of in (3.2) and the definition of in Section 2 that . By the monotonicity of and (3.10) we obtain

that is, (3.5) is true for .

Next, we make the inductive assumption that (3.5) is true for . Consider

for all . Let which is nonnegative and nondecreasing in each variable on . Then (3.16) is equivalent to

Since is nondecreasing and satisfies for and , from (3.17) we obtain, for all ,

where

On the other hand, by the Mean Value Theorem for integrals and by the monotonicity of and , for arbitrarily given there exists in the open interval such that

Therefore, it follows from (3.18) and (3.20) that

substituting with in (3.21) and summing both sides of (3.21) from to , we get, for all ,

where we note that . For convenience, let

From (3.17) and (3.22) we can get

the same form as (3.4) for , for all , where we note that for all . We are ready to use the inductive assumption for (3.24). In order to demonstrate the basic condition of monotonicity, let obviously which is a continuous and nondecreasing function on . Thus each is continuous and nondecreasing on and satisfies for . Moreover,

which is also continuous nondecreasing on and positive on . This implies that , for . Therefore, the inductive assumption for (3.5) can be used to (3.24) and we obtain, for all ,

where , is the inverse of (for ), is determined recursively by

and , are functions of such that lie on the boundary of the lattice

where denotes either if it converges or . Note that

Thus, from (3.17), (3.23), and (3.27), (3.26) can be equivalently written as

We further claim that the term is the same as , defined in (3.6), . For convenience, let . Obviously, it is that .

The remainder case is that for some . Let

where is an arbitrary small number. Obviously, for all . Using the same arguments as above and replacing with , we get

for all .

Considering continuities of and for as well as of in and letting , we obtain (2.4). This completes the proof.

We remark that , lie on the boundary of the lattice . In particular, (2.4) is true for all when every () satisfies . Therefore, we may take , .