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  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 ,
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 , .