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
,
.