### 2.1. General Approach

Examining the proof of Theorem 1.4 we observe that the same procedure can be again recursively applied. More precisely, let us start with the next double inequality

where

are two given functions. Assume that, by the same procedure as in the proof of Theorem 1.4 we have

with the following relationships

Reiterating successively the same, we then construct two sequences, denoted by

and

, satisfying the following inequalities:

where

and

are defined by the recursive relationships

The initial data

and

, which of course depend generally of the convex function

, are for the moment upper and lower bounds of inequality (1.1), respectively, and satisfying

Summarizing the previous approach, we may state the following results.

Theorem 2.1.

With the above, the sequence

is increasing and

is a decreasing one. Moreover, the following inequalities:

hold true for all
.

Proof.

Follows from the construction of
and
. It is also possible to prove the same by using the above recursive relationships defining
and
. The proof is complete.

Corollary 2.2.

The sequences

and

both converge and their limits are, respectively, the lower and upper bounds of

, that is,

Proof.

According to inequalities (2.7), the sequence
is increasing upper bounded by
while
is decreasing lower bounded by
. It follows that
and
both converge. Passing to the limits in inequalities (2.7) we obtain (2.8), which completes the proof.

Now, we will observe a question arising naturally from the above study: what is the explicit form of
(and
) in terms of
? The answer to this is given in the following result.

Theorem 2.3.

With the above, for all

, there hold

Proof.

Of course, it is sufficient to show the first formulae which follows from a simple induction with a manipulation on the summation indices. We omit the routine details.

After this, we can put the following question: what are the explicit limits of the sequences
and
? Before giving an answer to this question in a special case, we may state the following examples.

Example 2.4.

Of course, the first choice of

and

is to take the upper and lower bounds of (1.1), respectively, that is,

With this choice, we have

which, respectively, correspond to the lower and upper bounds of (1.8). By convexity of
, it is easy to see that the inequalities (2.6) are satisfied. In this case we will prove in the next subsection that
and
coincide with
and
, respectively, and so both converge to
.

Example 2.5.

Following Corollary 1.3 we can take

for fixed
,
. It is not hard to verify that the inequalities (2.6) are here satisfied. In this case, our above approach defines us two sequences which depend on the variable
. For this, such sequences of functions will be denoted by
and
. This example, which contains the above one, will be detailed in the following.

### 2.2. Case of Example 2.4

Choosing
and
as in Example 2.4, we first state the following result.

Proposition 2.6.

where
and
are given by (1.10).

Proof.

It is a simple verification from formulas (2.9) with (1.10).

Now, we will reproduce to prove that the sequences
and
both converge to
by adopting our technical approach. In fact, with (2.10) the sequences
and
can be relied by a unique interesting relationship which, as we will see later, will simplify the corresponding proofs. Precisely, we may state the following result.

Proposition 2.7.

Assume that, for

, one has (2.10). Then the following relation holds:

Proof.

It is a simple induction on
and we omit the details for the reader.

Now we are in position to state the following result which gives an answer to the above question when
and
are chosen as in Example 2.4.

Theorem 2.8.

With (2.10), the sequences

and

are adjacent with the limit

and the following error-estimations hold

Proof.

According to Corollary 2.2, the sequences

and

both converge and by the relation (2.14) their limits are equal. Now, by virtue of (2.14) again we can write

This, with the inequalities (2.7), yields

By a simple mathematical induction, we simultaneously obtain (2.15) and (2.16). Thus completes the proof.

Remark 2.9.

Starting from a general point of view, we have found again Theorem 1.5 under a new angle and via a technical approach. Furthermore, such approach stems its importance in what follows.

(i)As the reader can remark it, the proofs are here more simple as that of [4] for proving the monotonicity and computing the limit of the considered sequences. See [4, pages 3–5] for such comparison.

(ii)The sequences having
as limit are here defined by simple and recursive relationships which play interesting role in the theoretical study as in the computation context.

(iii)Some estimations improving those already stated in the literature are obtained here. In particular, inequalities (2.16) appear to be new for telling us that, in the numerical context, the convergence of
and
to
is with geometric-speed.

### 2.3. Case of Example 2.5

As pointed out before, we can take

for fixed

. The function sequences

and

are defined, for all

, by the recursive relationships

By induction, it is not hard to see that the maps
and
, for fixed
, are convex and increasing.

Similarly to the above, we obtain the next result.

Theorem 2.10.

With (2.19), the following assertions are met.

(1)The function sequences
and
, for fixed
, are, respectively, monotone increasing and decreasing.

(2)For fixed
, the functions
and
are (convex and) monotonic increasing.

(3)For all

and

, one has

Proof.

- (1)
By construction, as in the proof of Theorem 2.1.

- (2)
Comes from the recursive relationships defining
and
.

- (3)
By construction as in the above.

By virtue of the monotonicity of the sequences

,

in a part, and that of the maps

,

in another part, the double iterative-functional inequality (2.21) yields some improvements of refinements recalled in the above section. In particular, we immediately find the inequalities (1.3) and (1.6), respectively, by writing

for all

, and

for all
.

Open Question 2.3.

As we have seen, for every
, the sequences
and
both converge. What are their limits? To know if such convergence is uniform on
is not obvious and appears also to be interesting.