Iterative Refinements of the Hermite-Hadamard Inequality, Applications to the Standard Means
© Sever S. Dragomir and Mustapha Raïssouli. 2010
Received: 29 July 2010
Accepted: 19 October 2010
Published: 24 October 2010
Two adjacent recursive processes converging to the mean value of a real-valued convex function are given. Refinements of the Hermite-Hadamard inequality are obtained. Some applications to the special means are discussed. A brief extension for convex mappings with variables in a linear space is also provided.
is known in the literature as the Hermite-Hadamard inequality for convex functions. Such inequality is very useful in many mathematical contexts and contributes as a tool for establishing some interesting estimations.
In recent few years, many authors have been interested to give some refinements and extensions of the Hermite-Hadamard inequality (1.1), [1–4]. Dragomir  gave a refinement of the left side of (1.1) as summarized in the next result.
Yang and Hong  gave a refinement of the right side of (1.1) as itemized below.
From the above theorems we immediately deduce the following.
The following refinement of (1.1) is also well-known.
For the sake of completeness and in order to explain the key idea of our approach to the reader we will reproduce here the proof of the above known theorem.
The desired result (1.8) follows by adding the above obtained inequalities (1.9).
2. Iterative Refinements of the Hermite-Hadamard Inequality
Let be a nonempty convex subset of and let be a convex function. As already pointed out, our fundamental goal in the present section is to give some iterative refinements of (1.1) containing those recalled in the above. We start with our general viewpoint.
2.1. General Approach
Summarizing the previous approach, we may state the following results.
hold true for all .
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.
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.
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.
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 .
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.
where and are given by (1.10).
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.
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.
By a simple mathematical induction, we simultaneously obtain (2.15) and (2.16). Thus completes the proof.
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  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
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.
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.
By construction, as in the proof of Theorem 2.1.
Comes from the recursive relationships defining and .
By construction as in the above.
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.
3. Applications to Scalar Means
As already pointed out, this section will be devoted to display some applications of the above theoretical results. For this, we need some additional basic notions about special means.
3.1. Application 1: Approximation of the Logarithmic Mean
Applying the above theoretical result to this particular case we immediately obtain the following result.
The above theorem tells us that containing logarithm can be approached by an iterative algorithm involving only the elementary operations sum, product and inverse. Further, such algorithm is simple, recursive and practical for the numerical context, with a geometric-speed.
3.2. Application 2: Approximation of the Identric Mean
where the dual sequence is defined by a similar relationship as with the initial data . Our above approach allows us to announce the following interesting result.
It is immediate from the above general study. The details are left to the reader.
when goes to , is not obvious to establish directly. This proves again the interest of this work and the generality of our approach.
The identric mean having a transcendent expression is here approached by an algorithm, of algebraic type, utile for the theoretical study and simple for the numerical computation. Further as well-known, to define a non monotone operator mean, via Kubo-Ando theory , from the scalar case is not possible. Thus, our approach here could be the key idea for defining the identric mean involving operator and functional variables.
4. Extension for Real-Valued Function with Vector Variable
In general, the computation of the middle side integrals of the above inequalities is not always possible. So, approaching such integrals by recursive and practical algorithms appears to be very interesting. Our aim in this section is to state briefly an analogue of our above approach, with its related fundamental results, for convex functions . We start with the analogue of Theorem 1.4.
If we divide both inequalities with and add the obtained results we deduce the desired double inequality (4.3).
Now, the extension of our above study is itemized in the following statement.
Similar to that of real variables. We omit the details here.
Of course, the sequences and are relied by similar relation as (2.14) and explicitly given by analogue expressions of (2.9). In particular, we may state the following.
The Hermite-Hadamard inequality, together with some associate refinements, can be extended for nonreal-valued maps that are convex with respect to a given (partial) ordering. In this direction, we indicate the recent paper .
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