Open Access

Iterative Refinements of the Hermite-Hadamard Inequality, Applications to the Standard Means

Journal of Inequalities and Applications20102010:107950

DOI: 10.1155/2010/107950

Received: 29 July 2010

Accepted: 19 October 2010

Published: 24 October 2010

Abstract

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.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq1_HTML.gif be a nonempty convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq2_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq3_HTML.gif be a convex function. For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq4_HTML.gif , the following double inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ1_HTML.gif
(1.1)

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), [14]. Dragomir [1] gave a refinement of the left side of (1.1) as summarized in the next result.

Theorem 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq5_HTML.gif be a convex function and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq6_HTML.gif be defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ2_HTML.gif
(1.2)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq7_HTML.gif is convex increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq8_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq9_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ3_HTML.gif
(1.3)

Yang and Hong [3] gave a refinement of the right side of (1.1) as itemized below.

Theorem 1.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq10_HTML.gif be a convex function and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq11_HTML.gif be defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ4_HTML.gif
(1.4)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq12_HTML.gif is convex increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq13_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq14_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ5_HTML.gif
(1.5)

From the above theorems we immediately deduce the following.

Corollary 1.3.

With the above, there holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ6_HTML.gif
(1.6)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq15_HTML.gif , with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ7_HTML.gif
(1.7)

The following refinement of (1.1) is also well-known.

Theorem 1.4.

With the above, the following double inequality holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ8_HTML.gif
(1.8)

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.

Proof.

Applying (1.1) successively in the subintervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq17_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ9_HTML.gif
(1.9)

The desired result (1.8) follows by adding the above obtained inequalities (1.9).

In [4] Zabandan introduced an improvement of Theorem 1.4 as recited in the following. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq19_HTML.gif be the sequences defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ10_HTML.gif
(1.10)

Theorem 1.5.

With the above, one has the following inequalities:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ11_HTML.gif
(1.11)
with the relationship
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ12_HTML.gif
(1.12)

Notation 1.

Throughout this paper, and for the sake of presentation, the above expressions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq21_HTML.gif will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq23_HTML.gif , and the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq24_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq25_HTML.gif , respectively. Further, the middle member of inequality (1.1), usually known by the mean value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq26_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq27_HTML.gif , will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq28_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ13_HTML.gif
(1.13)

2. Iterative Refinements of the Hermite-Hadamard Inequality

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq29_HTML.gif be a nonempty convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq30_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq31_HTML.gif 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

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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ14_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq32_HTML.gif are two given functions. Assume that, by the same procedure as in the proof of Theorem 1.4 we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ15_HTML.gif
(2.2)
with the following relationships
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ16_HTML.gif
(2.3)
Reiterating successively the same, we then construct two sequences, denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq34_HTML.gif , satisfying the following inequalities:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ17_HTML.gif
(2.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq36_HTML.gif are defined by the recursive relationships
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ18_HTML.gif
(2.5)
The initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq38_HTML.gif , which of course depend generally of the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq39_HTML.gif , are for the moment upper and lower bounds of inequality (1.1), respectively, and satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ19_HTML.gif
(2.6)

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

Theorem 2.1.

With the above, the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq40_HTML.gif is increasing and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq41_HTML.gif is a decreasing one. Moreover, the following inequalities:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ20_HTML.gif
(2.7)

hold true for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq42_HTML.gif .

Proof.

Follows from the construction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq44_HTML.gif . It is also possible to prove the same by using the above recursive relationships defining https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq46_HTML.gif . The proof is complete.

Corollary 2.2.

The sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq48_HTML.gif both converge and their limits are, respectively, the lower and upper bounds of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq49_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ21_HTML.gif
(2.8)

Proof.

According to inequalities (2.7), the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq50_HTML.gif is increasing upper bounded by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq51_HTML.gif while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq52_HTML.gif is decreasing lower bounded by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq53_HTML.gif . It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq55_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq56_HTML.gif (and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq57_HTML.gif ) in terms of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq58_HTML.gif ? The answer to this is given in the following result.

Theorem 2.3.

With the above, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq59_HTML.gif , there hold
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ22_HTML.gif
(2.9)

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq61_HTML.gif ? 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq62_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq63_HTML.gif is to take the upper and lower bounds of (1.1), respectively, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ23_HTML.gif
(2.10)
With this choice, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ24_HTML.gif
(2.11)

which, respectively, correspond to the lower and upper bounds of (1.8). By convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq64_HTML.gif , it is easy to see that the inequalities (2.6) are satisfied. In this case we will prove in the next subsection that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq66_HTML.gif coincide with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq68_HTML.gif , respectively, and so both converge to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq69_HTML.gif .

Example 2.5.

Following Corollary 1.3 we can take
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ25_HTML.gif
(2.12)

for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq71_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq72_HTML.gif . For this, such sequences of functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq74_HTML.gif . This example, which contains the above one, will be detailed in the following.

2.2. Case of Example 2.4

Choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq75_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq76_HTML.gif as in Example 2.4, we first state the following result.

Proposition 2.6.

With (2.10), one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ26_HTML.gif
(2.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq78_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq80_HTML.gif both converge to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq81_HTML.gif by adopting our technical approach. In fact, with (2.10) the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq83_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq84_HTML.gif , one has (2.10). Then the following relation holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ27_HTML.gif
(2.14)

Proof.

It is a simple induction on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq85_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq86_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq87_HTML.gif are chosen as in Example 2.4.

Theorem 2.8.

With (2.10), the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq89_HTML.gif are adjacent with the limit
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ28_HTML.gif
(2.15)
and the following error-estimations hold
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ29_HTML.gif
(2.16)

Proof.

According to Corollary 2.2, the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq91_HTML.gif both converge and by the relation (2.14) their limits are equal. Now, by virtue of (2.14) again we can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ30_HTML.gif
(2.17)
This, with the inequalities (2.7), yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ31_HTML.gif
(2.18)

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq92_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq94_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq95_HTML.gif is with geometric-speed.

2.3. Case of Example 2.5

As pointed out before, we can take
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ32_HTML.gif
(2.19)
for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq96_HTML.gif . The function sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq98_HTML.gif are defined, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq99_HTML.gif , by the recursive relationships
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ33_HTML.gif
(2.20)

By induction, it is not hard to see that the maps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq101_HTML.gif , for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq102_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq104_HTML.gif , for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq105_HTML.gif , are, respectively, monotone increasing and decreasing.

(2)For fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq106_HTML.gif , the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq108_HTML.gif are (convex and) monotonic increasing.

(3)For all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq110_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ34_HTML.gif
(2.21)
Proof.
  1. (1)

    By construction, as in the proof of Theorem 2.1.

     
  2. (2)

    Comes from the recursive relationships defining https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq111_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq112_HTML.gif .

     
  3. (3)

    By construction as in the above.

     
By virtue of the monotonicity of the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq113_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq114_HTML.gif in a part, and that of the maps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq115_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq116_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ35_HTML.gif
(2.22)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq117_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ36_HTML.gif
(2.23)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq118_HTML.gif .

Open Question 2.3.

As we have seen, for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq119_HTML.gif , the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq120_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq121_HTML.gif both converge. What are their limits? To know if such convergence is uniform on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq122_HTML.gif 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.

For two nonnegative real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq123_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq124_HTML.gif , the arithmetic, geometric, harmonic, logarithmic, exponential (or identric) means of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq126_HTML.gif are, respectively, defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ37_HTML.gif
(3.1)
with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq127_HTML.gif . The following inequalities are well known in the literature
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ38_HTML.gif
(3.2)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq129_HTML.gif are given, the computations of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq132_HTML.gif are simple while that of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq133_HTML.gif and specially that of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq134_HTML.gif are not. So, approaching https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq136_HTML.gif by simple and practical algorithms appears to be interesting. That is the fundamental aim of what follows. In the following applications, we consider the choice (of Example 2.4),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ39_HTML.gif
(3.3)

3.1. Application 1: Approximation of the Logarithmic Mean

Consider the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq137_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq138_HTML.gif . Preserving the same notations as in the previous section, the associate sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq139_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq140_HTML.gif correspond to the initial data
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ40_HTML.gif
(3.4)

Applying the above theoretical result to this particular case we immediately obtain the following result.

Theorem 3.1.

The sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq142_HTML.gif , corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq143_HTML.gif , both converge to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq144_HTML.gif with the next estimation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ41_HTML.gif
(3.5)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq145_HTML.gif , and the following inequalities hold
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ42_HTML.gif
(3.6)

The above theorem tells us that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq146_HTML.gif 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

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq147_HTML.gif be the convex map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq148_HTML.gif . Writing explicitly the corresponding iterative process https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq149_HTML.gif we see that, for reason of simplicity, we may set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ43_HTML.gif
(3.7)
The auxiliary sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq150_HTML.gif is so recursively defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ44_HTML.gif
(3.8)
As for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq151_HTML.gif , it is easy to establish by a simple induction that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ45_HTML.gif
(3.9)

where the dual sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq152_HTML.gif is defined by a similar relationship as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq153_HTML.gif with the initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq154_HTML.gif . Our above approach allows us to announce the following interesting result.

Theorem 3.2.

The above sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq155_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq156_HTML.gif with the estimation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ46_HTML.gif
(3.10)
and the iterative inequalities hold
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ47_HTML.gif
(3.11)
Furthermore, one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ48_HTML.gif
(3.12)

Proof.

It is immediate from the above general study. The details are left to the reader.

Combining the inequalities of Theorems 3.1 and 3.2, with the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq157_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq158_HTML.gif , we simultaneously obtain the known inequalities (3.2). Further, the next result of convergence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ49_HTML.gif
(3.13)

when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq159_HTML.gif goes to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq160_HTML.gif , is not obvious to establish directly. This proves again the interest of this work and the generality of our approach.

Remark 3.3.

The identric mean https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq161_HTML.gif 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 [5], 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

As well known, the Hermite-Hadamard inequality has an extension for real-valued convex functions with variables in a linear vector space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq162_HTML.gif in the following sense: let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq163_HTML.gif be a nonempty convex of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq164_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq165_HTML.gif be a convex function, then for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq166_HTML.gif there holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ50_HTML.gif
(4.1)
In particular, in every linear normed space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq167_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ51_HTML.gif
(4.2)

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq168_HTML.gif . We start with the analogue of Theorem 1.4.

Theorem 4.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq169_HTML.gif be a convex function. Then, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq171_HTML.gif , there holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ52_HTML.gif
(4.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq173_HTML.gif are given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ53_HTML.gif
(4.4)

Proof.

On making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq174_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ54_HTML.gif
(4.5)
while for the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq175_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ55_HTML.gif
(4.6)
Now, applying the inequality (4.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ56_HTML.gif
(4.7)

If we divide both inequalities with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq176_HTML.gif and add the obtained results we deduce the desired double inequality (4.3).

Similarly, we set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ57_HTML.gif
(4.8)

Now, the extension of our above study is itemized in the following statement.

Theorem 4.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq177_HTML.gif be a nonempty convex subset of a linear space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq179_HTML.gif a convex function. For all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq180_HTML.gif , the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq181_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq182_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ58_HTML.gif
(4.9)
are, respectively, monotonic increasing and decreasing and both converge to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq183_HTML.gif with the following estimation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ59_HTML.gif
(4.10)

Proof.

Similar to that of real variables. We omit the details here.

Of course, the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq184_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq185_HTML.gif are relied by similar relation as (2.14) and explicitly given by analogue expressions of (2.9). In particular, we may state the following.

Example 4.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq186_HTML.gif be a real number and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq187_HTML.gif be the convex function defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq188_HTML.gif . In this case, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq189_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq190_HTML.gif are given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ60_HTML.gif
(4.11)
with the following inequalities:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ61_HTML.gif
(4.12)

Remark 4.4.

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 [6].

Authors’ Affiliations

(1)
Research Group in Mathematical Inequalities and Applications, School of Engineering and Science, Victoria University
(2)
School of Computational and Applied Mathematics, University of the Witwatersrand
(3)
Applied Functional Analysis Team, AFACSI Laboratory, Faculty of Science, Moulay Ismaïl University

References

  1. Dragomir SS: Two mappings in connection to Hadamard's inequalities. Journal of Mathematical Analysis and Applications 1992, 167(1):49–56. 10.1016/0022-247X(92)90233-4MathSciNetView ArticleMATH
  2. Dragomir SS, McAndrew A: Refinements of the Hermite-Hadamard inequality for convex functions. Journal of Inequalities in Pure and Applied Mathematics 2005., 6(5, article no. 140):
  3. Yang G-S, Hong M-C: A note on Hadamard's inequality. Tamkang Journal of Mathematics 1997, 28(1):33–37.MathSciNetMATH
  4. Zabandan G: A new refinement of the Hermite-Hadamard inequality for convex functions. Journal of Inequalities in Pure and Applied Mathematics 2009., 10(2, article no. 45):
  5. Kubo F, Ando T: Means of positive linear operators. Mathematische Annalen 1980, 246(3):205–224. 10.1007/BF01371042MathSciNetView ArticleMATH
  6. Dragomir SS, Raïssouli M: Jensen and Hermite-Hadamard inequalities for the Legendre-Fenchel duality, application to convex operator maps. Mathematica Slovaca, 2010, Submitted Mathematica Slovaca, 2010, Submitted

Copyright

© Sever S. Dragomir and Mustapha Raïssouli. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.