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

  • SeverS Dragomir1, 2 and

    Affiliated with

    • Mustapha Raïssouli3Email author

      Affiliated with

      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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq1_HTML.gif be a nonempty convex subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq2_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq3_HTML.gif be a convex function. For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq4_HTML.gif , the following double inequality
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq5_HTML.gif be a convex function and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq6_HTML.gif be defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ2_HTML.gif
      (1.2)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq7_HTML.gif is convex increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq8_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq9_HTML.gif , one has
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq10_HTML.gif be a convex function and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq11_HTML.gif be defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ4_HTML.gif
      (1.4)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq12_HTML.gif is convex increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq13_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq14_HTML.gif , one has
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ6_HTML.gif
      (1.6)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq15_HTML.gif , with
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq17_HTML.gif we obtain
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq19_HTML.gif be the sequences defined by
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ11_HTML.gif
      (1.11)
      with the relationship
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq21_HTML.gif will be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq22_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq23_HTML.gif , and the sequences http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq24_HTML.gif by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq26_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq27_HTML.gif , will be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq28_HTML.gif , that is,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq29_HTML.gif be a nonempty convex subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq30_HTML.gif and let http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ14_HTML.gif
      (2.1)
      where http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ15_HTML.gif
      (2.2)
      with the following relationships
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq34_HTML.gif , satisfying the following inequalities:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ17_HTML.gif
      (2.4)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq36_HTML.gif are defined by the recursive relationships
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ18_HTML.gif
      (2.5)
      The initial data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq37_HTML.gif and http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq40_HTML.gif is increasing and http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ20_HTML.gif
      (2.7)

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

      Proof.

      Follows from the construction of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq43_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq45_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq47_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq49_HTML.gif , that is,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq50_HTML.gif is increasing upper bounded by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq51_HTML.gif while http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq52_HTML.gif is decreasing lower bounded by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq53_HTML.gif . It follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq54_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq56_HTML.gif (and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq57_HTML.gif ) in terms of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq59_HTML.gif , there hold
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq60_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq62_HTML.gif and http://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,
      http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq66_HTML.gif coincide with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq67_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq68_HTML.gif , respectively, and so both converge to http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ25_HTML.gif
      (2.12)

      for fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq70_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq73_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq75_HTML.gif and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ26_HTML.gif
      (2.13)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq77_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq80_HTML.gif both converge to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq82_HTML.gif and http://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 http://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:
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq86_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq88_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq89_HTML.gif are adjacent with the limit
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq90_HTML.gif and http://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
      http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq94_HTML.gif to http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ32_HTML.gif
      (2.19)
      for fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq96_HTML.gif . The function sequences http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq98_HTML.gif are defined, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq99_HTML.gif , by the recursive relationships
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq101_HTML.gif , for fixed http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq104_HTML.gif , for fixed http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq106_HTML.gif , the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq107_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq109_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq110_HTML.gif , one has
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq111_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq113_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq115_HTML.gif , http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ35_HTML.gif
      (2.22)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq117_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ36_HTML.gif
      (2.23)

      for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq119_HTML.gif , the sequences http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq120_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq123_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq126_HTML.gif are, respectively, defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ37_HTML.gif
      (3.1)
      with http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ38_HTML.gif
      (3.2)
      When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq128_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq129_HTML.gif are given, the computations of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq132_HTML.gif are simple while that of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq133_HTML.gif and specially that of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq134_HTML.gif are not. So, approaching http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq135_HTML.gif and http://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),
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq137_HTML.gif defined by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq140_HTML.gif correspond to the initial data
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq142_HTML.gif , corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq143_HTML.gif , both converge to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq144_HTML.gif with the next estimation
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ41_HTML.gif
      (3.5)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq145_HTML.gif , and the following inequalities hold
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq147_HTML.gif be the convex map http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq148_HTML.gif . Writing explicitly the corresponding iterative process http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ43_HTML.gif
      (3.7)
      The auxiliary sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq150_HTML.gif is so recursively defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ44_HTML.gif
      (3.8)
      As for http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ45_HTML.gif
      (3.9)

      where the dual sequence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq153_HTML.gif with the initial data http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq155_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq156_HTML.gif with the estimation
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ47_HTML.gif
      (3.11)
      Furthermore, one has
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq157_HTML.gif for all http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ49_HTML.gif
      (3.13)

      when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq159_HTML.gif goes to http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq162_HTML.gif in the following sense: let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq163_HTML.gif be a nonempty convex of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq164_HTML.gif and let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq166_HTML.gif there holds
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq167_HTML.gif , we have
      http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq171_HTML.gif , there holds
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ52_HTML.gif
      (4.3)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq172_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq173_HTML.gif are given by
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq174_HTML.gif , we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq175_HTML.gif we have
      http://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
      http://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 http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq178_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq179_HTML.gif a convex function. For all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq180_HTML.gif , the sequences http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq181_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq182_HTML.gif defined by
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq183_HTML.gif with the following estimation
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq184_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq186_HTML.gif be a real number and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq187_HTML.gif be the convex function defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq188_HTML.gif . In this case, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq189_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_IEq190_HTML.gif are given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F107950/MediaObjects/13660_2010_Article_2053_Equ60_HTML.gif
      (4.11)
      with the following inequalities:
      http://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

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      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.MathSciNet
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      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.