Abstract and Applied AnalysisVolume 2011 (2011), Article ID 924319, 13 pagesdoi:10.1155/2011/924319

Research Article

Refinements of the Lower Bounds of the Jensen Functional

Iva Franjić,¹ Sadia Khalid,² and Josip Pečarić2,3

¹Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia²Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan³Faculty of Textile Technology, University of Zagreb, Prilaz Baruna Filipovića 28A, 10000 Zagreb, Croatia

Received 1 July 2011; Accepted 4 August 2011

Academic Editor: Wing-Sum Cheung

Copyright © 2011 Iva Franjić et al. 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.

Abstract.

The lower bounds of the functional defined as the difference of the right-hand and the left-hand side of the Jensen inequality are studied. Refinements of some previously known results are given by applying results from the theory of majorization. Furthermore, some interesting special cases are considered.

1. Introduction

The classical Jensen inequality states (see e.g., [1]).

Theorem 1.1 (see [2]). Let be an interval in , and let be a convex function. Let , , and let be a positive -tuple, that is, such that for , then where . If is strictly convex, then inequality (1.1) is strict unless .

In this work, the functional defined as the difference of the right-hand and the left-hand sides of the Jensen inequality is studied. More precisely, its lower bounds are investigated, together with various sets of assumptions under which they hold.

The lower bounds of were the topic of interest in many papers. For example, the following results were proved in [3] (see also [1, page 717]). In what follows, is an interval in .

Theorem 1.2. Let be a convex function, , and let be a positive -tuple, then

Theorem 1.3. Let be a convex function and . Let and be positive -tuples such that , that is, , then where and .

Further, in [4], the following theorem was given. An alternative proof of the same result was given in [5].

Theorem 1.4. Let be a convex function, , and . Let and be positive -tuples such that , then

For more related results, see [6–8]. The motivation for the research in this work were the following results presented in [9].

Lemma 1.5. Let be a convex function on , a positive -tuple such that and such that . For fixed , where , the Jensen functional defined in (1.2) is minimal when , that is, where

Lemma 1.6. Let be a convex function on , a positive -tuple such that and such that . For fixed , where , the Jensen functional defined in (1.2) is minimal when , that is, where

Theorem 1.7. Let be a convex function on , a positive -tuple such that and such that . For fixed and , where , the Jensen functional defined in (1.2) is minimal when that is, where are as in (1.7) and are as in (1.9).

The key step in proving these results was the following lemma presented in the same paper.

Lemma 1.8. Let be a convex function on , and let be nonnegative real numbers. If are such that and then

Note that for a monotonic -tuple , Theorem 1.7 is an improvement of Theorem 1.2, in a sense that (the maximum of) the right-hand side of (1.11) is greater than the middle part of (1.3), which follows directly from the Jensen inequality. The aim of this work is to give an improvement of Lemmas 1.5 and 1.6, and Theorem 1.7, in a sense that the condition of monotonicity imposed on the -tuple will be relaxed. Several sets of conditions under which (1.6), (1.8), and (1.11) hold shall be given. In our proofs, in addition to Lemma 1.8, the following result from the theory of majorization is needed. It was obtained in [10].

Lemma 1.9. Let be a convex function on , a positive -tuple, and such that If is a decreasing -tuple, then one has while if is an increasing -tuple, then we have If is strictly convex and , then (1.15) and (1.16) are strict.

Note that for , inequality (1.15) holds if and if (1.12) is valid, while inequality (1.16) holds if and if (1.12) is valid.

2. Main Results

In what follows, is as in (1.2), are as in (1.7), and , as in (1.9). Without any loss of generality, we assume that , since for positive -tuples such that results follow easily by substituting with . Furthermore, for , we introduce the following notation: Note that .

Theorem 2.1. Let be a convex function on and a positive -tuple such that . Let and , . If is such that then one has

Proof. The claim is that As a simple consequence of the Jensen inequality (1.1), we have Therefore, if we prove the claim will follow. The idea is to apply Lemma 1.8 for , , , , , and . Condition (1.12) is obviously satisfied. In addition, we need to check that Easy calculation shows that both of these conditions are valid if (2.2) holds. Thus, the claim follows from Lemma 1.8. Note that we could have taken , , , , , and , instead. In this case, the necessary conditions would follow from (2.3).

Theorem 2.2. Let the conditions of Theorem 2.1 hold. If is such that then inequality (2.4) holds.

Proof. Proof is analogous to the proof of Theorem 2.1. Instead of Lemma 1.8, we apply Lemma 1.9 for and the same choice of weights and points, or their obvious rearrangement.

Theorem 2.3. Let be a convex function on and a positive -tuple such that . Let and , . If is such that then one has

Proof. Similarly as in the proof of Theorem 2.1, after first applying the Jensen inequality to the sum on the left-hand side, the claim will follow if we prove We can apply Lemma 1.8 for , , , and , since condition (1.12) is obviously satisfied and (2.11) ensures that the rest of the necessary conditions are fulfilled, and thus the claim is proved. After the obvious rearrangement, applying Lemma 1.8 with (2.12), the claim is recaptured.

Theorem 2.4. Let the conditions of Theorem 2.3 hold. If is such that then inequality (2.13) holds.

Proof. It is analogous to the proof of Theorem 2.3. Instead of Lemma 1.8, we apply Lemma 1.9 for and the same choice of weights and points, or their obvious rearrangement.

Corollary 2.5. Let be a convex function on and a positive -tuple such that , . Let be a real -tuple and . If is such that and is such that either (2.2) or (2.3) holds, then one has If is such that and is such that either (2.11) or (2.12) holds, then one has

Proof. The first inequality in (2.19) follows from Theorem 2.3 for , and the second is a direct consequence of Theorem 2.1, while the first inequality in (2.22) follows from Theorem 2.1 for , and the second is a consequence of Theorem 2.3.

Corollary 2.6. Let the conditions of Corollary 2.5 hold. If is such that and is such that either (2.9) or (2.10) holds, then inequality (2.19) holds. If is such that and is such that either (2.15) or (2.16) holds, then inequality (2.22) holds.

Proof. The first inequality in (2.19) follows from Theorem 2.3 for , and the second is a direct consequence of Theorem 2.1, while the first inequality in (2.22) follows from Theorem 2.1 for , and the second is a consequence of Theorem 2.3.

Theorem 2.7. Let be a convex function on and a positive -tuple such that , . Let be a real -tuple, and let . If and are such that or then one has

Proof. The claim is that After applying the Jensen inequality to the two sums on the left-hand side, we need to prove Set , , , ,, , and . Assumption (2.27) ensures that the necessary conditions of Lemma 1.9 for are fulfilled, and so (2.31) follows from (1.15). By obvious rearrangement, utilizing (2.28), the inequality is recaptured.

Remark 2.8. Note that conditions (2.9) and (2.23) combined together give a condition while (2.15) and (2.25) combined together give both of which are more restricting than (2.27). The same is true for combining conditions (2.10) and (2.24), or (2.16) and (2.26), and comparing the result with (2.28).

Theorem 2.9. Let be a convex function on and a positive -tuple such that , . Let and , , then one has

Proof. The first inequality is an immediate consequence of the Jensen inequality. The other two follow immediately from (1.4).

Remark 2.10. Inequalities (2.19), (2.22), and (2.34) recapture results from Lemmas 1.5 and 1.6, and Theorem 1.7 as special cases, since an increasing -tuple fulfils conditions (2.2) and (2.17), that is, (2.11) and (2.20). A decreasing -tuple , on the other hand, fulfills conditions (2.3) and (2.18), that is, (2.12) and (2.21). The proofs of Theorem 2.9 and Corollary 2.5, that is, Theorems 2.1 and 2.3, are in fact analogous to the proofs of Theorem 1.7, Lemmas 1.5 and 1.6 from [9].

3. Some Special Cases

In this section, we consider some special cases of the presented results. The same special cases were considered in [9], but here we obtain them under more relaxed conditions on the -tuple . More precisely, Corollaries 2.5 and 2.6, or Theorem 2.7, after applying Theorem 2.9, yield

Corollary 3.1. Let the conditions of Corollaries 2.5 and 2.6, or Theorem 2.7 hold, then

Proof. This follows from (3.1) for , using notation .

Corollary 3.2. Let the conditions of Corollaries 2.5 and 2.6, or Theorem 2.7 hold, and let in addition , , then

Proof. Follows from (3.1) for .

Corollary 3.3. Let the conditions of Corollaries 2.5 and 2.6, or Theorem 2.7 hold, and let in addition , , then

Proof. This follows from (3.1) for .

In [9], additional bounds of , lower than those obtained in the previous corollaries, were derived for the case and . Now, note that from Theorem 2.9, under conditions of Corollaries 2.5 and 2.6, or Theorem 2.7, we have Next, we compare estimates obtained from (3.5) with those obtained in [9].

Case 1. For , using notation , inequality (3.5) takes the form In [9], under the assumption that is an increasing -tuple, the following inequality was obtained where Note that when , (3.6) recaptures this result. However, when , the constant is better, since .

Case 2. For and , inequality (3.5) takes the form In [9], under the assumption that is an increasing -tuple such that , the following inequality was obtained: where In order to compare these two estimates, first assume that . Since it follows that the estimate in (3.9) is better than the one in (3.10).Next, assume that . First, observe that Simple calculation reveals that and so we conclude that the estimate in (3.9) is better than the one in (3.10) when , while when , the estimate in (3.10) is better than the one in (3.9).Further, assume that . In this case, the estimate in (3.10) is better than the one in (3.9), that is, Namely, Finally, if , the estimate in (3.10) is again better than the one in (3.9), that is, This is equivalent to In this case, we have and since the claim follows.

Acknowledgments

This research work was partially funded by the Higher Education Commission, Pakistan. The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants nos. 058-1170889-1050 (first author) and 117-1170889-0888 (third author).

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