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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 924319, 13 pages
http://dx.doi.org/10.1155/2011/924319
Research Article

Refinements of the Lower Bounds of the Jensen Functional

1Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
2Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan
3Faculty 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 𝐼 be an interval in , and let 𝑓𝐼 be a convex function. Let 𝑛2, 𝐱=(𝑥1,,𝑥𝑛)𝐼𝑛, and let 𝐩=(𝑝1,,𝑝𝑛) be a positive 𝑛-tuple, that is, such that 𝑝𝑖>0 for 𝑖=1,,𝑛, then 𝑓1𝑃𝑛𝑛𝑖=1𝑝𝑖𝑥𝑖1𝑃𝑛𝑛𝑖=1𝑝𝑖𝑓𝑥𝑖,(1.1) where 𝑃𝑛=𝑛𝑖=1𝑝𝑖. If 𝑓 is strictly convex, then inequality (1.1) is strict unless 𝑥1==𝑥𝑛.

In this work, the functional1𝐽(𝐱,𝐩,𝑓)=𝑃𝑛𝑛𝑖=1𝑝𝑖𝑓𝑥𝑖1𝑓𝑃𝑛𝑛𝑖=1𝑝𝑖𝑥𝑖(1.2) 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 𝐽(𝐱,𝐩,𝑓) 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, 𝐼 is an interval in .

Theorem 1.2. Let 𝑓𝐼 be a convex function, 𝐱𝐼𝑛, and let 𝐩 be a positive 𝑛-tuple, then 𝑃𝑛𝐽(𝐱,𝐩,𝑓)max1𝑗𝑘𝑛𝑝𝑗𝑓𝑥𝑗+𝑝𝑘𝑓𝑥𝑘𝑝𝑗+𝑝𝑘𝑓𝑝𝑗𝑥𝑗+𝑝𝑘𝑥𝑘𝑝𝑗+𝑝𝑘0.(1.3)

Theorem 1.3. Let 𝑓𝐼 be a convex function and 𝐱𝐼𝑛. Let 𝐩 and 𝐫 be positive 𝑛-tuples such that 𝐩𝐫, that is, 𝑝𝑖𝑟𝑖,𝑖=1,,𝑛, then 𝑃𝑛𝐽(𝐱,𝐩,𝑓)𝑅𝑛𝐽(𝐱,𝐫,𝑓)0,(1.4) where 𝑃𝑛=𝑛𝑖=1𝑝𝑖 and 𝑅𝑛=𝑛𝑖=1𝑟𝑖.

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

Theorem 1.4. Let 𝑓𝐼 be a convex function, 𝑛2, and 𝐱𝐼𝑛. Let 𝐩 and 𝐪 be positive 𝑛-tuples such that 𝑛𝑖=1𝑝𝑖=𝑛𝑖=1𝑞𝑖=1, then max1𝑗𝑛𝑝𝑗𝑞𝑗𝐽(𝐱,𝐪,𝑓)𝐽(𝐱,𝐩,𝑓)min1𝑗n𝑝𝑗𝑞𝑗𝐽(𝐱,𝐪,𝑓)0.(1.5)

For more related results, see [68]. The motivation for the research in this work were the following results presented in [9].

Lemma 1.5. Let 𝑓 be a convex function on 𝐼, 𝐩 a positive 𝑛-tuple such that 𝑃𝑛=𝑛𝑖=1𝑝𝑖=1 and 𝑥1,𝑥2,,𝑥𝑛𝐼,𝑛3 such that 𝑥1𝑥2𝑥𝑛. For fixed 𝑥𝑗,𝑥𝑗+1,,𝑥𝑛, where 𝑗=2,3,,𝑛1, the Jensen functional 𝐽(𝐱,𝐩,𝑓) defined in (1.2) is minimal when 𝑥1=𝑥2==𝑥𝑗1=𝑥𝑗, that is, 𝐽(𝐱,𝐩,𝑓)𝑃𝑗𝑓𝑥𝑗+𝑛𝑖=𝑗+1𝑝𝑖𝑓𝑥𝑖𝑃𝑓𝑗𝑥𝑗+𝑛𝑖=𝑗+1𝑝𝑖𝑥𝑖,(1.6) where 𝑃𝑗=𝑗𝑖=1𝑝𝑖,𝑗=1,,𝑛.(1.7)

Lemma 1.6. Let 𝑓 be a convex function on 𝐼, 𝐩 a positive 𝑛-tuple such that 𝑃𝑛=𝑛𝑖=1𝑝𝑖=1 and 𝑥1,𝑥2,,𝑥𝑛𝐼,𝑛3 such that 𝑥1𝑥2𝑥𝑛. For fixed 𝑥1,𝑥2,,𝑥𝑘, where 𝑘=2,3,,𝑛1, the Jensen functional 𝐽(𝐱,𝐩,𝑓) defined in (1.2) is minimal when 𝑥𝑘=𝑥𝑘+1==𝑥𝑛1=𝑥𝑛, that is, 𝐽(𝐱,𝐩,𝑓)𝑘1𝑖=1𝑝𝑖𝑓𝑥𝑖+𝑄𝑘𝑓𝑥𝑘𝑓𝑘1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘,(1.8) where 𝑄𝑘=𝑛𝑖=𝑘𝑝𝑖,𝑘=1,,𝑛.(1.9)

Theorem 1.7. Let 𝑓 be a convex function on 𝐼, 𝐩 a positive 𝑛-tuple such that 𝑃𝑛=𝑛𝑖=1𝑝𝑖=1 and 𝑥1,𝑥2,,𝑥𝑛𝐼,𝑛3 such that 𝑥1𝑥2𝑥𝑛. For fixed 𝑥𝑗 and 𝑥𝑘, where 1𝑗<𝑘𝑛, the Jensen functional 𝐽(𝐱,𝐩,𝑓) defined in (1.2) is minimal when 𝑥1=𝑥2==𝑥𝑗,𝑥𝑘=𝑥𝑘+1==𝑥𝑛,𝑥𝑗+1=𝑥𝑗+2==𝑥𝑘1=𝑃𝑗𝑥j+𝑄𝑘𝑥𝑘𝑃𝑗+𝑄𝑘,(1.10) that is, 𝐽(𝐱,𝐩,𝑓)𝑃𝑗𝑓𝑥𝑗+𝑄𝑘𝑓𝑥𝑘𝑃𝑗+𝑄𝑘𝑓𝑃𝑗𝑥𝑗+𝑄𝑘𝑥𝑘𝑃𝑗+𝑄𝑘,(1.11) where 𝑃𝑗 are as in (1.7) and 𝑄𝑘 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 𝑓 be a convex function on 𝐼, and let 𝑝1,𝑝2 be nonnegative real numbers. If 𝑎1,𝑎2,𝑏1,𝑏2𝐼 are such that 𝑎1,𝑎2[𝑏1,𝑏2] and 𝑝1𝑎1+𝑝2𝑎2=𝑝1𝑏1+𝑝2𝑏2,(1.12) then 𝑝1𝑓𝑎1+𝑝2𝑓𝑎2𝑝1𝑓𝑏1+𝑝2𝑓𝑏2.(1.13)

Note that for a monotonic 𝑛-tuple 𝐱, 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 𝑛-tuple 𝐱 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 𝑓 be a convex function on 𝐼, 𝐩 a positive 𝑛-tuple, and 𝐚,𝐛𝐼𝑛 such that 𝑘𝑖=1𝑝𝑖𝑎𝑖𝑘𝑖=1𝑝𝑖𝑏𝑖for𝑘=1,2,,𝑛1,𝑛𝑖=1𝑝𝑖𝑎𝑖=𝑛𝑖=1𝑝𝑖𝑏𝑖.(1.14) If 𝐚 is a decreasing 𝑛-tuple, then one has 𝑛𝑖=1𝑝𝑖𝑓𝑎𝑖𝑛𝑖=1𝑝𝑖𝑓b𝑖,(1.15) while if 𝐛 is an increasing 𝑛-tuple, then we have 𝑛𝑖=1𝑝𝑖𝑓𝑏𝑖𝑛𝑖=1𝑝𝑖𝑓𝑎𝑖.(1.16) If 𝑓 is strictly convex and 𝐚𝐛, then (1.15) and (1.16) are strict.

Note that for 𝑛=2, inequality (1.15) holds if 𝑎2𝑎1𝑏1 and if (1.12) is valid, while inequality (1.16) holds if 𝑎1𝑏1𝑏2 and if (1.12) is valid.

2. Main Results

In what follows, 𝐽(𝐱,𝐩,𝑓) is as in (1.2), 𝑃𝑗 are as in (1.7), and 𝑄𝑘, as in (1.9). Without any loss of generality, we assume that 𝑃𝑛=1, since for positive 𝑛-tuples such that 𝑃𝑛1 results follow easily by substituting 𝑝𝑖 with 𝑝𝑖/𝑃𝑛. Furthermore, for 1𝑗<𝑘𝑛, we introduce the following notation: 𝐽min𝑃(𝐱,𝐩,𝑓)=min𝑗,𝑄𝑘𝑓𝑥𝑗𝑥+𝑓𝑘𝑥2𝑓𝑗+𝑥𝑘2,𝐽𝑗𝑘(𝐱,𝐩,𝑓)=𝑃𝑗𝑓𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑓𝑥𝑖+𝑄𝑘𝑓𝑥𝑘𝑃𝑓𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘.(2.1) Note that 𝐽1𝑛(𝐱,𝐩,𝑓)=𝐽(𝐱,𝐩,𝑓).

Theorem 2.1. Let 𝑓 be a convex function on 𝐼 and 𝐩 a positive 𝑛-tuple such that 𝑃𝑛=1,𝑛2. Let 1𝑗<𝑘𝑛 and 𝑥𝑖𝐼, 𝑖=1,,𝑘. If 𝑥𝑗 is such that 1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗1𝑄𝑗+1𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘,1(2.2)or𝑄𝑗+1𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘𝑥𝑗1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖,(2.3) then one has 𝐽1𝑘(𝐱,𝐩,𝑓)𝐽𝑗𝑘(𝐱,𝐩,𝑓).(2.4)

Proof. The claim is that 𝑗𝑖=1𝑝𝑖𝑓𝑥𝑖𝑓𝑘1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘𝑃𝑗𝑓𝑥𝑗𝑃𝑓𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘.(2.5) As a simple consequence of the Jensen inequality (1.1), we have 𝑗𝑖=1𝑝𝑖𝑓𝑥𝑖𝑃𝑗𝑓1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖.(2.6) Therefore, if we prove 𝑃𝑗𝑓1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑃+𝑓𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘𝑃𝑗𝑓𝑥𝑗+𝑓𝑘1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘,(2.7) the claim will follow. The idea is to apply Lemma 1.8 for 𝑝1=𝑃𝑗, 𝑝2=1, 𝑎1=𝑥𝑗, 𝑎2=𝑘1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘, 𝑏1=(1/𝑃𝑗)𝑗𝑖=1𝑝𝑖𝑥𝑖, and 𝑏2=𝑃𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘. Condition (1.12) is obviously satisfied. In addition, we need to check that 1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗𝑃𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘,1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑘1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘𝑃𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘.(2.8) 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 𝑝1=1, 𝑝2=𝑃𝑗, 𝑎1=𝑘1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘, 𝑎2=𝑥𝑗, 𝑏1=𝑃𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘, and 𝑏2=(1/𝑃𝑗)𝑗𝑖=1𝑝𝑖𝑥𝑖, instead. In this case, the necessary conditions would follow from (2.3).

Theorem 2.2. Let the conditions of Theorem 2.1 hold. If 𝑥𝑗 is such that 1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗k1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘,(2.9)or𝑘1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘𝑥𝑗1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖,(2.10) 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 𝑛=2 and the same choice of weights and points, or their obvious rearrangement.

Theorem 2.3. Let 𝑓 be a convex function on 𝐼 and 𝐩 a positive 𝑛-tuple such that 𝑃𝑛=1,𝑛2. Let 1𝑗<𝑘𝑛 and 𝑥𝑖𝐼, 𝑖=𝑗,,𝑛. If 𝑥𝑘 is such that 1𝑃𝑘1𝑃𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖𝑥𝑘1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖,1(2.11)or𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖𝑥𝑘1𝑃𝑘1𝑃𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖,(2.12) then one has 𝐽𝑗𝑛(𝐱,𝐩,𝑓)𝐽𝑗𝑘(𝐱,𝐩,𝑓).(2.13)

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 𝑄𝑘𝑓1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖𝑃+𝑓𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘𝑄𝑘𝑥𝑘𝑃+𝑓𝑗𝑥𝑗+𝑛𝑖=𝑗+1𝑝𝑖𝑥𝑖.(2.14) We can apply Lemma 1.8 for 𝑝1=1, 𝑝2=𝑄𝑘, 𝑎1=𝑃𝑗𝑥𝑗+𝑛𝑖=𝑗+1𝑝𝑖𝑥𝑖,𝑎2=𝑥𝑘,𝑏1=𝑃𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘, and 𝑏2=(1/𝑄𝑘)𝑛𝑖=𝑘𝑝𝑖𝑥𝑖, 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 𝑥𝑘 is such that 𝑃𝑗𝑥𝑗+𝑛𝑖=𝑗+1𝑝𝑖𝑥𝑖𝑥𝑘1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖,1(2.15)or𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖𝑥𝑘𝑃𝑗𝑥𝑗+𝑛𝑖=𝑗+1𝑝𝑖𝑥𝑖,(2.16) 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 𝑛=2 and the same choice of weights and points, or their obvious rearrangement.

Corollary 2.5. Let 𝑓 be a convex function on 𝐼 and 𝐩 a positive 𝑛-tuple such that 𝑃𝑛=1, 𝑛2. Let 𝐱𝐼𝑛 be a real 𝑛-tuple and 1𝑗<𝑘𝑛.
If 𝑥𝑘 is such that 1𝑃𝑘1𝑘1𝑖=1𝑝𝑖𝑥𝑖𝑥𝑘1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖,1(2.17)or𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖𝑥𝑘1𝑃𝑘1𝑘1𝑖=1𝑝𝑖𝑥𝑖,(2.18) and 𝑥𝑗 is such that either (2.2) or (2.3) holds, then one has 𝐽(𝐱,𝐩,𝑓)𝐽1𝑘(𝐱,𝐩,𝑓)𝐽𝑗k(𝐱,𝐩,𝑓).(2.19)
If 𝑥𝑗 is such that 1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗1𝑄𝑛𝑗+1𝑖=𝑗+1𝑝𝑖𝑥𝑖,1(2.20)or𝑄𝑛𝑗+1𝑖=𝑗+1𝑝𝑖𝑥𝑖𝑥𝑗1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖,(2.21) and 𝑥𝑘 is such that either (2.11) or (2.12) holds, then one has 𝐽(𝐱,𝐩,𝑓)𝐽𝑗𝑛(𝐱,𝐩,𝑓)𝐽𝑗𝑘(𝐱,𝐩,𝑓).(2.22)

Proof. The first inequality in (2.19) follows from Theorem 2.3 for 𝑗=1, and the second is a direct consequence of Theorem 2.1, while the first inequality in (2.22) follows from Theorem 2.1 for 𝑘=𝑛, and the second is a consequence of Theorem 2.3.

Corollary 2.6. Let the conditions of Corollary 2.5 hold.
If 𝑥𝑘 is such that 𝑛𝑖=1𝑝𝑖𝑥𝑖𝑥𝑘1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖,1(2.23)or𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖𝑥𝑘𝑛𝑖=1𝑝𝑖𝑥𝑖,(2.24) and 𝑥𝑗 is such that either (2.9) or (2.10) holds, then inequality (2.19) holds.
If 𝑥𝑗 is such that 1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗𝑛𝑖=1𝑝𝑖𝑥𝑖,(2.25)or𝑛𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖,(2.26) and 𝑥𝑘 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 𝑗=1, and the second is a direct consequence of Theorem 2.1, while the first inequality in (2.22) follows from Theorem 2.1 for 𝑘=𝑛, and the second is a consequence of Theorem 2.3.

Theorem 2.7. Let 𝑓 be a convex function on 𝐼 and 𝐩 a positive 𝑛-tuple such that 𝑃𝑛=1, 𝑛2. Let 𝐱𝐼𝑛 be a real 𝑛-tuple, and let 1𝑗<𝑘𝑛. If 𝑥𝑗 and 𝑥𝑘 are such that 1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗𝑛𝑖=1𝑝𝑖𝑥𝑖𝑥𝑘1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖,(2.27) or 1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖𝑥𝑘𝑛𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖,(2.28) then one has 𝐽(𝐱,𝐩,𝑓)𝐽𝑗𝑘(𝐱,𝐩,𝑓).(2.29)

Proof. The claim is that 𝑗𝑖=1𝑝𝑖𝑓𝑥𝑖+𝑛𝑖=𝑘𝑝𝑖𝑓𝑥𝑖𝑓𝑛𝑖=1𝑝𝑖𝑥𝑖𝑃𝑗𝑓𝑥𝑗+𝑄𝑘𝑓𝑥𝑘𝑃𝑓𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘.(2.30) After applying the Jensen inequality to the two sums on the left-hand side, we need to prove 𝑄𝑘𝑓1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖𝑃+𝑓𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘+𝑃𝑗𝑓1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑄𝑘𝑓𝑥𝑘+𝑓𝑛𝑖=1𝑝𝑖𝑥𝑖+𝑃𝑗𝑓𝑥𝑗.(2.31) Set 𝑝1=𝑄𝑘, 𝑝2=1, 𝑝3=𝑃𝑗, 𝑎1=𝑥𝑘,𝑎2=𝑛𝑖=1𝑝𝑖𝑥𝑖, 𝑎3=𝑥𝑗,𝑏1=(1/𝑄𝑘)𝑛𝑖=𝑘𝑝𝑖𝑥𝑖,𝑏2=𝑃𝑗𝑥𝑗+𝑘1𝑖=𝑗+1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘, and 𝑏3=(1/𝑃𝑗)𝑗𝑖=1𝑝𝑖𝑥𝑖. Assumption (2.27) ensures that the necessary conditions of Lemma 1.9 for 𝑛=3 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 1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗𝑘1𝑖=1𝑝𝑖𝑥𝑖+𝑄𝑘𝑥𝑘𝑛𝑖=1𝑝𝑖𝑥𝑖𝑥𝑘1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖,(2.32) while (2.15) and (2.25) combined together give 1𝑃𝑗𝑗𝑖=1𝑝𝑖𝑥𝑖𝑥𝑗𝑛𝑖=1𝑝𝑖𝑥𝑖𝑃𝑗𝑥𝑗+𝑛𝑖=𝑗+1𝑝𝑖𝑥𝑖𝑥𝑘1𝑄𝑘𝑛𝑖=𝑘𝑝𝑖𝑥𝑖,(2.33) 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 𝑓 be a convex function on 𝐼 and 𝐩 a positive 𝑛-tuple such that 𝑃𝑛=1, 𝑛2. Let 1𝑗<𝑘𝑛 and 𝑥𝑖𝐼, 𝑖=𝑗,,𝑘, then one has 𝐽𝑗𝑘(𝐱,𝐩,𝑓)𝑃𝑗𝑓𝑥𝑗+𝑄𝑘𝑓𝑥𝑘𝑃𝑗+𝑄𝑘𝑓𝑃𝑗𝑥𝑗+𝑄𝑘𝑥𝑘𝑃𝑗+𝑄𝑘𝐽min(𝐱,𝐩,𝑓)0.(2.34)

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 𝑛-tuple 𝐱 fulfils conditions (2.2) and (2.17), that is, (2.11) and (2.20). A decreasing 𝑛-tuple 𝐱, 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 𝑛-tuple 𝐱. More precisely, Corollaries 2.5 and 2.6, or Theorem 2.7, after applying Theorem 2.9, yield𝐽(𝐱,𝐩,𝑓)𝑃𝑗𝑓𝑥𝑗+𝑄𝑘𝑓𝑥𝑘𝑃𝑗+𝑄𝑘𝑓𝑃𝑗𝑥𝑗+𝑄𝑘𝑥𝑘𝑃𝑗+𝑄𝑘.(3.1)

Corollary 3.1. Let the conditions of Corollaries 2.5 and 2.6, or Theorem 2.7 hold, then 𝑛𝑖=1𝑝𝑖𝑎𝑖𝑛𝑖=1𝑎𝑝𝑖𝑖𝑃𝑗𝑎𝑗+𝑄𝑘𝑎𝑘𝑃𝑗+𝑄𝑘𝑎𝑃𝑗/(𝑃𝑗+𝑄𝑘)𝑗𝑎𝑄𝑘/(𝑃𝑗+𝑄𝑘)𝑘.(3.2)

Proof. This follows from (3.1) for 𝑓(𝑥)=𝑒𝑥, using notation 𝑎𝑖=𝑒𝑥𝑖.

Corollary 3.2. Let the conditions of Corollaries 2.5 and 2.6, or Theorem 2.7 hold, and let in addition 𝑥𝑖>0, 𝑖=1,,𝑛, then 𝑛𝑖=1𝑝𝑖𝑥𝑖𝑛𝑖=1𝑥𝑝𝑖𝑖1𝑥𝑃𝑗𝑗𝑥𝑄𝑘𝑘𝑃𝑗𝑥𝑗+𝑄𝑘𝑥𝑘𝑃𝑗+𝑄𝑘𝑃𝑗+𝑄𝑘.(3.3)

Proof. Follows from (3.1) for 𝑓(𝑥)=ln𝑥.

Corollary 3.3. Let the conditions of Corollaries 2.5 and 2.6, or Theorem 2.7 hold, and let in addition 𝑥𝑖>0, 𝑖=1,,𝑛, then 𝑛𝑖=1𝑝𝑖𝑥𝑖1𝑛𝑖=1𝑝𝑖𝑥𝑖𝑃𝑗𝑄𝑘𝑥𝑘𝑥𝑗2𝑥𝑗𝑥𝑘𝑃𝑗𝑥𝑗+𝑄𝑘𝑥𝑘.(3.4)

Proof. This follows from (3.1) for 𝑓(𝑥)=1/𝑥.

In [9], additional bounds of 𝐽(𝐱,𝐩,𝑓), lower than those obtained in the previous corollaries, were derived for the case 𝑓(𝑥)=𝑒𝑥 and 𝑓(𝑥)=1/𝑥. Now, note that from Theorem 2.9, under conditions of Corollaries 2.5 and 2.6, or Theorem 2.7, we have𝐽(𝐱,𝐩,𝑓)𝐽min(𝐱,𝐩,𝑓)0.(3.5) Next, we compare estimates obtained from (3.5) with those obtained in [9].

Case 1. For f(x)=ex, using notation ai=exi, inequality (3.5) takes the form 𝑛𝑖=1𝑝𝑖𝑎𝑖𝑛𝑖=1𝑎𝑝𝑖𝑖𝑃min𝑗,𝑄𝑘𝑎𝑘𝑎𝑗2.(3.6) In [9], under the assumption that 𝐚 is an increasing 𝑛-tuple, the following inequality was obtained 𝑛𝑖=1𝑝𝑖𝑎𝑖𝑛𝑖=1𝑎𝑝𝑖𝑖𝐶𝑎𝑘𝑎𝑗2,(3.7) where 𝐶=2𝑃𝑗𝑄𝑘𝑃𝑗+𝑄𝑘,𝑃𝑗𝑄𝑘,𝑄𝑘,𝑃𝑗𝑄𝑘.(3.8) Note that when 𝑃𝑗𝑄𝑘, (3.6) recaptures this result. However, when 𝑃𝑗𝑄𝑘, the constant 𝐶 is better, since 2𝑃𝑗𝑄𝑘/(𝑃𝑗+𝑄𝑘)𝑃𝑗.

Case 2. For f(x)=1/x and xi>0,𝑖=1,,𝑛, inequality (3.5) takes the form 𝑛𝑖=1𝑝𝑖𝑥𝑖1𝑛𝑖=1𝑝𝑖𝑥𝑖𝑃min𝑗,𝑄𝑘𝑥𝑘𝑥𝑗2𝑥𝑗𝑥𝑘𝑥𝑗+𝑥𝑘.(3.9) In [9], under the assumption that 𝐱 is an increasing 𝑛-tuple such that 𝑥1>0, the following inequality was obtained: 𝑛𝑖=1𝑝𝑖𝑥𝑖1𝑛𝑖=1𝑝𝑖𝑥𝑖𝐶𝑥𝑘𝑥𝑗2𝑥𝑗𝑥𝑘,(3.10) where 𝑃𝐶=𝑗,𝑃𝑗3𝑄𝑘,4𝑃𝑗𝑄𝑘𝑃𝑗+𝑄𝑘,𝑃𝑗3𝑄𝑘.(3.11) In order to compare these two estimates, first assume that 𝑃𝑗𝑄𝑘. Since 𝑃𝑗𝑥𝑘𝑥𝑗2𝑥𝑗𝑥𝑘𝑥𝑗+𝑥𝑘𝑃𝑗𝑥𝑘𝑥𝑗2𝑥𝑗𝑥𝑘𝑥𝑘+𝑥𝑗2𝑥𝑗+𝑥𝑘,(3.12) it follows that the estimate in (3.9) is better than the one in (3.10).
Next, assume that 𝑄𝑘𝑃𝑗2𝑄𝑘. First, observe that 𝑄𝑘𝑥𝑘+𝑥𝑗2𝑃𝑗𝑥𝑘+𝑥𝑗𝑃𝑗𝑄𝑘𝑥𝑘+𝑥𝑗2𝑥𝑘+𝑥𝑗.(3.13) Simple calculation reveals that 1𝑥𝑘+𝑥𝑗2𝑥𝑘+𝑥𝑗2,(3.14) and so we conclude that the estimate in (3.9) is better than the one in (3.10) when 𝑄𝑘𝑃𝑗𝑄𝑘((𝑥𝑘+𝑥𝑗)2/(𝑥𝑘+𝑥𝑗)), while when 𝑄𝑘((𝑥𝑘+𝑥𝑗)2/(𝑥𝑘+𝑥𝑗))𝑃𝑗2𝑄𝑘, the estimate in (3.10) is better than the one in (3.9).
Further, assume that 2𝑄𝑘𝑃𝑗3𝑄𝑘. In this case, the estimate in (3.10) is better than the one in (3.9), that is, 𝑃𝑗𝑥𝑘+𝑥𝑗𝑄𝑘𝑥𝑘+𝑥𝑗2.(3.15) Namely, 𝑃𝑗𝑥𝑘+𝑥𝑗2𝑄𝑘𝑥𝑘+𝑥𝑗,2𝑄𝑘𝑥𝑘+𝑥𝑗𝑄𝑘𝑥𝑘+𝑥𝑗2𝑥𝑘𝑥𝑗20.(3.16)
Finally, if 3𝑄𝑘𝑃𝑗, the estimate in (3.10) is again better than the one in (3.9), that is, 4𝑃𝑗𝑄𝑘𝑃𝑗+𝑄𝑘𝑥𝑗+𝑥𝑘𝑄𝑘𝑥𝑘+𝑥𝑗2.(3.17) This is equivalent to 𝑃𝑗3𝑥𝑗+3𝑥𝑘2𝑥𝑗𝑥𝑘𝑄𝑘𝑥𝑘+𝑥𝑗2.(3.18) In this case, we have 𝑃𝑗3𝑥𝑗+3𝑥𝑘2𝑥𝑗𝑥𝑘𝑄𝑘3𝑥𝑗+3𝑥𝑘2𝑥𝑗𝑥𝑘,(3.19) and since 𝑄𝑘3𝑥𝑗+3𝑥𝑘2𝑥𝑗𝑥𝑘𝑄𝑘𝑥𝑘+𝑥𝑗2𝑥𝑘𝑥𝑗20,(3.20) 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).

References

  1. D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
  2. J. L. W. V. Jensen, “Sur les fonctions convexes et les inégalités entre les valeurs moyennes,” Acta Mathematica, vol. 30, no. 1, pp. 175–193, 1906. View at Publisher · View at Google Scholar
  3. S. S. Dragomir, J. Pečarić, and L. E. Persson, “Properties of some functionals related to Jensen's inequality,” Acta Mathematica Hungarica, vol. 70, no. 1-2, pp. 129–143, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. S. S. Dragomir, “Bounds for the normalised Jensen functional,” Bulletin of the Australian Mathematical Society, vol. 74, no. 3, pp. 471–478, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. Barić, M. Matić, and J. E. Pečarić, “On the bounds for the normalized Jensen functional and Jensen-Steffensen inequality,” Mathematical Inequalities & Applications, vol. 12, no. 2, pp. 413–432, 2009. View at Google Scholar · View at Zentralblatt MATH
  6. S. Abramovich, S. Ivelić, and J. E. Pečarić, “Improvement of Jensen-Steffensen's inequality for superquadratic functions,” Banach Journal of Mathematical Analysis, vol. 4, no. 1, pp. 159–169, 2010. View at Google Scholar · View at Zentralblatt MATH
  7. S. Ivelić, A. Matković, and J. E. Pečarić, “On a Jensen-Mercer operator inequality,” Banach Journal of Mathematical Analysis, vol. 5, no. 1, pp. 19–28, 2011. View at Google Scholar
  8. M. Khosravi, J. S. Aujla, S. S. Dragomir, and M. S. Moslehian, “Refinements of Choi-Davis-Jensen's inequality,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 2, pp. 127–133, 2011. View at Google Scholar
  9. V. Cirtoaje, “The best lower bound depended on two fixed variables for Jensen's inequality with ordered variables,” Journal of Inequalities and Applications, vol. 2010, Article ID 128258, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. N. Latif, J. Pečarić, and I. Perić, “On majorization of vectors, Favard and Berwald inequalities,” In press.