Refinements of the Lower Bounds of the Jensen Functional

and Applied Analysis 3 where Pj j ∑ i 1 pi, j 1, . . . , n. 1.7 Lemma 1.6. Let f be a convex function on I, p a positive n-tuple such that Pn ∑n i 1 pi 1 and x1, x2, . . . , xn ∈ I, n ≥ 3 such that x1 ≤ x2 ≤ · · · ≤ xn. For fixed x1, x2, . . . , xk, where k 2, 3, . . . , n− 1, the Jensen functional J x,p, f defined in 1.2 is minimal when xk xk 1 · · · xn−1 xn, that is, J ( x,p, f ) ≥ k−1 ∑ i 1 pif xi Qkf xk − f ( k−1 ∑ i 1 pixi Qkxk ) , 1.8


Introduction
The classical Jensen inequality states see e.g., 1 .
Theorem 1.1 see 2 .Let I be an interval in R, and let f : I → R be a convex function.Let n ≥ 2, x x 1 , . . ., x n ∈ I n , and let p p 1 , . . ., p n be a positive n-tuple, that is, such that p i > 0 for i 1, . . ., n, then where P n n i 1 p i .If f is strictly convex, then inequality 1.1 is strict unless

Abstract and Applied Analysis
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 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 R.
Theorem 1.2.Let f : I → R be a convex function, x ∈ I n , and let p be a positive n-tuple, then P n • J x, p, f ≥ max 1≤j≤k≤n p j f x j p k f x k − p j p k f p j x j p k x k p j p k ≥ 0. 1.3 Theorem 1.3.Let f : I → R be a convex function and x ∈ I n .Let p and r be positive n-tuples such that p ≥ r, that is, p i ≥ r i , i 1, . . ., n, then where P n n i 1 p i and R n n i 1 r i .
Further, in 4 , the following theorem was given.An alternative proof of the same result was given in 5 .
Theorem 1.4.Let f : I → R be a convex function, n ≥ 2, and x ∈ I n .Let p and q be positive n-tuples such that n i 1 p i n i 1 q i 1, then max 1≤j≤n p j q j J x, q, f ≥ J x, p, f ≥ min 1≤j≤n p j q j J x, q, f ≥ 0. 1.5 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 f be a convex function on I, p a positive n-tuple such that
For fixed x j , x j 1 , . . ., x n , where j 2, 3, . . ., n − 1, the Jensen functional J x, p, f defined in 1.2 is minimal when where Lemma 1.6.Let f be a convex function on I, p a positive n-tuple such that P n n i 1 p i 1 and where Theorem 1.7.Let f be a convex function on I, p a positive n-tuple such that P n n i 1 p i 1 and For fixed x j and x k , where 1 ≤ j < k ≤ n, the Jensen functional J x, p, f defined in 1.2 is minimal when where P j are as in 1.7 and Q k 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 p 1 , p 2 be nonnegative real numbers.If 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, b ∈ I n such that If a is a decreasing n-tuple, then one has while if b is an increasing n-tuple, then we have If f is strictly convex and a / b, then 1.15 and 1.16 are strict.
Note that for n 2, inequality 1.15 holds if a 2 ≤ a 1 ≤ b 1 and if 1.12 is valid, while inequality 1.16 holds if a 1 ≤ b 1 ≤ b 2 and if 1.12 is valid.

Main Results
In what follows, J x, p, f is as in 1.2 , P j are as in 1.7 , and Q k , as in 1.9 .Without any loss of generality, we assume that P n 1, since for positive n-tuples such that P n / 1 results follow easily by substituting p i with p i /P n .Furthermore, for 1 ≤ j < k ≤ n, we introduce the following notation:

2.1
Note that J 1n 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 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 12 is obviously satisfied.In addition, we need to check that

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 p 1 1, 2.9 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 P n 1, n ≥ 2. Let 1 ≤ j < k ≤ n and x i ∈ I, i j, . . ., n.If x k is such that 11 2.12 then one has J jn x, p, f ≥ J jk 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

2.14
We can apply Lemma 1.8 for p 1 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 x k 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 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 P n 1, n ≥ 2. Let x ∈ I n be a real n-tuple and 1 ≤ j < k ≤ n.If x k is such that 18 and x j is such that either 2.2 or 2.3 holds, then one has J x, p, f ≥ J 1k x, p, f ≥ J jk x, p, f .

2.19
If x j is such that and x k is such that either 2.11 or 2.12 holds, then one has J x, p, f ≥ J jn x, p, f ≥ J jk 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 x k is such that
24 and x j is such that either 2.9 or 2.10 holds, then inequality 2.19 holds.If x j is such that 25 and x k 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 P n 1, n ≥ 2. Let x ∈ I n be a real n-tuple, and let 1 ≤ j < k ≤ n.If x j and x k are such that then one has J x, p, f ≥ J jk x, p, f .

2.29
Proof.The claim is that

2.30
Abstract and Applied Analysis 9 After applying the Jensen inequality to the two sums on the left-hand side, we need to prove

2.31
Set , and b 3 1/P j j i 1 p i x i .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 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 f be a convex function on I and p a positive n-tuple such that P n 1, n ≥ 2. Let 1 ≤ j < k ≤ n and x i ∈ I, i j, . . ., k, then one has

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 .

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, yield Proof.This follows from 3.1 for f x e x , using notation a i e x i .
Corollary 3.2.Let the conditions of Corollaries 2.5 and 2.6, or Theorem 2.7 hold, and let in addition Proof.Follows from 3.1 for f x − ln x.
Corollary 3.3.Let the conditions of Corollaries 2.5 and 2.6, or Theorem 2.7 hold, and let in addition x i > 0, i 1, . . ., n, then 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 e x 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 have J x, p, f ≥ J min x, p, f ≥ 0. 3.5 Next, we compare estimates obtained from 3.5 with those obtained in 9 .
Case 1.For f x e x , using notation a i e x i , inequality 3.5 takes the form

3.6
In 9 , under the assumption that a is an increasing n-tuple, the following inequality was obtained where 3.8 Note that when P j ≥ Q k , 3.6 recaptures this result.However, when Case 2. For f x 1/x and x i > 0, i 1, . . ., n, inequality 3.5 takes the form 3.9 In 9 , under the assumption that x is an increasing n-tuple such that x 1 > 0, the following inequality was obtained:

3.11
In order to compare these two estimates, first assume that P j ≤ Q k .Since 12 it follows that the estimate in 3.9 is better than the one in 3.10 .Next, assume that Q k ≤ P j ≤ 2Q k .First, observe that x k x j .

3.13
Simple calculation reveals that x k x j ≤ 2, 3.14 and so we conclude that the estimate in 3.9 is better than the one in 3.10 when Q k ≤ P j ≤ Q k √ x k √ x j 2 / x k x j , while when Q k √ x k √ x j 2 / x k x j ≤ P j ≤ 2Q k , the estimate in 3.10 is better than the one in 3.9 .Further, assume that 2Q k ≤ P j ≤ 3Q k .In this case, the estimate in 3.10 is better than the one in 3.9 , that is,

3.16
Finally, if 3Q k ≤ P j , the estimate in 3.10 is again better than the one in 3.9 , that is, 3.17 This is equivalent to

3.18
In this case, we have and since ≥ 0, 3.20 the claim follows.
and b 2 1/P j j i 1 p i x i , instead.In this case, the necessary conditions would follow from 2.3 .Let the conditions of Theorem 2.1 hold.If x j is such that