# Refinements of the Lower Bounds of the Jensen Functional

Authors: Iva Franjić; Sadia Khalid; Josip Pečarić
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 1. Introduction

The classical Jensen inequality states (see e.g., [1]).Theorem 1.1 (see [2]).
LetI be an interval in  ℝ, and let  f:I→ℝ be a convex  function. Let  n≥2,  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
(1.1)f(1Pn∑i=1npixi)≤1Pn∑i=1npif(xi),
where Pn=∑i=1npi. If f is strictly convex, then inequality (1.1) is strict unless x1=⋯=xn.In this work, the functional(1.2)J(x,p,f)=1Pn∑i=1npif(xi)-f(1Pn∑i=1npixi)
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 ofJ(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.
Letf:I→ℝ be a convex function,  x∈In, and let  p be a positive  n-tuple, then
(1.3)Pn⋅J(x,p,f)≥max⁡1≤j≤k≤n{pjf(xj)+pkf(xk)-(pj+pk)f(pjxj+pkxkpj+pk)}≥0.Theorem 1.3.
Letf:I→ℝ be a convex function and  x∈In. Let  p and  r be positive  n-tuples such that  p≥r, that is,  pi≥ri,i=1,…,n, then
(1.4)Pn⋅J(x,p,f)≥Rn⋅J(x,r,f)≥0,
where Pn=∑i=1npi and Rn=∑i=1nri.Further, in [4], the following theorem was given. An alternative proof of the same result was given in [5].Theorem 1.4.
Letf:I→ℝ be a convex function,  n≥2, and  x∈In. Let  p and  q be positive  n-tuples such that  ∑i=1npi=∑i=1nqi=1, then
(1.5)max⁡1≤j≤n{pjqj}J(x,q,f)≥J(x,p,f)≥min⁡1≤j≤n{pjqj}J(x,q,f)≥0.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.
Letf be a convex function on I, p a positive n-tuple such that Pn=∑i=1npi=1 and x1,x2,…,xn∈I,n≥3 such that x1≤x2≤⋯≤xn. For fixed xj,xj+1,…,xn, where j=2,3,…,n-1, the Jensen functional J(x,p,f) defined in (1.2) is minimal when x1=x2=⋯=xj-1=xj, that is,
(1.6)J(x,p,f)≥Pjf(xj)+∑i=j+1npif(xi)-f(Pjxj+∑i=j+1npixi),
where
(1.7)Pj=∑i=1jpi,j=1,…,n.Lemma 1.6.
Letf be a convex function on I, p a positive n-tuple such that Pn=∑i=1npi=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,
(1.8)J(x,p,f)≥∑i=1k-1pif(xi)+Qkf(xk)-f(∑i=1k-1pixi+Qkxk),
where
(1.9)Qk=∑i=knpi,k=1,…,n.Theorem 1.7.
Letf be a convex function on  I,  p a positive  n-tuple such that  Pn=∑i=1npi=1 and  x1,x2,…,xn∈I,n≥3 such that  x1≤x2≤⋯≤xn. For fixed  xj and  xk, where  1≤j<k≤n, the Jensen functional  J(x,p,f) defined in (1.2) is minimal when
(1.10)x1=x2=⋯=xj,xk=xk+1=⋯=xn,xj+1=xj+2=⋯=xk-1=Pjxj+QkxkPj+Qk,
that is,
(1.11)J(x,p,f)≥Pjf(xj)+Qkf(xk)-(Pj+Qk)f(Pjxj+QkxkPj+Qk),
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.
Letf be a convex function on I, and let p1,p2 be nonnegative real numbers. If a1,a2,b1,b2∈I are such that a1,a2∈[b1,b2] and
(1.12)p1a1+p2a2=p1b1+p2b2,
then
(1.13)p1f(a1)+p2f(a2)≤p1f(b1)+p2f(b2).Note that for a monotonicn-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.
Letf be a convex function on I, p a positive n-tuple, and a,b∈In such that
(1.14)∑i=1kpiai≤∑i=1kpibifork=1,2,…,n-1,∑i=1npiai=∑i=1npibi.
If a is a decreasing n-tuple, then one has
(1.15)∑i=1npif(ai)≤∑i=1npif(bi),
while if b is an increasing n-tuple, then we have
(1.16)∑i=1npif(bi)≤∑i=1npif(ai).
If f is strictly convex and a≠b, then (1.15) and (1.16) are strict.Note that forn=2, inequality (1.15) holds if a2≤a1≤b1 and if (1.12) is valid, while inequality (1.16) holds if a1≤b1≤b2 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 Pn≠1 results follow easily by substituting pi with pi/Pn. Furthermore, for 1≤j<k≤n, we introduce the following notation: (2.1)Jmin⁡(x,p,f)=min⁡{Pj,Qk}(f(xj)+f(xk)-2f(xj+xk2)),Jjk(x,p,f)=Pjf(xj)+∑i=j+1k-1pif(xi)+Qkf(xk)-f(Pjxj+∑i=j+1k-1pixi+Qkxk).
Note that J1n(x,p,f)=J(x,p,f).Theorem 2.1.
Letf be a convex function on  I and  p a positive  n-tuple such that  Pn=1,n≥2. Let  1≤j<k≤n and  xi∈I,  i=1,…,k. If  xj is such that
(2.2)1Pj∑i=1jpixi≤xj≤1Qj+1(∑i=j+1k-1pixi+Qkxk),(2.3)or1Qj+1(∑i=j+1k-1pixi+Qkxk)≤xj≤1Pj∑i=1jpixi,
then one has
(2.4)J1k(x,p,f)≥Jjk(x,p,f).Proof.
The claim is that(2.5)∑i=1jpif(xi)-f(∑i=1k-1pixi+Qkxk)≥Pjf(xj)-f(Pjxj+∑i=j+1k-1pixi+Qkxk).
As a simple consequence of the Jensen inequality (1.1), we have
(2.6)∑i=1jpif(xi)≥Pjf(1Pj∑i=1jpixi).
Therefore, if we prove
(2.7)Pjf(1Pj∑i=1jpixi)+f(Pjxj+∑i=j+1k-1pixi+Qkxk)≥Pjf(xj)+f(∑i=1k-1pixi+Qkxk),
the claim will follow. The idea is to apply Lemma 1.8 for p1=Pj, p2=1, a1=xj, a2=∑i=1k-1pixi+Qkxk,  b1=(1/Pj)∑i=1jpixi,  and b2=Pjxj+∑i=j+1k-1pixi+Qkxk. Condition (1.12) is obviously satisfied. In addition, we need to check that
(2.8)1Pj∑i=1jpixi≤xj≤Pjxj+∑i=j+1k-1pixi+Qkxk,1Pj∑i=1jpixi≤∑i=1k-1pixi+Qkxk≤Pjxj+∑i=j+1k-1pixi+Qkxk.
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=∑i=1k-1pixi+Qkxk,  a2=xj,  b1=Pjxj+∑i=j+1k-1pixi+Qkxk, and b2=(1/Pj)∑i=1jpixi, instead. In this case, the necessary conditions would follow from (2.3).Theorem 2.2.
Let the conditions of Theorem2.1 hold. If  xj is such that
(2.9)1Pj∑i=1jpixi≤xj≤∑i=1k-1pixi+Qkxk,(2.10)or∑i=1k-1pixi+Qkxk≤xj≤1Pj∑i=1jpixi,
then inequality (2.4) holds.Proof.
Proof is analogous to the proof of Theorem2.1. 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.
Letf be a convex function on  I and  p a positive  n-tuple such that  Pn=1,n≥2. Let  1≤j<k≤n and  xi∈I,  i=j,…,n. If  xk is such that
(2.11)1Pk-1(Pjxj+∑i=j+1k-1pixi)≤xk≤1Qk∑i=knpixi,(2.12)or1Qk∑i=knpixi≤xk≤1Pk-1(Pjxj+∑i=j+1k-1pixi),
then one has
(2.13)Jjn(x,p,f)≥Jjk(x,p,f).Proof.
Similarly as in the proof of Theorem2.1, after first applying the Jensen inequality to the sum on the left-hand side, the claim will follow if we prove
(2.14)Qkf(1Qk∑i=knpixi)+f(Pjxj+∑i=j+1k-1pixi+Qkxk)≥Qk(xk)+f(Pjxj+∑i=j+1npixi).
We can apply Lemma 1.8 for p1=1, p2=Qk, a1=Pjxj+∑i=j+1npixi,a2=xk,b1=Pjxj+∑i=j+1k-1pixi+Qkxk, and b2=(1/Qk)∑i=knpixi, 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 Theorem2.3 hold. If  xk is such that
(2.15)Pjxj+∑i=j+1npixi≤xk≤1Qk∑i=knpixi,(2.16)or1Qk∑i=knpixi≤xk≤Pjxj+∑i=j+1npixi,
then inequality (2.13) holds.Proof.
It is analogous to the proof of Theorem2.3. 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.
Letf be a convex function on I and p a positive n-tuple such that Pn=1, n≥2. Let x∈In be a real n-tuple and 1≤j<k≤n.
Ifxk is such that
(2.17)1Pk-1∑i=1k-1pixi≤xk≤1Qk∑i=knpixi,(2.18)or1Qk∑i=knpixi≤xk≤1Pk-1∑i=1k-1pixi,
and xj is such that either (2.2) or (2.3) holds, then one has
(2.19)J(x,p,f)≥J1k(x,p,f)≥Jjk(x,p,f).
Ifxj is such that
(2.20)1Pj∑i=1jpixi≤xj≤1Qj+1∑i=j+1npixi,(2.21)or1Qj+1∑i=j+1npixi≤xj≤1Pj∑i=1jpixi,
and xk is such that either (2.11) or (2.12) holds, then one has
(2.22)J(x,p,f)≥Jjn(x,p,f)≥Jjk(x,p,f).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 Corollary2.5 hold.
Ifxk is such that
(2.23)∑i=1npixi≤xk≤1Qk∑i=knpixi,(2.24)or1Qk∑i=knpixi≤xk≤∑i=1npixi,
and xj is such that either (2.9) or (2.10) holds, then inequality (2.19) holds.
Ifxj is such that
(2.25)1Pj∑i=1jpixi≤xj≤∑i=1npixi,(2.26)or∑i=1npixi≤xj≤1Pj∑i=1jpixi,
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.
Letf be a convex function on  I and  p a positive  n-tuple such that  Pn=1,  n≥2. Let  x∈In be a real  n-tuple, and let  1≤j<k≤n. If  xj and  xk are such that
(2.27)1Pj∑i=1jpixi≤xj≤∑i=1npixi≤xk≤1Qk∑i=knpixi,
or
(2.28)1Qk∑i=knpixi≤xk≤∑i=1npixi≤xj≤1Pj∑i=1jpixi,
then one has
(2.29)J(x,p,f)≥Jjk(x,p,f).Proof.
The claim is that(2.30)∑i=1jpif(xi)+∑i=knpif(xi)-f(∑i=1npixi)≥Pjf(xj)+Qkf(xk)-f(Pjxj+∑i=j+1k-1pixi+Qkxk).
After applying the Jensen inequality to the two sums on the left-hand side, we need to prove
(2.31)Qkf(1Qk∑i=knpixi)+f(Pjxj+∑i=j+1k-1pixi+Qkxk)+Pjf(1Pj∑i=1jpixi)≥Qkf(xk)+f(∑i=1npixi)+Pjf(xj).
Set p1=Qk, p2=1, p3=Pj, a1=xk,a2=∑i=1npixi, a3=xj,b1=(1/Qk)∑i=knpixi,b2=Pjxj+∑i=j+1k-1pixi+Qkxk, and b3=(1/Pj)∑i=1jpixi. 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
(2.32)1Pj∑i=1jpixi≤xj≤∑i=1k-1pixi+Qkxk≤∑i=1npixi≤xk≤1Qk∑i=knpixi,
while (2.15) and (2.25) combined together give
(2.33)1Pj∑i=1jpixi≤xj≤∑i=1npixi≤Pjxj+∑i=j+1npixi≤xk≤1Qk∑i=knpixi,
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.
Letf be a convex function on  I and  p a positive  n-tuple such that  Pn=1,  n≥2. Let  1≤j<k≤n and  xi∈I,  i=j,…,k, then one has
(2.34)Jjk(x,p,f)≥Pjf(xj)+Qkf(xk)-(Pj+Qk)f(Pjxj+QkxkPj+Qk)≥Jmin⁡(x,p,f)≥0.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, yield(3.1)J(x,p,f)≥Pjf(xj)+Qkf(xk)-(Pj+Qk)f(Pjxj+QkxkPj+Qk).Corollary 3.1.
Let the conditions of Corollaries2.5 and 2.6, or Theorem 2.7 hold, then
(3.2)∑i=1npiai-∏i=1naipi≥Pjaj+Qkak-(Pj+Qk)ajPj/(Pj+Qk)akQk/(Pj+Qk).Proof.
This follows from (3.1) for f(x)=ex, using notation ai=exi.Corollary 3.2.
Let the conditions of Corollaries2.5 and 2.6, or Theorem 2.7 hold, and let in addition xi>0, i=1,…,n, then
(3.3)∑i=1npixi∏i=1nxipi≥1xjPjxkQk(Pjxj+QkxkPj+Qk)Pj+Qk.Proof.
Follows from (3.1) for f(x)=-lnx.Corollary 3.3.
Let the conditions of Corollaries2.5 and 2.6, or Theorem 2.7 hold, and let in addition xi>0, i=1,…,n, then
(3.4)∑i=1npixi-1∑i=1npixi≥PjQk(xk-xj)2xjxk(Pjxj+Qkxk).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 have(3.5)J(x,p,f)≥Jmin⁡(x,p,f)≥0.
Next, we compare estimates obtained from (3.5) with those obtained in [9].Case 1.
Forf(x)=ex, using notation ai=exi, inequality (3.5) takes the form
(3.6)∑i=1npiai-∏i=1naipi≥min⁡{Pj,Qk}(ak-aj)2.
In [9], under the assumption that a is an increasing n-tuple, the following inequality was obtained
(3.7)∑i=1npiai-∏i=1naipi≥C(ak-aj)2,
where
(3.8)C={2PjQkPj+Qk,Pj≤Qk,Qk,Pj≥Qk.
Note that when Pj≥Qk, (3.6) recaptures this result. However, when Pj≤Qk, the constant C is better, since 2PjQk/(Pj+Qk)≥Pj.Case 2.
Forf(x)=1/x and xi>0,i=1,…,n, inequality (3.5) takes the form
(3.9)∑i=1npixi-1∑i=1npixi≥min⁡{Pj,Qk}(xk-xj)2xjxk(xj+xk).
In [9], under the assumption that x is an increasing n-tuple such that x1>0, the following inequality was obtained:
(3.10)∑i=1npixi-1∑i=1npixi≥C(xk-xj)2xjxk,
where
(3.11)C={Pj,Pj≤3Qk,4PjQkPj+Qk,Pj≥3Qk.
In order to compare these two estimates, first assume that Pj≤Qk. Since
(3.12)Pj(xk-xj)2xjxk(xj+xk)≥Pj(xk-xj)2xjxk⟺(xk+xj)2≥xj+xk,
it follows that the estimate in (3.9) is better than the one in (3.10).
Next, assume thatQk≤Pj≤2Qk. First, observe that
(3.13)Qk(xk+xj)2≥Pj(xk+xj)⟺Pj≤Qk(xk+xj)2xk+xj.
Simple calculation reveals that
(3.14)1≤(xk+xj)2xk+xj≤2,
and so we conclude that the estimate in (3.9) is better than the one in (3.10) when Qk≤Pj≤Qk((xk+xj)2/(xk+xj)), while when Qk((xk+xj)2/(xk+xj))≤Pj≤2Qk, the estimate in (3.10) is better than the one in (3.9).
Further, assume that2Qk≤Pj≤3Qk. In this case, the estimate in (3.10) is better than the one in (3.9), that is,
(3.15)Pj(xk+xj)≥Qk(xk+xj)2.
Namely,
(3.16)Pj(xk+xj)≥2Qk(xk+xj),2Qk(xk+xj)≥Qk(xk+xj)2⟺(xk-xj)2≥0.
Finally, if3Qk≤Pj, the estimate in (3.10) is again better than the one in (3.9), that is,
(3.17)4PjQkPj+Qk(xj+xk)≥Qk(xk+xj)2.
This is equivalent to
(3.18)Pj(3xj+3xk-2xjxk)≥Qk(xk+xj)2.
In this case, we have
(3.19)Pj(3xj+3xk-2xjxk)≥Qk(3xj+3xk-2xjxk),
and since
(3.20)Qk(3xj+3xk-2xjxk)≥Qk(xk+xj)2⟺(xk-xj)2≥0,
the claim follows.



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