Journal of Inequalities in Pure and
Applied Mathematics
ON A REVERSE OF JESSEN’S INEQUALITY FOR ISOTONIC
LINEAR FUNCTIONALS
volume 2, issue 3, article 36,
2001.
S.S. DRAGOMIR
School of Communications and Informatics
Victoria University of Technology
PO Box 14428
Melbourne City MC
8001 Victoria, Australia
EMail: sever@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Received 23 April, 2001;
accepted 28 May, 2001.
Communicated by: A. Lupas
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
047-01
Quit
Abstract
A reverse of Jessen’s inequality and its version for m − Ψ−convex and M −
Ψ−convex functions are obtained. Some applications for particular cases are
also pointed out.
2000 Mathematics Subject Classification: 26D15, 26D99
Key words: Jessen’s Inequality, Isotonic Linear Functionals.
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The Concepts of m − Ψ−Convex and M − Ψ− Convex
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
A Reverse Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4
A Further Result for m − Ψ−Convex and M − Ψ−Convex
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5
Some Applications For Bullen’s Inequality . . . . . . . . . . . . . . . 25
References
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
1.
Introduction
Let L be a linear class of real-valued functions g : E → R having the properties
(L1) f, g ∈ L imply (αf + βg) ∈ L for all α, β ∈ R;
(L2) 1 ∈ L, i.e., if f0 (t) = 1, t ∈ E then f0 ∈ L.
An isotonic linear functional A : L → R is a functional satisfying
(A1) A (αf + βg) = αA (f ) + βA (g) for all f, g ∈ L and α, β ∈ R.
(A2) If f ∈ L and f ≥ 0, then A (f ) ≥ 0.
The mapping A is said to be normalised if
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
(A3) A (1) = 1.
Isotonic, that is, order-preserving, linear functionals are natural objects in
analysis which enjoy a number of convenient properties. Thus, they provide, for
example, Jessen’s inequality, which is a functional form of Jensen’s inequality
(see [2] and [10]).
We recall Jessen’s inequality (see also [8]).
Theorem 1.1. Let φ : I ⊆ R → R (I is an interval), be a convex function and
f : E → I such that φ ◦ f , f ∈ L. If A : L → R is an isotonic linear and
normalised functional, then
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
(1.1)
φ (A (f )) ≤ A (φ ◦ f ) .
A counterpart of this result was proved by Beesack and Pečarić in [2] for
compact intervals I = [α, β].
Page 3 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
Theorem 1.2. Let φ : [α, β] ⊂ R → R be a convex function and f : E → [α, β]
such that φ ◦ f , f ∈ L. If A : L → R is an isotonic linear and normalised
functional, then
(1.2)
A (φ ◦ f ) ≤
β − A (f )
A (f ) − α
φ (α) +
φ (β) .
β−α
β−α
Remark 1.1. Note that (1.2) is a generalisation of the inequality
(1.3)
A (φ) ≤
b − A (e1 )
A (e1 ) − a
φ (a) +
φ (b)
b−a
b−a
due to Lupaş [9] (see for example [2, Theorem A]), which assumed E = [a, b],
L satisfies (L1), (L2), A : L → R satisfies (A1), (A2), A (1) = 1, φ is convex on
E and φ ∈ L, e1 ∈ L, where e1 (x) = x, x ∈ [a, b].
The following inequality is well known in the literature as the HermiteHadamard inequality
Z b
a+b
1
ϕ (a) + ϕ (b)
≤
ϕ (t) dt ≤
,
(1.4)
ϕ
2
b−a a
2
provided that ϕ : [a, b] → R is a convex function.
Using Theorem 1.1 and Theorem 1.2, we may state the following generalisation of the Hermite-Hadamard inequality for isotonic linear functionals ([11]
and [12]).
Theorem 1.3. Let φ : [a, b] ⊂ R → R be a convex function and e : E → [a, b]
with e, φ ◦ e ∈ L. If A → R is an isotonic linear and normalised functional,
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
with A (e) =
(1.5)
a+b
,
2
then
ϕ (a) + ϕ (b)
a+b
ϕ
≤ A (φ ◦ e) ≤
.
2
2
For other results concerning convex functions and isotonic linear functionals,
see [3] – [6], [12] – [14] and the recent monograph [7].
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
2.
The Concepts of m − Ψ−Convex and M − Ψ−
Convex Functions
Assume that the mapping Ψ : I ⊆ R → R (I is an interval) is convex on I and
m ∈ R. We shall say that the mapping φ : I → R is m − Ψ− lower convex
if φ − mΨ is a convex mapping on I (see [4]). We may introduce the class of
functions
(2.1)
L (I, m, Ψ) := {φ : I → R|φ − mΨ is convex on I} .
Similarly, for M ∈ R and Ψ as above, we can introduce the class of M −
Ψ−upper convex functions by (see [4])
(2.2)
U (I, M, Ψ) := {φ : I → R|M Ψ − φ is convex on I} .
The intersection of these two classes will be called the class of (m, M ) −
Ψ−convex functions and will be denoted by
(2.3)
B (I, m, M, Ψ) := L (I, m, Ψ) ∩ U (I, M, Ψ) .
Remark 2.1. If Ψ ∈ B (I, m, M, Ψ), then φ − mΨ and M Ψ − φ are convex
and then (φ − mΨ) + (M Ψ − φ) is also convex which shows that (M − m) Ψ
is convex, implying that M ≥ m (as Ψ is assumed not to be the trivial convex
function Ψ (t) = 0, t ∈ I).
The above concepts may be introduced in the general case of a convex subset
in a real linear space, but we do not consider this extension here.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
In [6], S.S. Dragomir and N.M. Ionescu introduced the concept of g−convex
dominated mappings, for a mapping f : I → R. We recall this, by saying, for
a given convex function g : I → R, the function f : I → R is g−convex
dominated iff g + f and g − f are convex mappings on I. In [6], the authors
pointed out a number of inequalities for convex dominated functions related to
Jensen’s, Fuchs’, Pečarić’s, Barlow-Marshall-Proschan and Vasić-Mijalković
results, etc.
We observe that the concept of g−convex dominated functions can be obtained as a particular case from (m, M ) − Ψ−convex functions by choosing
m = −1, M = 1 and Ψ = g.
The following lemma holds (see also [4]).
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Lemma 2.1. Let Ψ, φ : I ⊆ R → R be differentiable functions on I̊ and Ψ is a
convex function on I̊.
(i) For m ∈ R, the function φ ∈ L I̊, m, Ψ iff
0
0
(2.4) m [Ψ (x) − Ψ (y) − Ψ (y) (x − y)] ≤ φ (x)−φ (y)−φ (y) (x − y)
for all x, y ∈I̊.
(ii) For M ∈ R, the function φ ∈ U I̊, M, Ψ iff
(2.5)
φ (x) − φ (y) − φ0 (y) (x − y) ≤ M [Ψ (x) − Ψ (y) − Ψ0 (y) (x − y)]
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
for all x, y ∈I̊.
(iii) For M, m ∈ R with M ≥ m, the function φ ∈ B I̊, m, M, Ψ iff both
(2.4) and (2.5) hold.
Page 7 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
Proof. Follows by the fact that a differentiable mapping h : I → R is convex
on I̊ iff h (x) − h (y) ≥ h0 (y) (x − y) for all x, y ∈I̊.
Proposition 2.3.
Another elementary fact for twice differentiable functions also holds (see
also [4]).
Lemma 2.2. Let Ψ, φ : I ⊆ R → R be twice differentiable on I̊ and Ψ is convex
on I̊.
(i) For m ∈ R, the function φ ∈ L I̊, m, Ψ iff
(2.6)
mΨ00 (t) ≤ φ00 (t) for all t ∈ I̊.
(ii) For M ∈ R, the function φ ∈ U I̊, M, Ψ iff
(2.7)
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
φ00 (t) ≤ M Ψ00 (t) for all t ∈ I̊.
Title Page
(iii) For M, m ∈ R with M ≥ m, the function φ ∈ B I̊, m, M, Ψ iff both
(2.6) and (2.7) hold.
Proof. Follows by the fact that a twice differentiable function h : I → R is
convex on I̊ iff h00 (t) ≥ 0 for all t ∈I̊.
We consider the p−logarithmic mean of two positive numbers given by

a
if b = a,



p+1
1
Lp (a, b) :=

b
− ap+1 p


if a 6= b
(p + 1) (b − a)
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
and p ∈ R {−1, 0}.
The following proposition holds (see also [4]).
Let φ : (0, ∞) → R be a differentiable mapping.
(i) For m ∈ R, the function φ ∈ L ((0, ∞) , m, (·)p ) with p ∈ (−∞, 0) ∪
(1, ∞) iff
p−1
(2.8) mp (x − y) Lp−1
≤ φ (x) − φ (y) − φ0 (y) (x − y)
p−1 (x, y) − y
for all x, y ∈ (0, ∞) .
(ii) For M ∈ R, the function φ ∈ U ((0, ∞) , M, (·)p ) with p ∈ (−∞, 0) ∪
(1, ∞) iff
p−1
(2.9) φ (x) − φ (y) − φ0 (y) (x − y) ≤ M p (x − y) Lp−1
p−1 (x, y) − y
for all x, y ∈ (0, ∞) .
(iii) For M, m ∈ R with M ≥ m, the function φ ∈ B ((0, ∞) , M, (·)p ) with
p ∈ (−∞, 0) ∪ (1, ∞) iff both (2.8) and (2.9) hold.
The proof follows by Lemma 2.1 applied for the convex mapping Ψ (t) = tp ,
p ∈ (−∞, 0) ∪ (4, ∞) and via some elementary computation. We omit the
details.
The following corollary is useful in practice.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 30
Corollary 2.4. Let φ : (0, ∞) → R be a differentiable function.
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
(i) For m ∈ R, the function φ is m−quadratic-lower convex (i.e., for p = 2)
iff
(2.10)
m (x − y)2 ≤ φ (x) − φ (y) − φ0 (y) (x − y)
for all x, y ∈ (0, ∞).
(ii) For M ∈ R, the function φ is M −quadratic-upper convex iff
(2.11)
φ (x) − φ (y) − φ0 (y) (x − y) ≤ M (x − y)2
for all x, y ∈ (0, ∞).
(iii) For m, M ∈ R with M ≥ m, the function φ is (m, M ) −quadratic convex
if both (2.10) and (2.11) hold.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
The following proposition holds (see also [4]).
Contents
Proposition 2.5. Let φ : (0, ∞) → R be a twice differentiable function.
p
(i) For m ∈ R, the function φ ∈ L ((0, ∞) , m, (·) ) with p ∈ (−∞, 0) ∪
(1, ∞) iff
(2.12)
p−2
p (p − 1) mt
00
≤ φ (t) for all t ∈ (0, ∞) .
(ii) For M ∈ R, the function φ ∈ U ((0, ∞) , M, (·)p ) with p ∈ (−∞, 0) ∪
(1, ∞) iff
(2.13)
φ00 (t) ≤ p (p − 1) M tp−2 for all t ∈ (0, ∞) .
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
(iii) For m, M ∈ R with M ≥ m, the function φ ∈ B ((0, ∞) , m, M, (·)p ) with
p ∈ (−∞, 0) ∪ (1, ∞) iff both (2.12) and (2.13) hold.
As can be easily seen, the above proposition offers the practical criterion
of deciding when a twice differentiable mapping is (·)p −lower or (·)p −upper
convex and which weights the constant m and M are.
The following corollary is useful in practice.
Corollary 2.6. Assume that the mapping φ : (a, b) ⊆ R → R is twice differentiable.
(i) If inf φ00 (t) = k > −∞, then φ is k2 −quadratic lower convex on (a, b) ;
t∈(a,b)
(ii) If sup φ00 (t) = K < ∞, then φ is
t∈(a,b)
K
−quadratic
2
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
upper convex on (a, b) .
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
3.
A Reverse Inequality
We start with the following result which gives another counterpart for A (φ ◦ f ),
as did the Lupaş-Beesack-Pečarić result (1.2).
Theorem 3.1. Let φ : (α, β) ⊆ R → R be a differentiable convex function on
(α, β), f : E → (α, β) such that φ ◦ f , f , φ0 ◦ f , φ0 ◦ f · f ∈ L. If A : L → R
is an isotonic linear and normalised functional, then
(3.1)
0 ≤ A (φ ◦ f ) − φ (A (f ))
≤ A (φ0 ◦ f · f ) − A (f ) · A (φ0 ◦ f )
1 0
≤
[φ (β) − φ0 (α)] (β − α) (if α, β are finite).
4
Proof. As φ is differentiable convex on (α, β), we may write that
(3.2)
φ (x) − φ (y) ≥ φ0 (y) (x − y) , for all x, y ∈ (α, β) ,
from where we obtain
(3.3)
φ (A (f )) − (φ ◦ f ) (t) ≥ (φ0 ◦ f ) (t) (A (f ) − f (t))
for all t ∈ E, as, obviously, A (f ) ∈ (α, β).
If we apply to (3.3) the functional A, we may write
0
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
0
φ (A (f )) − A (φ ◦ f ) ≥ A (f ) · A (φ ◦ f ) − A (φ ◦ f · f ) ,
which is clearly equivalent to the first inequality in (3.1).
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
Page 12 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
It is well known that the following Grüss inequality for isotonic linear and
normalised functionals holds (see [1])
(3.4)
|A (hk) − A (h) A (k)| ≤
1
(M − m) (N − n) ,
4
provided that h, k ∈ L, hk ∈ L and −∞ < m ≤ h (t) ≤ M < ∞, −∞ < n ≤
k (t) ≤ N < ∞, for all t ∈ E.
Taking into account that for finite α,β we have α < f (t) < β with φ0 being
monotonic on (α, β), we have φ0 (α) ≤ φ0 ◦ f ≤ φ0 (β), and then by the Grüss
inequality, we may state that
A (φ0 ◦ f · f ) − A (f ) · A (φ0 ◦ f ) ≤
1 0
[φ (β) − φ0 (α)] (β − α)
4
and the theorem is completely proved.
The following corollary holds.
Corollary 3.2. Let φ : [a, b] ⊂I̊⊆ R → R be a differentiable convex function on
I̊. If φ, e1 , φ0 , φ0 · e1 ∈ L (e1 (x) = x, x ∈ [a, b]) and A : L → R is an isotonic
linear and normalised functional, then:
(3.5)
0 ≤ A (φ) − φ (A (e1 ))
≤ A (φ0 · e1 ) − A (e1 ) · A (φ0 )
1 0
[φ (b) − φ0 (a)] (b − a) .
≤
4
There are some particular cases which can naturally be considered.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
1. Let φ (x) = ln x, x > 0. If ln f , f , f1 ∈ L and A : L → R is an isotonic
linear and normalised functional, then:
1
(3.6)
0 ≤ ln [A (f )] − A [ln (f )] ≤ A (f ) A
− 1,
f
provided that f (t) > 0 for all t ∈ E and A (f ) > 0.
If 0 < m ≤ f (t) ≤ M < ∞, t ∈ E, then, by the second part of (3.1) we
have:
1
(M − m)2
(3.7) A (f ) A
−1≤
(which is a known result).
f
4mM
Note that the inequality (3.6) is equivalent to
1
A (f )
(3.8)
1≤
≤ exp A (f ) A
−1 .
exp [A [ln (f )]]
f
2. Let φ (x) = exp (x), x ∈ R. If exp (f ), f , f · exp (f ) ∈ L and A : L → R
is an isotonic linear and normalised functional, then
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
(3.9) 0 ≤ A [exp (f )] − exp [A (f )]
≤ A [f exp (f )] − A (f ) exp [A (f )]
1
≤ [exp (M ) − exp (m)] (M − m) (if m ≤ f ≤ M on E).
4
Close
Quit
Page 14 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
4.
A Further Result for m − Ψ−Convex and M −
Ψ−Convex Functions
In [4], S.S. Dragomir proved the following inequality of Jessen’s type for m −
Ψ−convex and M − Ψ−convex functions.
Theorem 4.1. Let Ψ : I ⊆ R → R be a convex function and f : E → I
such that Ψ ◦ f , f ∈ L and A : L → R be an isotonic linear and normalised
functional.
(i) If φ ∈ L (I, m, Ψ) and φ ◦ f ∈ L, then we have the inequality
(4.1)
m [A (Ψ ◦ f ) − Ψ (A (f ))] ≤ A (φ ◦ f ) − φ (A (f )) .
(ii) If φ ∈ U (I, M, Ψ) and φ ◦ f ∈ L, then we have the inequality
(4.2)
A (φ ◦ f ) − φ (A (f )) ≤ M [A (Ψ ◦ f ) − Ψ (A (f ))] .
(iii) If φ ∈ B (I, m, M, Ψ) and φ ◦ f ∈ L, then both (4.1) and (4.2) hold.
The following corollary is useful in practice.
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Corollary 4.2. Let Ψ : I ⊆ R → R be a twice differentiable convex function
on I̊, f : E → I such that Ψ ◦ f , f ∈ L and A : L → R be an isotonic linear
and normalised functional.
00
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
Close
Quit
Page 15 of 30
00
(i) If φ : I → R is twice differentiable and φ (t) ≥ mΨ (t), t ∈I̊ (where m
is a given real number), then (4.1) holds, provided that φ ◦ f ∈ L.
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
(ii) If φ : I → R is twice differentiable and φ00 (t) ≤ M Ψ00 (t), t ∈I̊ (where M
is a given real number), then (4.2) holds, provided that φ ◦ f ∈ L.
(iii) If φ : I → R is twice differentiable and mΨ00 (t) ≤ φ00 (t) ≤ M Ψ00 (t),
t ∈I̊, then both (4.1) and (4.2) hold, provided φ ◦ f ∈ L.
In [5], S.S. Dragomir obtained the following result of Lupaş-Beesack-Pečarić
type for m − Ψ−convex and M − Ψ−convex functions.
Theorem 4.3. Let Ψ : [α, β] ⊂ R → R be a convex function and f : I → [α, β]
such that Ψ ◦ f , f ∈ L and A : L → R is an isotonic linear and normalised
functional.
(i) If φ ∈ L (I, m, Ψ) and φ ◦ f ∈ L, then we have the inequality
β − A (f )
A (f ) − α
(4.3) m
Ψ (α) +
Ψ (β) − A (Ψ ◦ f )
β−α
β−α
β − A (f )
A (f ) − α
≤
φ (α) +
φ (β) − A (φ ◦ f ) .
β−α
β−α
(ii) If φ ∈ U (I, M, Ψ) and φ ◦ f ∈ L, then
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
β − A (f )
A (f ) − α
(4.4)
φ (α) +
φ (β) − A (φ ◦ f )
β−α
β−α
A (f ) − α
β − A (f )
Ψ (α) +
Ψ (β) − A (Ψ ◦ f ) .
≤M
β−α
β−α
(iii) If φ ∈ B (I, m, M, Ψ) and φ ◦ f ∈ L, then both (4.3) and (4.4) hold.
Close
Quit
Page 16 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
The following corollary is useful in practice.
Corollary 4.4. Let Ψ : I ⊆ R → R be a twice differentiable convex function
on I̊, f : E → I such that Ψ ◦ f , f ∈ L and A : L → R is an isotonic linear
and normalised functional.
(i) If φ : I → R is twice differentiable, φ ◦ f ∈ L and φ00 (t) ≥ mΨ00 (t), t ∈I̊
(where m is a given real number), then (4.3) holds.
(ii) If φ : I → R is twice differentiable, φ ◦ f ∈ L and φ00 (t) ≤ M Ψ00 (t), t ∈I̊
(where m is a given real number), then (4.4) holds.
(iii) If mΨ00 (t) ≤ φ00 (t) ≤ M Ψ00 (t), t ∈I̊, then both (4.3) and (4.4) hold.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
We now prove the following new result.
Theorem 4.5. Let Ψ : I ⊆ R → R be differentiable convex function and f :
E → I such that Ψ ◦ f , Ψ0 ◦ f , Ψ0 ◦ f · f , f ∈ L and A : L → R be an isotonic
linear and normalised functional.
(i) If φ is differentiable, φ ∈ L I̊, m, Ψ and φ ◦ f , φ0 ◦ f , φ0 ◦ f · f ∈ L, then
we have the inequality
(4.5) m [A (Ψ0 ◦ f · f ) + Ψ (A (f )) − A (f ) · A (Ψ0 ◦ f ) − A (Ψ ◦ f )]
≤ A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f ) .
(ii) If φ is differentiable, φ ∈ U I̊, M, Ψ and φ ◦ f , φ0 ◦ f , φ0 ◦ f · f ∈ L, then
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
we have the inequality
(4.6) A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f )
≤ M [A (Ψ0 ◦ f · f ) + Ψ (A (f ))
−A (f ) · A (Ψ0 ◦ f ) − A (Ψ ◦ f )] .
(iii) If φ is differentiable, φ ∈ B I̊, m, M, Ψ and φ ◦ f , φ0 ◦ f , φ0 ◦ f · f ∈ L,
then both (4.5) and (4.6) hold.
Proof. The proof is as follows.
(i) As φ ∈ L (I, m, Ψ), then φ − mΨ is convex and we can apply the first part
of the inequality (3.1) for φ − mΨ getting
(4.7) A [(φ − mΨ) ◦ f ] − (φ − mΨ) (A (f ))
≤ A (φ − mΨ)0 ◦ f · f − A (f ) A (φ − mΨ)0 ◦ f .
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
However,
A [(φ − mΨ) ◦ f ] = A (φ ◦ f ) − mA (Ψ ◦ f ) ,
(φ − mΨ) (A (f )) = φ (A (f )) − mΨ (A (f )) ,
A (φ − mΨ)0 ◦ f · f = A (φ0 ◦ f · f ) − mA (Ψ0 ◦ f · f )
II
I
Go Back
Close
Quit
and
A (φ − mΨ)0 ◦ f = A (φ0 ◦ f ) − mA (Ψ0 ◦ f )
and then, by (4.7), we deduce the desired inequality (4.5).
Page 18 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
(ii) Goes likewise and we omit the details.
(iii) Follows by (i) and (ii).
The following corollary is useful in practice,
Corollary 4.6. Let Ψ : I ⊆ R → R be a twice differentiable convex function
on I̊, f : E → I such that Ψ ◦ f , Ψ0 ◦ f , Ψ0 ◦ f · f , f ∈ L and A : L → R be
an isotonic linear and normalised functional.
(i) If φ : I → R is twice differentiable, φ ◦ f , φ0 ◦ f , φ0 ◦ f · f ∈ L and φ00 (t) ≥
mΨ00 (t), t ∈I̊, (where m is a given real number), then the inequality (4.5)
holds.
(ii) With the same assumptions, but if φ00 (t) ≤ M Ψ00 (t), t ∈I̊, (where M is a
given real number), then the inequality (4.6) holds.
(iii) If mΨ00 (t) ≤ φ00 (t) ≤ M Ψ00 (t), t ∈I̊, then both (4.5) and (4.6) hold.
Some particular important cases of the above corollary are embodied in the
following proposition.
Proposition 4.7. Assume that the mapping φ : I ⊆ R → R is twice differentiable on I̊.
(i) If inf φ00 (t) = k > −∞, then we have the inequality
t∈I̊
1 (4.8)
k A f 2 − [A (f )]2
2
≤ A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f ) ,
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
provided that φ ◦ f , φ0 ◦ f , φ0 ◦ f · f, f 2 ∈ L.
(ii) If sup φ00 (t) = K < ∞, then we have the inequality
t∈I̊
(4.9) A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f )
1 ≤ K A f 2 − [A (f )]2 .
2
(iii) If −∞ < k ≤ φ00 (t) ≤ K < ∞, t ∈I̊, then both (4.8) and (4.9) hold.
The proof follows by Corollary 4.6 applied for Ψ (t) =
M = K.
Another result is the following one.
1 2
t
2
and m = k,
Proposition 4.8. Assume that the mapping φ : I ⊆ (0, ∞) → R is twice
differentiable on I̊. Let p ∈ (−∞, 0) ∪ (1, ∞) and define gp : I → R, gp (t) =
φ00 (t) t2−p .
(i) If inf gp (t) = γ > −∞, then we have the inequality
t∈I̊
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
γ
(4.10)
[(p − 1) [A (f p ) − [A (f )]p ]
p (p − 1)
−pA (f ) A f p−1 − [A (f )]p−1
≤ A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f ) ,
provided that φ ◦ f , φ0 ◦ f , φ0 ◦ f · f, f p , f p−1 ∈ L.
Close
Quit
Page 20 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
(ii) If sup gp (t) = Γ < ∞, then we have the inequality
t∈I̊
(4.11) A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f )
Γ
≤
[(p − 1) [A (f p ) − [A (f )]p ]
p (p − 1)
−pA (f ) A f p−1 − [A (f )]p−1 .
(iii) If −∞ < γ ≤ gp (t) ≤ Γ < ∞, t ∈I̊, then both (4.10) and (4.11) hold.
Proof. The proof is as follows.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
(i) We have for the auxiliary mapping hp (t) = φ (t) −
γ
tp
p(p−1)
that
h00p (t) = φ00 (t) − γtp−2 = tp−2 t2−p φ00 (t) − γ
= tp−2 (gp (t) − γ) ≥ 0.
γ
That is, hp is convex or, equivalently, φ ∈ L I, p(p−1)
, (·)p . Applying
Corollary 4.6, we get
γ
pA (f p ) + [A (f )]p − pA (f ) A f p−1 − A (f p )
p (p − 1)
≤ A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f ) ,
which is clearly equivalent to (4.10).
(ii) Goes similarly.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
(iii) Follows by (i) and (ii).
The following proposition also holds.
Proposition 4.9. Assume that the mapping φ : I ⊆ (0, ∞) → R is twice
differentiable on I̊. Define l (t) = t2 φ00 (t), t ∈ I.
(i) If inf l (t) = s > −∞, then we have the inequality
t∈I̊
1
(4.12) s A (f ) A
− 1 − (ln [A (f )] − A [ln (f )])
f
≤ A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f ) ,
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
−1
provided that φ ◦ f, φ
−1
◦ f, φ
◦ f · f,
1
, ln f
f
∈ L and A (f ) > 0.
(ii) If sup l (t) = S < ∞, then we have the inequality
t∈I̊
(4.13) A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f )
1
≤ S A (f ) A
− 1 − (ln [A (f )] − A [ln (f )]) .
f
Contents
JJ
J
II
I
Go Back
Close
Quit
(iii) If −∞ < s ≤ l (t) ≤ S < ∞ for t ∈I̊, then both (4.12) and (4.13) hold.
Proof. The proof is as follows.
Page 22 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
(i) Define the auxiliary function h (t) = φ (t) + s ln t. Then
h00 (t) = φ00 (t) −
s
1
= 2 φ00 (t) t2 − s ≥ 0
2
t
t
which shows that h is convex, or, equivalently, φ ∈ L (I, s, − ln (·)). Applying Corollary 4.6, we may write
1
s −A (1) − ln A (f ) + A (f ) A
+ A (ln (f ))
f
≤ A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f ) ,
which is clearly equivalent to (4.12).
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
(ii) Goes similarly.
(iii) Follows by (i) and (ii).
Title Page
Contents
Finally, the following result also holds.
Proposition 4.10. Assume that the mapping φ : I ⊆ (0, ∞) → R is twice
differentiable on I̊. Define I˜ (t) = tφ00 (t), t ∈ I.
(i) If inf I˜ (t) = δ > −∞, then we have the inequality
t∈I̊
(4.14) δA (f ) [ln [A (f )] − A (ln (f ))]
≤ A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f ) ,
provided that φ ◦ f, φ0 ◦ f, φ0 ◦ f · f, ln f, f ∈ L and A (f ) > 0.
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
(ii) If sup I˜ (t) = ∆ < ∞, then we have the inequality
t∈I̊
(4.15) A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f )
≤ ∆A (f ) [ln [A (f )] − A (ln (f ))] .
(iii) If −∞ < δ ≤ I˜ (t) ≤ ∆ < ∞ for t ∈I̊, then both (4.14) and (4.15) hold.
Proof. The proof is as follows.
(i) Define the auxiliary mapping h (t) = φ (t) − δt ln t, t ∈ I. Then
i
1 h˜
δ
1 00
h (t) = φ (t) − = 2 [φ (t) t − δ] =
I (t) − δ ≥ 0
t
t
t
00
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
00
which shows that h is convex or equivalently, φ ∈ L (I, δ, (·) ln (·)). Applying Corollary 4.6, we get
δ [A [(ln f + 1) f ] + A (f ) ln A (f ) − A (f ) A (ln f + 1) − A (f ln f )]
≤ A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f )
Title Page
Contents
JJ
J
II
I
Go Back
which is equivalent with (4.14).
(ii) Goes similarly.
(iii) Follows by (i) and (ii).
Close
Quit
Page 24 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
5.
Some Applications For Bullen’s Inequality
The following inequality is well known in the literature as Bullen’s inequality
(see for example [7, p. 10])
Z b
1
1 φ (a) + φ (b)
a+b
(5.1)
φ (t) dt ≤
+φ
,
b−a a
2
2
2
provided that φ : [a, b] → R is a convex function on [a, b]. In other words, as
(5.1) is equivalent to:
Z b
Z b
1
a+b
φ (a) + φ (b)
1
φ (t) dt−φ
≤
−
φ (t) dt
(5.2) 0 ≤
b−a a
2
2
b−a a
we can conclude that in the Hermite-Hadamard inequality
Z b
φ (a) + φ (b)
1
a+b
(5.3)
≥
φ (t) dt ≥ φ
2
b−a a
2
Rb
1
the integral mean b−a
φ (t) dt is closer to φ 2 than to φ(a)+φ(b)
.
2
a
Using some of the results pointed out in the previous sections, we may upper
and lower bound the Bullen difference:
Z b
1 φ (a) + φ (b)
a+b
1
+φ
−
φ (t) dt
B (φ; a, b) :=
2
2
2
b−a a
a+b
(which is positive for convex functions) for different classes of twice differentiable functions φ.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 25 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
Rb
1
Now, if we assume that A (f ) := b−a
f (t) dt, then for f = e, e (x) = x,
a
x ∈ [a, b], we have, for a differentiable function φ, that
A (φ0 ◦ f · f ) + φ (A (f )) − A (f ) · A (φ0 ◦ f ) − A (φ ◦ f )
Z b
a+b
1
0
=
xφ (x) dx + φ
b−a a
2
Z b
Z b
a+b
1
1
0
−
·
φ (x) dx −
φ (x) dx
2
b−a a
b−a a
Z b
1
a+b
=
bφ (b) − aφ (a) −
φ (x) dx + φ
b−a
2
a
Z b
1
a + b φ (b) − φ (a)
−
·
−
φ (x) dx
2
b−a
b−a a
Z b
φ (a) + φ (b)
a+b
2
+φ
−
φ (x) dx
=
2
2
b−a a
= 2B (φ; a, b) .
a) Assume that φ : [a, b] ⊂ R → R is a twice differentiable function satisfying
the property that −∞ < k ≤ φ00 (t) ≤ K < ∞. Then by Proposition 4.7, we
may state the inequality
1
1
(b − a)2 k ≤ B (φ; a, b) ≤
(b − a)2 K.
48
48
This follows by Proposition 4.7 on taking into account that
2
Z b
Z b
1
1
(b − a)2
2
x dx −
xdx =
.
b−a a
b−a a
12
(5.4)
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
b) Now, assume that the twice differentiable function φ : [a, b] ⊂ (0, ∞) → R
satisfies the property that −∞ < γ ≤ t2−p φ00 (t) ≤ Γ < ∞, t ∈ (a, b), p ∈
(−∞, 0) ∪ (1, ∞). Then by Proposition 4.8 and taking into account that
p
p
A (f ) − (A (f ))
p
Z b
Z b
1
1
p
=
x dx −
xdx
b−a a
b−a a
= Lpp (a, b) − Ap (a, b) ,
and
p−1
A f p−1 − (A (f ))p−1 = Lp−1
(a, b) ,
p−1 (a, b) − A
we may state the inequality
(5.5)
γ
(p − 1) Lpp (a, b) − Ap (a, b)
p (p − 1)
p−1
−pA (a, b) Lp−1
(a, b)
p−1 (a, b) − A
≤ B (φ; a, b)
Γ
≤
(p − 1) Lpp (a, b) − Ap (a, b)
p (p − 1)
p−1
−pA (a, b) Lp−1
(a, b) .
p−1 (a, b) − A
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
c) Assume that the twice differentiable function φ : [a, b] ⊂ (0, ∞) → R
satisfies the property that −∞ < s ≤ t2 φ00 (t) ≤ S < ∞, t ∈ (a, b), then by
Quit
Page 27 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
Proposition 4.9, and taking into account that
A (f ) A f −1 − 1 − ln [A (f )] + A ln (f )
A (a, b)
=
− 1 − ln A (a, b) + I (a, b)
L (a, b)
I (a, b)
A (a, b) − L (a, b)
= ln
· exp
,
A (a, b)
L (a, b)
we get the inequality
I (a, b)
A (a, b) − L (a, b)
s
(5.6)
ln
· exp
2
A (a, b)
L (a, b)
≤ B (φ; a, b)
I (a, b)
A (a, b) − L (a, b)
S
≤ ln
· exp
.
2
A (a, b)
L (a, b)
00
d) Finally, if φ satisfies the condition −∞ < δ ≤ tφ (t) ≤ ∆ < ∞, then by
Proposition 4.10, we may state the inequality
A (a, b)
A (a, b)
(5.7)
δA (a, b) ln
≤ B (φ; a, b) ≤ ∆A (a, b) ln
.
I (a, b)
I (a, b)
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 28 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
References
[1] D. ANDRICA AND C. BADEA, Grüss’ inequality for positive linear functionals, Periodica Math. Hung., 19 (1998), 155–167.
[2] P.R. BEESACK AND J.E. PEČARIĆ, On Jessen’s inequality for convex
functions, J. Math. Anal. Appl., 110 (1985), 536–552.
[3] S.S. DRAGOMIR, A refinement of Hadamard’s inequality for isotonic linear functionals, Tamkang J. Math (Taiwan), 24 (1992), 101–106.
[4] S.S. DRAGOMIR, On the Jessen’s inequality for isotonic linear functionals, submitted.
[5] S.S. DRAGOMIR, On the Lupaş-Beesack-Pečarić inequality for isotonic
linear functionals, Nonlinear Functional Analysis and Applications, in
press.
[6] S.S. DRAGOMIR AND N.M. IONESCU, On some inequalities for
convex-dominated functions, L’Anal. Num. Théor. L’Approx., 19(1)
(1990), 21–27.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
[7] S.S. DRAGOMIR AND C.E.M. PEARCE, Selected Topics
on
Hermite-Hadamard
Inequalities
and
Applications,
RGMIA
Monographs,
Victoria
University,
2000.
http://rgmia.vu.edu.au/monographs.html
Go Back
[8] S.S. DRAGOMIR, C.E.M. PEARCE AND J.E. PEČARIĆ, On Jessen’s
and related inequalities for isotonic sublinear functionals, Acta. Sci. Math.
(Szeged), 61 (1995), 373–382.
Page 29 of 30
Close
Quit
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au
[9] A. LUPAŞ, A generalisation of Hadamard’s inequalities for convex functions, Univ. Beograd. Elek. Fak., 577–579 (1976), 115–121.
[10] J.E. PEČARIĆ, On Jessen’s inequality for convex functions (III), J. Math.
Anal. Appl., 156 (1991), 231–239.
[11] J.E. PEČARIĆ AND P.R. BEESACK, On Jessen’s inequality for convex
functions (II), J. Math. Anal. Appl., 156 (1991), 231–239.
[12] J.E. PEČARIĆ AND S.S. DRAGOMIR, A generalisation of Hadamard’s
inequality for isotonic linear functionals, Radovi Mat. (Sarajevo), 7 (1991),
103–107.
[13] J.E. PEČARIĆ AND I. RAŞA, On Jessen’s inequality, Acta. Sci. Math.
(Szeged), 56 (1992), 305–309.
[14] G. TOADER AND S.S. DRAGOMIR, Refinement of Jessen’s inequality,
Demonstratio Mathematica, 28 (1995), 329–334.
On a Reverse of Jessen’s
Inequality for Isotonic Linear
Functionals
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 30 of 30
J. Ineq. Pure and Appl. Math. 2(3) Art. 36, 2001
http://jipam.vu.edu.au