Journal of Inequalities in Pure and
Applied Mathematics
REFINEMENTS OF THE HERMITE-HADAMARD INEQUALITY
FOR CONVEX FUNCTIONS
volume 6, issue 5, article 140,
2005.
S.S. DRAGOMIR AND A. McANDREW
School of Computer Science and Mathematics
Victoria University
PO Box 14428, MCMC 8001
VIC, Australia.
EMail: sever@csm.vu.edu.au
URL: http://rgmia.vu.edu.au/dragomir/
EMail: Alasdair.Mcandrew@vu.edu.au
URL: http://sci.vu.edu.au/˜amca/
Received 06 April, 2005;
accepted 20 September, 2005.
Communicated by: A. Lupaş
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
106-05
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Abstract
New refinements for the celebrated Hermite-Hadamard inequality for convex
functions are obtained. Applications for special means are pointed out as well.
2000 Mathematics Subject Classification: Primary 26D15; Secondary 26D10.
Key words: Hermite-Hadamard Inequality.
This paper is based on the talk given by the second author within the “International
Conference of Mathematical Inequalities and their Applications, I”, December 0608, 2004, Victoria University, Melbourne, Australia [http://rgmia.vu.edu.au/
conference]
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Applications for Special Means . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
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1.
Introduction
The following result is well known in the literature as the Hermite-Hadamard
integral inequality:
Z b
a+b
1
f (a) + f (b)
(1.1)
f
≤
f (t) dt ≤
,
2
b−a a
2
provided that f : [a, b] → R is a convex function on [a, b] .
The following refinements of the H· − H· inequality were obtained in [2]
1
(1.2)
b−a
b
a+b
f (x) dx − f
2
a
Z b
f (x) + f (a + b − x) 1
a + b ≥
dx − f
≥ 0.
b−a a 2
2
Z
and
b
f (a) + f (b)
1
−
f (t) dt
2
b−a a
 Rb
1

|f
(a)|
−
|f
(x)|
dx
if f (a) = f (b)

b−a a

≥

R f (b)
Rb

1
1
 |x|
dx
−
|f
(x)|
dx
if f (a) 6= f (b)
f (b)−f (a) f (a)
b−a a
Z
(1.3)
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
for the general case of convex functions f : [a, b] → R.
S.S. Dragomir and A. McAndrew
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If one would assume differentiability of f on (a, b) , then the following
bounds in terms of its derivative holds (see [3, pp. 30-31])
Z b
f (a) + f (b)
1
(1.4)
−
f (t) dt ≥ max {|A| , |B| , |C|} ≥ 0
2
b−a a
where
Z b
Z b
x − a + b |f 0 (x)| dx − 1
|f 0 (x)| dx,
2
4
a
a
"Z a+b
#
Z b
2
1
f (b) − f (a)
f (x) dx −
f (x) dx
−
B :=
a+b
4
b−a a
2
1
A :=
b−a
and
1
C :=
b−a
Z b
a
a+b
x−
2
S.S. Dragomir and A. McAndrew
Title Page
|f 0 (x)| dx.
Contents
A different approach considered in [1] led to the following lower bounds
Z b
f (a) + f (b)
1
−
f (t) dt ≥ max {|D| , |E| , |F |} ≥ 0,
(1.5)
2
b−a a
where
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Z
b
Z
b
Z
b
1
1
1
|xf 0 (x)| dx −
|f 0 (x)| dx ·
|x| dx,
b−a a
b−a a
b−a a
Z b
Z b
1
a+b
1
0
E :=
·
x |f (x)| dx −
|f 0 (x)| xdx
b−a a
2
b−a a
D :=
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
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and
1
F :=
b−a
Z
a
b
1
f (b) − f (a)
|x| f (x) dx −
·
b−a
b−a
0
Z
b
|x| dx.
a
For other results connected to the H· − H· inequality see the recent monograph on line [3].
In the present paper, we use a different method to obtain other refinements of
the H· − H· inequality. Applications for special means are pointed out as well.
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
Title Page
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2.
The Results
The following refinement of the Hermite-Hadamard inequality for differentiable
convex functions holds.
Theorem 2.1. Assume that f : [a, b] → R is differentiable convex on (a, b) .
Then one has the inequality:
Z b
a+b
1
(2.1)
f (t) dt − f
b−a a
2
Z b
1
0 a + b a
+
b
b
−
a
f (x) − f
dx −
≥ 0.
≥ · f
b−a a 2
4
2
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
Proof. Since f is differentiable convex on (a, b) , then for each x, y ∈ (a, b) one
has the inequality
S.S. Dragomir and A. McAndrew
f (x) − f (y) ≥ (x − y) f 0 (y) .
Title Page
(2.2)
Using the properties of modulus, we have
(2.3)
f (x) − f (y) − (x − y) f 0 (y) = |f (x) − f (y) − (x − y) f 0 (y)|
≥ ||f (x) − f (y)| − |x − y| |f 0 (y)||
for each x, y ∈ (a, b) .
If we choose y = a+b
in (2.3) we get
2
a+b
a+b
a+b
0
(2.4) f (x) − f
− x−
f
2
2
2
0 a + b a
+
b
a
+
b
− x −
f
≥ f (x) − f
2
2 2
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for any x ∈ (a, b) .
Integrating (2.4) on [a, b] , dividing by (b − a) and using the properties of
modulus, we have
Z b
Z b
a+b
1
a+b
1
a+b
0
f (x) dx − f
−f
·
x−
dx
b−a a
2
2
b−a a
2
Z b 0 a + b a
+
b
a
+
b
1
f (x) − f
− x −
f
dx
≥
b − a a 2
2 2
Z b
1
a + b ≥
f (x) − f
dx
b−a a 2
Z b
0 a+b 1
a
+
b
x −
dx
− f
2
b−a a
2 Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
Title Page
and since
Z b
(2.5)
a
a+b
x−
2
Z b
2
a
+
b
x −
dx = (b − a) ,
dx = 0,
2 4
a
we deduce by (2.5) the desired result (2.1).
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The second result is embodied in the following theorem.
Theorem 2.2. Assume that f : [a, b] → R is differentiable convex on (a, b) .
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Then one has the inequality
Z b
1 f (a) + f (b)
a+b
1
(2.6)
+f
−
f (x) dx
2
2
2
b−a a
Z b
a
+
b
1 1
f (x) − f
dx
≥ 2 b−a a
2
Z b
1
a + b 0
≥ 0.
−
x
−
|f
(x)|
dx
b−a a 2 Proof. We choose x =
(2.7)
f
a+b
2
a+b
2
in (2.3) to get
a+b
− f (y) −
− y f 0 (y)
2
a + b
0
a+b
≥ f
− f (y) − − y |f (y)| .
2
2
Integrating (2.7) over y, dividing by (b − a) and using the modulus properties,
we get
(2.8) f
a+b
2
Z b
Z b
1
a+b
−
f (y) dy −
− y f 0 (y) dy
b−a a
2
a
Z b 1
a
+
b
f
dy
≥ −
f
(y)
b−a a 2
Z b
a + b
0
1
−
− y |f (y)| dy .
b−a a
2
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
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Since
Z b
a
a+b
y−
2
f (a) + f (b)
f (y) dy =
(b − a) −
2
0
Z
b
f (t) dt,
a
then by (2.8) we deduce
f
a+b
2
Z b
f (a) + f (b)
2
+
−
f (y) dy
2
b−a a
Z b
1
a + b f (y) − f
≥
dy
b−a a 2
Z b
0
a
+
b
1
y −
|f (y)| dy −
b−a a
2
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
Title Page
which is clearly equivalent to (2.6).
The following result holding for the subclass of monotonic and convex functions is whort to mention.
Theorem 2.3. Assume that f : [a, b] → R is monotonic and convex on (a, b) .
Then we have:
Z b
1 f (a) + f (b)
a+b
1
(2.9)
+f
−
f (x) dx
2
2
2
b−a a
Z b
1
1
a
+
b
≥ [f (b) − f (a)] +
sgn
− x f (x) dx .
4
b−a a
2
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Proof. Since the class of differentiable convex functions in (a, b) is dense in
uniform topology in the class of all convex functions defined on (a, b) , we may
assume, without loss of generality, that f is differentiable convex and monotonic
on (a, b) .
Firstly, assume that f is monotonic nondecreasing on [a, b] . Then
Z b
Z
f (x) − f a + b dx =
2
a
a+b
2
a
Z
b
=
a+b
f
− f (x) dx
2
Z b a+b
dx
+
f (x) − f
a+b
2
2
Z a+b
2
f (x) dx −
f (x) dx,
a+b
2
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
a
Title Page
Z b
0
a
+
b
x −
|f (x)| dx
2 a
Z a+b Z b 2
a+b
a+b
0
=
− x f (x) dx +
x−
f 0 (x) dx
a+b
2
2
a
2
a+b Z a+b
2
2
a+b
=
− x f (x)
+
f (x) dx
2
a
a
b
Z b
a+b
f (x)
−
+ x−
f (x) dx
a+b
2
a+b
2
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b−a
=−
f (a) +
2
Z
a
a+b
2
b−a
f (x) dx +
f (b) −
2
Z
b
f (x) dx.
a+b
2
Using (2.6) we have
Z b
1 f (a) + f (b)
a+b
1
+f
−
f (x) dx
2
2
2
b−a a
Z
Z a+b
b
2
1
f (x) dx −
f (x) dx
≥
2 (b − a) a+b
a
2
#
"
Z b
Z a+b
2
b−a
b−a
f (x) dx −
f (x) dx −
f (b) −
f (a) +
a+b
2
2
a
2
Z
Z a+b
b
2
b−a
1
f (x) dx − 2
f (x) dx −
[f (b) − f (a)] ,
=
2
2
2 (b − a) a+b
a
2
which is clearly equivalent to (2.9).
A similar argument may be done if f is monotonic nonincreasing and we
omit the details.
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
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3.
Applications for Special Means
Let us recall the following means:
a) The arithmetic mean
A (a, b) :=
b) The geometric mean
a+b
, a, b > 0,
2
√
G (a, b) :=
ab; a, b ≥ 0,
c) The harmonic mean
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
H (a, b) :=
1
a
2
; a, b > 0,
+ 1b
d) The identric mean
Contents
 1

1 bb b−a


if b 6= a
e aa
I (a, b) :=
; a, b > 0



a
if b = a
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e) The logarithmic mean
L (a, b) :=
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


b−a
if b 6= a
ln b − ln a
; a, b > 0


a
if b = a
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f) The p−logarithmic mean
 1

bp+1 −ap+1 p
 (p+1)(b−a)
if b 6= a, p ∈ R\ {−1, 0}
Lp (a, b) :=
; a, b > 0.


a
if b = a
It is well known that, if, on denoting L−1 (a, b) := L (a, b) and L0 (a, b) :=
I (a, b) , then the function R 3 p → Lp (a, b) is strictly monotonic increasing
and, in particular, the following classical inequalities are valid
(3.1)
min {a, b} ≤ H (a, b) ≤ G (a, b)
≤ L (a, b) ≤ I (a, b) ≤ A (a, b) ≤ max {a, b}
for any a, b > 0.
The following proposition holds:
Proposition 3.1. Let 0 < a < b < ∞. Then we have the following refinement
for the inequality A ≥ L :
" #
2
2
AL
G
G
(3.2)
A−L≥
− ln
− 1 ≥ 0.
b−a
A
A
The proof follows by Theorem 2.1 on choosing f : [a, b] → (0, ∞), f (t) =
1/t and we omit the details.
The following proposition contains a refinementof the following well known
inequality
1 −1
A + H −1 ≥ L−1 .
2
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
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Proposition 3.2. With the above assumption for a and b we have
" #
2
2
1 −1
A
A
1
(3.3)
A + H −1 − L−1 ≥
− ln
− 1 ≥ 0.
2
b−a
G
G
The proof follows by Theorem 2.3 for the same funcion f : [a, b] → (0, ∞),
f (t) = 1/t, which is monotonic and convex on [a, b] , and the details are omitted.
One may state other similar results that improve classical inequalities for
means by choosing appropriate convex functions f. However, they will not be
stated below.
Refinements of the
Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
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References
[1] S.S. DRAGOMIR AND S. MABZELA, Some error estimates in the trapezoidal quadrature rule, Tamsui Oxford J. of Math. Sci., 16(2) (2000), 259–
272.
[2] S.S. DRAGOMIR, Refinements of the Hermite-Hadamard inequality for
convex functions, Tamsui Oxford J. of Math. Sci., 17(2) (2001), 131–137.
[3] S.S. DRAGOMIR AND C.E.M. PEARCE, Selected Topics on HermiteHadamard Type Inequalities and Applications, RGMIA, Monographs,
2000.
[ONLINE:
http://rgmia.vu.edu.au/monographs.
html].
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Hermite-Hadamard Inequality
for Convex Functions
S.S. Dragomir and A. McAndrew
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