HADAMARD TYPE INEQUALITIES FOR
m-CONVEX AND (α, m)-CONVEX FUNCTIONS
M. KLARIČIĆ BAKULA
M. E. ÖZDEMIR
Department of Mathematics
Faculty of Science
University of Split
Teslina 12, 21000 Split, Croatia
EMail: milica@pmfst.hr
Atatürk University
K. K. Education Faculty
Department of Mathematics
25240 Kampüs, Erzurum, Turkey
EMail: emos@atauni.edu.tr
J. PEČARIĆ
Faculty of Textile Technology
University of Zagreb
Pierottijeva 6, 10000 Zagreb, Croatia
EMail: pecaric@hazu.hr
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 1 of 25
Go Back
Received:
03 March, 2008
Accepted:
31 July, 2008
Communicated by:
E. Neuman
2000 AMS Sub. Class.:
26D15, 26A51.
Key words:
m-convex functions, (α, m)-convex functions, Hadamard’s inequalities
Full Screen
Close
Abstract:
In this paper we establish several Hadamard type inequalities for differentiable m-convex and (α, m)-convex functions. We also establish
Hadamard type inequalities for products of two m-convex or (α, m)convex functions. Our results generalize some results of B.G. Pachpatte
as well as some results of C.E.M. Pearce and J. Pečarić.
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 2 of 25
Go Back
Full Screen
Close
Contents
1
Introduction
4
2
Inequalities for m-Convex Functions
9
3
Inequalities for (α, m)-Convex Functions
18
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 3 of 25
Go Back
Full Screen
Close
1.
Introduction
The following definitions are well known in literature.
Let [0, b] , where b is greater than 0, be an interval of the real line R, and let K (b)
denote the class of all functions f : [0, b] → R which are continuous and nonnegative
on [0, b] and such that f (0) = 0.
We say that the function f is convex on [0, b] if
f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f (y)
holds for all x, y ∈ [0, b] and t ∈ [0, 1] . Let KC (b) denote the class of all functions
f ∈ K (b) convex on [0, b] , and let KF (b) be the class of all functions f ∈ K (b)
convex in mean on [0, b] , that is, the class of all functions f ∈ K (b) for which
F ∈ KC (b) , where the mean function F of the function f ∈ K (b) is defined by
( 1 Rx
f (t) dt, x ∈ (0, b] ;
x 0
F (x) =
0,
x = 0.
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Let KS (b) denote the class of all functions f ∈ K (b) which are starshaped with
respect to the origin on [0, b] , that is, the class of all functions f with the property
that
f (tx) ≤ tf (x)
Page 4 of 25
holds for all x ∈ [0, b] and t ∈ [0, 1] . In [1] Bruckner and Ostrow, among others,
proved that
KC (b) ⊂ KF (b) ⊂ KS (b) .
Close
In [9] G. Toader defined m-convexity: another intermediate between the usual
convexity and starshaped convexity.
Go Back
Full Screen
Definition 1.1. The function f : [0, b] → R, b > 0, is said to be m-convex, where
m ∈ [0, 1] , if we have
f (tx + m (1 − t) y) ≤ tf (x) + m (1 − t) f (y)
for all x, y ∈ [0, b] and t ∈ [0, 1] . We say that f is m-concave if −f is m-convex.
Denote by Km (b) the class of all m-convex functions on [0, b] for which f (0) ≤ 0.
Obviously, for m = 1 Definition 1.1 recaptures the concept of standard convex
functions on [0, b] , and for m = 0 the concept of starshaped functions.
The following lemmas hold (see [10]).
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Lemma A. If f is in the class Km (b) , then it is starshaped.
Lemma B. If f is in the class Km (b) and 0 < n < m ≤ 1, then f is in the class
Kn (b).
From Lemma A and Lemma B it follows that
K1 (b) ⊂ Km (b) ⊂ K0 (b) ,
whenever m ∈ (0, 1) . Note that in the class K1 (b) are only convex functions f :
[0, b] → R for which f (0) ≤ 0, that is, K1 (b) is a proper subclass of the class of
convex functions on [0, b] .
It is interesting to point out that for any m ∈ (0, 1) there are continuous and differentiable functions which are m-convex, but which are not convex in the standard
sense (see [11]).
In [3] S.S. Dragomir and G. Toader proved the following Hadamard type inequality for m-convex functions.
Title Page
Contents
JJ
II
J
I
Page 5 of 25
Go Back
Full Screen
Close
Theorem A. Let f : [0, ∞) → R be an m-convex function with m ∈ (0, 1] . If
0 ≤ a < b < ∞ and f ∈ L1 ([a, b]) then
(
)
Z b
a
f (a) + mf mb f (b) + mf m
1
.
(1.1)
f (x)dx ≤ min
,
b−a a
2
2
Some generalizations of this result can be found in [4].
The notion of m-convexity has been further generalized in [5] as it is stated in the
following definition:
Definition 1.2. The function f : [0, b] → R, b > 0, is said to be (α, m)-convex,
where (α, m) ∈ [0, 1]2 , if we have
f (tx + m (1 − t) y) ≤ tα f (x) + m (1 − tα ) f (y)
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
for all x, y ∈ [0, b] and t ∈ [0, 1] .
α
Denote by Km
(b) the class of all (α, m)-convex functions on [0, b] for which
f (0) ≤ 0.
It can be easily seen that for (α, m) ∈ {(0, 0) , (α, 0) , (1, 0) , (1, m) , (1, 1) , (α, 1)}
one obtains the following classes of functions: increasing, α-starshaped, starshaped,
m-convex, convex and α-convex functions respectively. Note that in the class K11 (b)
are only convex functions f : [0, b] → R for which f (0) ≤ 0, that is K11 (b) is a
proper subclass of the class of all convex functions on [0, b] . The interested reader
can find more about partial ordering of convexity in [8, p. 8, 280].
In [2] in order to prove some inequalities related to Hadamard’s inequality S. S.
Dragomir and R. P. Agarwal used the following lemma.
JJ
II
J
I
Page 6 of 25
Go Back
Full Screen
Close
Lemma C. Let f : I → R, I ⊂ R, be a differentiable mapping on I̊, and a, b ∈ I,
where a < b. If f 0 ∈ L1 ([a, b]) , then
Z b
Z
f (a) + f (b)
1
b−a 1
−
(1.2)
f (x)dx =
(1 − 2t) f 0 (ta + (1 − t) b)dt.
2
b−a a
2
0
Here I̊ denotes the interior of I.
In [7], using the same Lemma C, C.E.M. Pearce and J. Pečarić proved the following theorem.
Theorem B. Let f : I → R, I ⊂ R, be a differentiable mapping on I 0 , and a, b ∈ I,
where a < b. If |f 0 |q is convex on [a, b] for some q ≥ 1, then
1
Z b
f (a) + f (b)
b − a |f 0 (a)|q + |f 0 (b)|q q
1
−
f (x)dx ≤
2
b−a a
4
2
and
1
Z b
b − a |f 0 (a)|q + |f 0 (b)|q q
f a + b − 1
f (x)dx ≤
.
2
b−a a
4
2
In [6] B. G. Pachpatte established two new Hadamard type inequalities for products of convex functions. They are given in the next theorem.
Theorem C. Let f, g : [a, b] → [0, ∞) be convex functions on [a, b] ⊂ R, where
a < b. Then
Z b
1
1
1
(1.3)
f (x) g (x) dx ≤ M (a, b) + N (a, b) ,
b−a a
3
6
where M (a, b) = f (a) g (a) + f (b) g (b) and N (a, b) = f (a) g (b) + f (b) g (a).
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 7 of 25
Go Back
Full Screen
Close
The main purpose of this paper is to establish new inequalities like those given
in Theorems A, B and C, but now for the classes of m-convex functions (Section 2)
and (α, m)-convex functions (Section 3).
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 8 of 25
Go Back
Full Screen
Close
2.
Inequalities for m-Convex Functions
Theorem 2.1. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R
be a differentiable function on I such that f 0 ∈ L1 ([a, b]) , where 0 ≤ a < b < ∞.
If |f 0 |q is m-convex on [a, b] for some fixed m ∈ (0, 1] and q ∈ [1, ∞) , then
Z b
f (a) + f (b)
1
−
f (x)dx
2
b−a a

 |f 0 (a)|q + m f 0
b−a
≤
min

2
4
b
m
q ! 1q
,
m f 0
a
m
!1 
q
+ |f 0 (b)|q q 
2
Proof. Suppose that q = 1. From Lemma C we have
Z b
f (a) + f (b)
1
(2.1)
−
f (x)dx
2
b−a a
Z
b−a 1
≤
|1 − 2t| |f 0 (ta + (1 − t) b)| dt.
2
0
Since |f 0 | is m-convex on [a, b] we know that for any t ∈ [0, 1]
0
b 0
|f (ta + (1 − t) b)| = f ta + m (1 − t)
m b 0
≤ t |f (a)| + m (1 − t) f 0
,
m Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
.
vol. 9, iss. 4, art. 96, 2008

Title Page
Contents
JJ
II
J
I
Page 9 of 25
Go Back
Full Screen
Close
hence
Z b
f (a) + f (b)
1
−
f
(x)dx
2
b−a a
Z 1
b−a
b 0
≤
|1 − 2t| t |f (a)| + m (1 − t) f 0
dt
2
m 0
Z b−a 1
b =
dt
t |1 − 2t| |f 0 (a)| + m (1 − t) |1 − 2t| f 0
2
m 0
(Z 1 2
b b−a
t (1 − 2t) |f 0 (a)| + m (1 − t) (1 − 2t) f 0
dt
=
m 2
0
)
Z 1
b t (2t − 1) |f 0 (a)| + m (1 − t) (2t − 1) f 0
dt
+
1
m 2
b−a
b 0
|f (a)| + m f 0
.
=
8
m Analogously we obtain
Z b
f (a) + f (b)
b − a 0 a 1
0
≤
−
f
(x)dx
m
f
+
|f
(b)|
,
2
b−a a
8
m
which completes the proof for this case.
Suppose now that q > 1. Using the well known Hölder inequality for q and
p = q/ (q − 1) we obtain
Z 1
|1 − 2t| |f 0 (ta + (1 − t) b)| dt
0
Z 1
1
1
=
|1 − 2t|1− q |1 − 2t| q |f 0 (ta + (1 − t) b)| dt
0
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 10 of 25
Go Back
Full Screen
Close
Z
≤
(2.2)
1
q−1
Z
q
|1 − 2t| dt
0
1
1q
|1 − 2t| |f 0 (ta + (1 − t) b)| dt .
q
0
0 q
Since |f | is m-convex on [a, b] we know that for every t ∈ [0, 1]
q
b q
q
0
0
(2.3)
|f (ta + (1 − t) b)| ≤ t |f (a)| + m (1 − t) f 0
,
m hence from (2.1) , (2.2) and (2.3) we have
Z b
f (a) + f (b)
1
−
f
(x)dx
2
b−a a
q−1
q 1q
Z 1
q Z 1
0
b
b−a
dt
≤
|1 − 2t| dt
|1 − 2t| f ta + m (1 − t)
m
2
0
0
q 1
Z 1
q−1
q
q
1
b−a
b q
0
≤
|1 − 2t| dt
|f (a)| + m f 0
2
4
m 0
1
!
q q
q
b − a m |f 0 (a)| + m f 0 mb =
4
2
which completes the proof.
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 11 of 25
Go Back
Full Screen
and analogously
Z b
b−a
f (a) + f (b)
1
≤
−
f
(x)dx
2
b−a a
4
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
m f 0
a
m
!1
q
+ |f 0 (b)|q q
2
Close
,
Theorem 2.2. Suppose that all the assumptions of Theorem 2.1 are satisfied. Then
Z b
a
+
b
1
f
−
f (x)dx
2
b−a a

! 1q 
0 b q ! 1q
0 a q
q
q


0
0
|f (a)| + m f m
m f m + |f (b)|
b−a
≤
min
,
.


4
2
2
Proof. Our starting point here is the identity (see [7, Theorem 2])
Z b
Z b
a+b
1
1
−
f (x)dx =
S (x) f 0 (x)dx,
f
2
b−a a
b−a a
where

 x − a, x ∈ a, a+b
;
2
S (x) =
 x − b, x ∈ a+b , b .
2
We have
Z b
a
+
b
1
f
−
f
(x)dx
2
b−a a
"Z a+b
#
Z b
2
1
(b − x) |f 0 (x)| dx
≤
(x − a) |f 0 (x)| dx +
a+b
b−a a
2
"Z 1
#
Z 1
2
= (b − a)
t |f 0 (ta + (1 − t) b)| dt +
(1 − t) |f 0 (ta + (1 − t) b)| dt
0
1
2
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 12 of 25
Go Back
Full Screen
Close
"Z
≤ (b − a)
1
2
0
b t t |f (a)| + m (1 − t) f 0
dt
m 0
#
b +
(1 − t) t |f 0 (a)| + m (1 − t) f 0
dt
1
m
2
b−a
b 0
=
|f (a)| + m f 0
,
8
m Z
1
and analogously
Z b
b − a 0 a 0
f a + b − 1
≤
f
(x)dx
m
f
+
|f
(b)|
.
2
b−a a
8
m
This completes the proof for the case q = 1.
An argument similar to the one used in the proof of Theorem 2.1 gives the proof
for the case q ∈ (1, ∞) .
As a special case of Theorem 2.1 for m = 1, that is for |f 0 |q convex on [a, b] , we
obtain the first inequality in Theorem B. Similarly, as a special case of Theorem 2.2
we obtain the second inequality in Theorem B.
Theorem 2.3. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R
be a differentiable function on I such that f 0 ∈ L1 ([a, b]) , where 0 ≤ a < b < ∞.
If |f 0 |q is m-convex on [a, b] for some fixed m ∈ (0, 1] and q ∈ (1, ∞) , then
q−1
1
Z b
1
q
f (a) + f (b)
b−a q−1
1
q
q
−
f (x)dx ≤
µ1 + µ2
2
b−a a
4
2q − 1
1
1
b−a
q
q
µ1 + µ2 ,
(2.4)
≤
4
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 13 of 25
Go Back
Full Screen
Close
where
(
µ1 = min
(
µ2 = min
|f 0 (a)|q + m f 0
2
a+b
2m
|f 0 (b)|q + m f 0
2
a+b
2m
q 0
f
,
a+b
2
q 0
f
,
a+b
2
q
+ m f 0
a
m
q )
b
m
q )
2
q
+ m f 0
2
,
.
Proof. If |f 0 |q is m-convex from Theorem A we have
Z 1
q
|f 0 (ta + (1 − t) b)| dt ≤ µ1 ,
2
Z
1
2
1
2
2
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
0
q
|f (ta + (1 − t) b)| dt ≤ µ2 ,
Contents
0
hence
Z b
f (a) + f (b)
1
−
f
(x)dx
2
b−a a
Z 1
b−a
≤
|1 − 2t| |f 0 (ta + (1 − t))| dt
2
#
"0Z 1
Z 1
2
b−a
(1 − 2t) |f 0 (ta + (1 − t) b)| dt +
(2t − 1) |f 0 (ta + (1 − t) b)| dt .
=
1
2
0
2
Using Hölder’s inequality for q ∈ (1, ∞) and p = q/ (q − 1) we obtain
Z b
f (a) + f (b)
1
−
f
(x)dx
2
b−a a
JJ
II
J
I
Page 14 of 25
Go Back
Full Screen
Close

b−a
≤
2
1
2
Z
(1 − 2t)
! q−1
q
q
q−1
Z
dt
0
1
! q−1
q
q
1
2
b−a
4
q
|f 0 (ta + (1 − t) b)| dt
q−1
2q − 1
! 1q 
1
Z
q
|f 0 (ta + (1 − t) b)| dt
(2t − 1) q−1 dt
+
≤
! 1q
0
Z
1
2
1
2
q−1
q

Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
1
1
µ1q + µ2q
,
vol. 9, iss. 4, art. 96, 2008
since
Z
1
2
(1 − 2t)
q
q−1
1
Z
q
(2t − 1) q−1 dt =
dt =
1
2
0
q−1
.
2 (2q − 1)
This completes the proof of the first inequality in (2.4) . The second inequality in
(2.4) follows from the fact
1
<
2
q−1
2q − 1
q−1
q
< 1,
q ∈ (1, ∞) .
Title Page
Contents
JJ
II
J
I
Page 15 of 25
Go Back
Theorem 2.4. Let f, g : [0, ∞) → [0, ∞) be such that f g is in L1 ([a, b]), where
0 ≤ a < b < ∞. If f is m1 -convex and g is m2 -convex on [a, b] for some fixed
m1 , m2 ∈ (0, 1] , then
Z b
1
f (x)g (x) dx ≤ min {M1 , M2 } ,
b−a a
Full Screen
Close
where
1
b
b
M1 =
f (a) g (a) + m1 m2 f
g
3
m1
m2
1
b
b
+
+ m1 f
g (a) ,
m2 f (a) g
6
m2
m1
a
a
1
M2 =
f (b) g (b) + m1 m2 f
g
3
m1
m
2
a
a
1
+
m2 f (b) g
+ m1 f
g (b) .
6
m2
m1
Proof. We have
b
b
f ta + m1 (1 − t)
≤ tf (a) + m1 (1 − t) f
,
m1
m1
b
b
g ta + m2 (1 − t)
≤ tg(a) + m2 (1 − t) g
,
m2
m2
for all t ∈ [0, 1] . f and g are nonnegative, hence
b
b
f ta + m1 (1 − t)
g ta + m2 (1 − t)
m1
m
2
b
b
2
≤ t f (a) g(a) + m2 t (1 − t) f (a) g
+ m1 t (1 − t) f
g(a)
m2
m1
b
b
2
+ m1 m2 (1 − t) f
g
.
m1
m2
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 16 of 25
Go Back
Full Screen
Close
Integrating both sides of the above inequality over [0, 1] we obtain
Z 1
f (ta + (1 − t) b)g (ta + (1 − t) b) dt
0
Z b
1
=
f (x)g (x) dx
b−a a
b
b
1
≤
f (a) g (a) + m1 m2 f
g
3
m1
m2
b
b
1
+
m2 f (a) g
+ m1 f
g (a) .
6
m2
m1
Analogously we obtain
Z b
1
a
a
1
f (x)g (x) dx ≤
f (b) g (b) + m1 m2 f
g
b−a a
3
m
m2
1
1
a
a
+
m2 f (b) g
+ m1 f
g (b) ,
6
m2
m1
hence
1
b−a
Z
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 17 of 25
b
f (x)g (x) dx ≤ min {M1 , M2 } .
a
Go Back
Full Screen
Remark 1. If in Theorem 2.4 we choose a 1-convex (convex) function g : [0, ∞) →
[0, ∞) defined by g (x) = 1 for all x ∈ [0, ∞), we obtain
(
)
Z b
a
f (a) + mf mb f (b) + mf m
1
,
,
f (x) dx ≤ min
b−a a
2
2
which is (1.1) . If the functions f and g are 1-convex we obtain (1.3) .
Close
3.
Inequalities for (α, m)-Convex Functions
In this section on two examples we ilustrate how the same inequalities as in Section
2 can be obtained for the class of (α, m)-convex functions.
Theorem 3.1. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R
be a differentiable function on I such that f 0 ∈ L1 ([a, b]) , where 0 ≤ a < b < ∞.
If |f 0 |q is (α, m)-convex on [a, b] for some fixed α, m ∈ (0, 1] and q ∈ [1, ∞) , then
Z b
f (a) + f (b)
1
−
f
(x)dx
2
b−a a
(
q 1
q−1
b−a 1 q
b q
q
0
,
≤
· min
ν1 |f (a)| + ν2 m f 0
2
2
m a q 1 q
q
0
ν1 |f (b)| + ν2 m f 0
,
m
where
α 1
1
ν1 =
α+
,
(α + 1) (α + 2)
2
2
α 1
α +α+2
1
ν2 =
−
.
(α + 1) (α + 2)
2
2
Proof. Suppose that q = 1. From Lemma A we have
Z b
f (a) + f (b)
1
−
f (x)dx
(3.1)
2
b−a a
Z
b−a 1
≤
|1 − 2t| |f 0 (ta + (1 − t) b)| dt.
2
0
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 18 of 25
Go Back
Full Screen
Close
Since |f 0 | is (α, m)-convex on [a, b] we know that for any t ∈ [0, 1]
0
b
α
0
α
≤ t |f (a)| + m (1 − t ) f 0 b ,
f ta + m (1 − t)
m
m thus we have
Z b
f (a) + f (b)
1
−
f (x)dx
2
b−a a
Z 1
b b−a
α
0
α 0
≤
|1 − 2t| t |f (a)| + m (1 − t ) f
dt
2
m 0
Z b−a 1 α
b 0
α
=
t |1 − 2t| |f (a)| + m (1 − t ) |1 − 2t| f 0
dt.
2
m 0
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
We have
α 1
1
t |1 − 2t| dt =
α+
= ν1 ,
(α + 1) (α + 2)
2
0
2
α Z 1
1
α +α+2
1
α
−
= ν2 ,
(1 − t ) |1 − 2t| dt =
(α + 1) (α + 2)
2
2
0
hence
Z b
b−a
f (a) + f (b)
1
b 0
−
f (x)dx ≤
ν1 |f (a)| + ν2 m f 0
.
2
b−a a
2
m Z
1
α
Analogously we obtain
Z b
a b−a
f (a) + f (b)
1
0
−
f (x)dx ≤
ν1 |f (b)| + ν2 m f 0
,
2
b−a a
2
m
JJ
II
J
I
Page 19 of 25
Go Back
Full Screen
Close
which completes the proof for this case.
Suppose now that q ∈ (1, ∞) . Similarly to Theorem 2.1 we have
Z
1
(3.2)
|1 − 2t| |f 0 (ta + (1 − t) b)| dt
0
Z
≤
1
q−1
Z
q
|1 − 2t| dt
0
1
1q
|1 − 2t| |f (ta + (1 − t) b)| dt .
0
q
0
0 q
Since |f | is (α, m)-convex on [a, b] we know that for every t ∈ [0, 1]
q
q
0
b
q
α
0
α
f ta + m (1 − t)
≤ t |f (a)| + m (1 − t ) f 0 b ,
(3.3)
m m hence from (3.1) , (3.2) and (3.3) we obtain
Z b
f (a) + f (b)
1
−
f (x)dx
2
b−a a
Z 1
q−1
Z 1
q 1q
q
0
b−a
b
dt
≤
|1 − 2t| dt
|1 − 2t| f ta + m (1 − t)
2
m 0
0
q 1
q−1 b−a 1 q
b q
q
≤
ν1 |f 0 (a)| + ν2 m f 0
2
2
m and analogously
q−1
Z b
a q 1
f (a) + f (b)
b−a 1 q 1
q
q
0
−
f (x)dx ≤
ν1 |f (b)| + ν2 m f 0
,
2
b−a a
2
2
m
which completes the proof.
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 20 of 25
Go Back
Full Screen
Close
Observe that if in Theorem 3.1 we have α = 1 the statement of Theorem 3.1
becomes the statement of Theorem 2.1.
Theorem 3.2. Let f, g : [0, ∞) → [0, ∞) be such that f g is in L1 ([a, b]), where
0 ≤ a < b < ∞. If f is (α1 , m1 )-convex and g is (α2 , m2 )-convex on [a, b] for some
fixed α1 , m1 , α2 , m2 ∈ (0, 1] , then
Z b
1
f (x)g (x) dx ≤ min {N1 , N2 } ,
b−a a
where
1
1
b
f (a) g (a)
N1 =
+ m2
−
f (a) g
α1 + α2 + 1
α1 + 1 α1 + α2 + 1
m2
1
1
b
+ m1
−
f
g(a)
α2 + 1 α1 + α2 + 1
m1
1
1
1
b
b
+ m1 m2 1 −
−
+
f
g
,
α1 + 1 α2 + 1 α1 + α2 + 1
m1
m2
and
f (b) g (b)
1
1
a
N2 =
+ m2
−
f (b) g
α1 + α2 + 1
α1 + 1 α1 + α2 + 1
m2
1
1
a
+ m1
−
f
g(b)
α2 + 1 α1 + α2 + 1
m1
1
1
a
a
1
+ m1 m2 1 −
−
+
f
g
.
α1 + 1 α2 + 1 α1 + α2 + 1
m1
m2
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 21 of 25
Go Back
Full Screen
Close
Proof. Since f is (α1 , m1 )-convex and g is (α2 , m2 )-convex on [a, b] we have
b
b
α1
α1
f ta + m1 (1 − t)
≤ t f (a) + m1 (1 − t ) f
,
m1
m1
b
b
α2
α2
g ta + m2 (1 − t)
≤ t g(a) + m2 (1 − t ) g
,
m2
m2
for all t ∈ [0, 1] . The functions f and g are nonnegative, hence
f (ta + (1 − t) b)g(ta + (1 − t) b) ≤ tα1 +α2 f (a) g(a)
b
b
α1
α2
α2
α1
+ m2 t (1 − t ) f (a) g
+ m1 t (1 − t ) f
g(a)
m2
m1
b
b
α1
α2
+ m1 m2 (1 − t ) (1 − t ) f
g
.
m1
m2
Integrating both sides of the above inequality over [0, 1] we obtain
Z 1
f (ta + (1 − t) b)g (ta + (1 − t) b) dt
0
Z b
1
=
f (x)g (x) dx
b−a a
f (a) g (a)
1
1
b
≤
+ m2
−
f (a) g
α1 + α2 + 1
α1 + 1 α1 + α2 + 1
m2
1
b
1
+ m1
−
f
g(a)
α2 + 1 α1 + α2 + 1
m1
1
1
1
b
b
+ m1 m2 1 −
−
+
f
g
.
α1 + 1 α2 + 1 α1 + α2 + 1
m1
m2
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 22 of 25
Go Back
Full Screen
Close
Analogously we have
Z b
1
f (x)g (x) dx
b−a a
1
a
f (b) g (b)
1
+ m2
−
f (b) g
≤
α1 + α2 + 1
α1 + 1 α1 + α2 + 1
m2
1
1
a
+ m1
−
f
g(b)
α2 + 1 α1 + α2 + 1
m1
1
1
1
a
a
+ m1 m2 1 −
−
+
f
g
,
α1 + 1 α2 + 1 α1 + α2 + 1
m1
m2
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
which completes the proof.
Title Page
If in Theorem 3.2 we have α1 = α2 = 1, the statement of Theorem 3.2 becomes
the statement of Theorem 2.4.
Contents
JJ
II
J
I
Page 23 of 25
Go Back
Full Screen
Close
References
[1] A.M. BRUCKNER AND E. OSTROW, Some function classes related to the
class of convex functions, Pacific J. Math., 12 (1962), 1203–1215.
[2] S.S. DRAGOMIR AND R.P. AGARWAL, Two inequalities for differentiable
mappings and applications to special means of real numbers and trapezoidal
formula, Appl. Math. Lett., 11(5) (1998), 91–95.
[3] S.S. DRAGOMIR AND G. TOADER, Some inequalities for m-convex functions, Studia Univ. Babeş-Bolyai Math., 38(1) (1993), 21–28.
[4] M. KLARIČIĆ BAKULA, J. PEČARIĆ AND M. RIBIČIĆ, Companion inequalities to Jensen’s inequality for m-convex and (α, m)-convex functions,
J. Inequal. Pure & Appl. Math., 7(5) (2006), Art. 194. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=811].
[5] V.G. MIHEŞAN, A generalization of the convexity, Seminar on Functional
Equations, Approx. and Convex., Cluj-Napoca (Romania) (1993).
[6] B.G. PACHPATTE, On some inequalities for convex functions, RGMIA Res.
Rep.Coll., 6(E) (2003), [ONLINE: http://rgmia.vu.edu.au/v6(E)
.html].
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 24 of 25
Go Back
Full Screen
[7] C.E.M. PEARCE AND J. PEČARIĆ, Inequalities for differentiable mappings
with application to special means and quadrature formulae, Appl. Math. Lett.,
13(2) (2000), 51–55.
[8] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Partial
Orderings, and Statistical Applications, Academic Press, Inc. (1992).
Close
[9] G. TOADER, Some generalizations of the convexity, Proceedings of the Colloquium on Approximation and Optimization, Univ. Cluj-Napoca, Cluj-Napoca,
1985, 329-338.
[10] G. TOADER, On a generalization of the convexity, Mathematica, 30 (53)
(1988), 83-87.
[11] S. TOADER, The order of a star-convex function, Bull. Applied & Comp.
Math., 85-B (1998), BAM-1473, 347–350.
Inequalities for m-Convex and
(α, m)-Convex Functions
M. Klaričić Bakula, M. E. Özdemir
and J. Pečarić
vol. 9, iss. 4, art. 96, 2008
Title Page
Contents
JJ
II
J
I
Page 25 of 25
Go Back
Full Screen
Close