NEW OSTROWSKI TYPE INEQUALITIES FOR m

Hacettepe Journal of Mathematics and Statistics
Volume 40 (2) (2011), 135 – 145
NEW OSTROWSKI TYPE INEQUALITIES
FOR m-CONVEX FUNCTIONS
AND APPLICATIONS
Havva Kavurmacı∗†, M. Emin Özdemir∗ and Merve Avcı∗
Received 09 : 06 : 2010 : Accepted 21 : 11 : 2010
Abstract
In this paper we establish new inequalities of Ostrowski type, for functions whose derivatives in absolute value are m-convex. We also give
some applications to special means of positive real numbers. Finally,
we obtain some error estimates for the midpoint formula.
Keywords: m-convex function, Starshaped function, Convex function, Ostrowski inequality, Hermite-Hadamard inequality, Hölder inequality, Power Mean inequality, Special means, The midpoint formula, Lipschitzian mapping.
2010 AMS Classification: 26 A 51, 26 D 10, 26 D 15.
Communicated by Alex Goncharov
1. Introduction
Let f : I ⊂ [0, ∞) → R be a differentiable mapping on I ◦ , the interior of the interval
I, such that f ′ ∈ L ([a, b]) where a, b ∈ I with a < b. If |f ′ (x)| ≤ M , then the following
inequality holds (see [2]):
Z b
(x − a)2 + (b − x)2
f (x) − 1
≤ M
f
(u)du
.
b−a a
b−a
2
This inequality is well known in the literature as the Ostrowski inequality. For some
results which generalize, improve, and extend the above inequality, see [2, 5, 6, 8, 10],
and references therein.
In [14], G. Toader defined m-convexity, an intermediate between usual convexity and
the starshaped property, as the following:
∗Atatürk University, K. K. Education Faculty, Department of Mathematics, 25240 Campus, Erzurum, Turkey. E-mail: (H. Kavurmacı) havva.kvrmc@yahoo.com (M. E. Özdemir)
emos@atauni.edu.tr (M. Avcı) merveavci@ymail.com
†
Corresponding Author.
136
H. Kavurmacı, M. E. Özdemir, M. Avcı
1.1. Definition. 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].
Denote by Km (b) the set of m-convex functions on [0, b] for which f (0) ≤ 0.
1.2. Definition. The function f : [0, b] → R, b > 0, is said to be starshaped if for every
x ∈ [0, b] and t ∈ [0, 1] we have:
f (tx) ≤ tf (x).
For m = 1, we recapture the concept of convex functions defined on [0, b], and for
m = 0 the concept of starshaped functions on [0, b].
The following theorem contains the Hermite-Hadamard integral inequality (see [7]).
1.3. Theorem. Let f : I ⊂ R → R be an M-Lipschitzian mapping on I, and a, b ∈ I
with a < b. Then we have the inequality:
Z b
f a + b − 1
≤ M (b − a) .
f
(x)
dx
(1.1)
2
b−a a
4
In [13], E. Set, M. E. Özdemir and M.Z. Sarikaya established the following theorem.
1.4. Theorem. Let f : I ◦ ⊂ [0, b∗ ] → R, b∗ > 0, be a differentiable mapping on I ◦ ,
q
a, b ∈ I ◦ with a < b. If |f ′ | is m-convex on [a, b], q > 1 and m ∈ (0, 1], then the
following inequality holds:
(1.2)
where
b
m
Z b
f a + b − 1
f (x) dx ≤ (b − a)
2
b−a a
1
1− q
3
< b∗ .
8
!
′ 1
f (a) + m q f ′ b ,
m
In [11], U. Kirmaci proved the following theorem.
1.5. Theorem. Let f : I ◦ ⊂ R → R be a differentiable mapping on I ◦ , a, b ∈ I ◦ with
a < b. If the mapping |f ′ | is convex on [a, b], then we have
(1.3)
Z b
b − a ′ ′ f a + b − 1
f (x) dx ≤
f (a) + f (b) .
2
b−a a
8
S.S. Dragomir and G. Toader proved the following Hermite-Hadamard type inequality
for m-convex functions, see [9, p.7].
(
)
Z b
b
a
f (a) + mf m
f (b) + mf m
1
(1.4)
f (x) dx ≤ min
,
.
b−a a
2
2
Some generalizations of this result can be found in [4].
Inequalities for m-Convex Functions
137
In [3], M. K. Bakula, M. E. Özdemir and J. Pečarić proved the following theorems.
1.6. Theorem. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R be
q
a differentiable function on I such that f ′ ∈ L([a, b]), where 0 ≤ a < b < ∞. If |f ′ | is
m-convex on [a, b] for some fixed m ∈ (0, 1] and q ∈ [1, ∞), then
1
Z b
1
f (a) + f (b)
1
≤ b − a µ1q + µ2q ,
(1.5)
−
f
(x)
dx
2
b−a a
4
where
q q
)
q
a q
|f ′ (a)| + m f ′ ( a+b
) f ′ ( a+b
) + m f ′ ( m
)
2m
2
µ1 = min
,
,
2
2
(
q q
)
q
b q
|f ′ (b)| + m f ′ ( a+b
) f ′ ( a+b
) + m f ′ ( m
)
2m
2
,
.
µ2 = min
2
2
(
1.7. Theorem. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R be
q
a differentiable function on I such that f ′ ∈ L([a, b]), where 0 ≤ a < b < ∞. If |f ′ | is
m-convex on [a, b] for some fixed m ∈ (0, 1] and q ∈ [1, ∞), then
Z b
f a + b − 1
f
(x)
dx
2
b−a a

!1
!1
′ a q
(1.6)
 |f ′ (a)|q + m f ′ ( b )q q
f ( ) + |f ′ (b)|q q 
m
b−a
m
m
≤
min
,
. 

4
2
2
The main purpose of this paper is to establish new Ostrowski type inequalities for
functions whose derivatives in absolute value are m-convex. Using these results we give
some applications to special means of positive real numbers, and obtain some error estimates for the midpoint formula.
2. The results
In [1], in order to prove some inequalities related to the Ostrowski inequality, M. Alomari and M. Darus used essentially the following lemma, in which however the constant
(b − a) has been changed to (a − b) in the formulation of equality (2.1).
2.1. Lemma. Let f : I ⊂ R → R be a differentiable mapping on I ◦ , where a, b ∈ I with
a < b. If f ′ ∈ L ([a, b]), then the following equality holds:
Z b
Z 1
1
(2.1)
f (x) −
f (u) du = (a − b)
p(t)f ′ (ta + (1 − t)b) dt
b−a a
0
for each t ∈ [0, 1], where

i
h
t
if t ∈ 0, b−x
,
b−a
i
p(t) =
b−x
t − 1 if t ∈
,1 ,
b−a
for all x ∈ [a, b].
2.2. Theorem. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R be
a differentiable function on I such that f ′ ∈ L([a, b]), where 0 ≤ a < b < ∞. If |f ′ | is
138
H. Kavurmacı, M. E. Özdemir, M. Avcı
m-convex on [a, b] for some fixed m ∈ (0, 1], then the following inequality holds:
Z b
f (x) − 1
f
(u)du
b−a a
2
3 1
1
b−x
2
b−x
≤ (b − a) min
−
+
|f ′ (a)|
6
2
b−a
3
b−a
2
3
3 f ′
(2.2)
− 13 b−x
+ 31 x−a
+ m 12 b−x
b−a
b−a
b−a
2
3 1
1
b−x
2
b−x
−
+
|f ′ (b)|
6
2
b−a
3
b−a
2
3
3 1
b−x
1
x−a
f ′
+ m 12 b−x
−
+
b−a
3
b−a
3
b−a
b
m
,
a
m
for each x ∈ [a, b].
Proof. By Lemma 2.1, we have
Z b
f (x) − 1
f
(u)
du
b−a a
Z b−x
b−a
t f ′ (ta + (1 − t)b) dt
≤ (b − a)
0
+ (b − a)
Z
1
b−x
b−a
(1 − t) f ′ (ta + (1 − t)b) dt.
Since |f ′ | is m-convex on [a, b] we know that for any t ∈ [0, 1],
′
f (ta + (1 − t)b) = f ′ (ta + m(1 − t) b )
m b ≤ t f ′ (a) + m(1 − t) f ′
.
m Hence,
Z b
f (x) − 1
f
(u)
du
b−a a
b−x
Z b−a b ≤ (b − a)
t tf ′ (a) + m(1 − t) f ′
dt
m
0
Z 1
b (1 − t) tf ′ (a) + m(1 − t) f ′
dt
+ (b − a)
b−x
m b−a
2
3 1
f ′ (a)
= (b − a)
− 1 b−x + 2 b−x
6
2
b−a
+m
1
2
3
b−x
b−a
2
b−a
−
1
3
b−x
b−a
where we use the facts that
Z b−x
b−a
′ b t t f (a) + m(1 − t) f ′
dt
m
0
3 2
f ′ (a) + m 1 b−x −
= 13 b−x
b−a
2
b−a
3
1
3
+
1
3
b−x
b−a
x−a
b−a
3 f ′
3 f ′
b
m
,
b
m
,
Inequalities for m-Convex Functions
139
and
Z
1
′ b (1 − t) t f (a) + m(1 − t) f ′
dt
b−x
m b−a
2
3 3 1
b−x
f ′ (a) + m 1 x−a f ′
= 16 − 12 b−x
+
b−a
3
b−a
3
b−a
b
m
.
Analogously we obtain
Z b
f (x) − 1
f
(u)
du
b−a a
2
3 1
b−x
2
b−x
1
−
+
|f ′ (b)|
≤ (b − a)
6
2
b−a
3
b−a
2
3
3 a ′
1
b−x
1
b−x
1
x−a
+ m 2 b−a − 3 b−a + 3 b−a
f
,
m
and the proof is completed.
2.3. Remark. Suppose that all the assumptions of Theorem 2.2 are satisfied. If we
choose x = a+b
, then we have
2
Z b
a + b
1
f
−
f
(u)
du
2
b−a a
a ′ b−a
b ′ ≤
min f (a) + m f ′
, f (b) + m f ′
,
8
m
m
which is the inequality (1.6) with q = 1.
2.4. Remark. Suppose that all the assumptions of Theorem 2.2 are satisfied. Then
(A) If we choose m = 1 and x = a+b
, we obtain
2
Z b
a+b
1
b − a ′ ′ f
−
f (u) du ≤
f (a) + f (b) ,
2
b−a
8
a
which is the inequality (1.3).
(B) If in addition we choose f ′ (x) ≤ M , M > 0 in (A), then:
Z b
a + b
1
f
≤ M (b − a) ,
−
f
(u)
du
2
b−a
4
a
which is the inequality (1.1).
2.5. Theorem. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R be
p
a differentiable function on I such that f ′ ∈ L([a, b]), where 0 ≤ a < b < ∞. If |f ′ | p−1
is m-convex on [a, b] for some fixed m ∈ (0, 1] and p > 1, p1 + 1q = 1, then the following
inequality holds:
Z b
1
f (x) − 1
f (u) du ≤
1
b−a a
(p + 1) p
(
"
(
)# 1q
q
q
x q
b q
|f ′ (b)| + m f ′ m
|f ′ (x)| + m f ′ m
(b − x)2
×
min
,
(2.3)
b−a
2
2
"
(
)# q1 )
q
q
x q
a q
|f ′ (a)| + m f ′ m
|f ′ (x)| + m f ′ m
(x − a)2
min
,
+
b−a
2
2
for each x ∈ [a, b].
140
H. Kavurmacı, M. E. Özdemir, M. Avcı
Proof. From Lemma 2.1, and using the Hölder inequality, we have
Z b
f (x) − 1
f
(u)du
b−a a
Z b−x
b−a
≤ (b − a)
t f ′ (ta + (1 − t)b) dt
0
+ (b − a)
≤ (b − a)
Z
b−x
b−a
Z
tp dt
0
+ (b − a)
1
b−x
b−a
(1 − t) f ′ (ta + (1 − t)b) dt
1 Z
p
Z
1
b−x
b−a
b−x
b−a
0
′
f (ta + (1 − t)b)q dt
(1 − t)p dt
1 Z
p
1
b−x
b−a
1
q
′
1
f (ta + (1 − t)b)q dt q
p+1 1 1
p
b−x p
1
b−x q
b−a
p+1
b−a
′ x q
1
′
q
q
b q q
|f (b)| + m f m |f ′ (x)| + m f ′ m
× min
,
2
2
p+1 1 1
p
x−a p
1
x−a q
+ (b − a)
b−a
p+1
b−a
′ x q
′
q
q
a q 1
q
|f (a)| + m f m
|f ′ (x)| + m f ′ m
× min
,
2
2
1
1
=
1
(p + 1) p b − a
(
′
q
q
x q
b q 1
q
|f (b)| + m f ′ m
|f ′ (x)| + m f ′ m
× (b − x)2 min
,
2
2
)
′ x q
′
q
q
′
a q 1
q
|f (a)| + m f m |f (x)| + m f ′ m
2
+ (x − a) min
,
,
2
2
≤ (b − a)
where we use the facts that
p+1
p+1
Z b−x
Z 1
b−a
x−a
b−x
1
1
(1 − t)p dt =
tp dt =
,
,
b−x
b−a
p+1
b−a
p+1
0
b−a
and by (1.4) we get
Z b−x
q
b − a b−a ′
f (ta + (1 − t)b) dt
b−x 0
(
q
|f ′ (b)| + m f ′
≤ min
2
Z 1
′
b−a
f (ta + (1 − t)b)q dt
x − a b−x
b−a
(
q
|f ′ (a)| + m f ′
≤ min
2
The proof is completed.
x
m
q
|f ′ (x)|q + m f ′
,
2
x
m
q
q
|f ′ (x)| + m f ′
,
2
b
m
q )
a
m
,
q )
.
Inequalities for m-Convex Functions
141
2.6. Corollary. Suppose that all the assumptions of Theorem 2.5 are satisfied. If we
choose |f ′ (x)| ≤ M , M > 0, then we have
Z b
f (x) − 1
f
(u)du
b−a a
≤
1
1
(p + 1) p
!
1+m
2
1
q
M
(b − x)2 + (x − a)2
.
b−a
2.7. Corollary. Suppose that all the assumptions of Theorem 2.5 are satisfied. If we
1
p
1
1
choose x = a+b
and
<
< 1, then we have
2
2
p+1
where
1
Z b
1
b−a
q
q
f a + b − 1
f (u) du ≤
µ1 + µ2 ,
2
b−a a
4
q
q
)
|f ′ (b)| + m f ′ ( a+b
2m
,
2
(
q
q
|f ′ (a)| + m f ′ ( a+b
)
2m
,
µ2 = min
2
µ1 = min
(
′ a+b q
)
b q
f (
) + m f ′ m
2
,
2
′ a+b q
)
a q
f (
) + m f ′ ( m
)
2
.
2
2.8. Remark. Corollary 2.7 is similar to the inequality (1.5), but for the left-hand side
of the Hermite-Hadamard inequality.
2.9. Theorem. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R be
q
a differentiable function on I such that f ′ ∈ L([a, b]), where 0 ≤ a < b < ∞. If |f ′ | is
m-convex on [a, b] for some fixed m ∈ (0, 1] and q ∈ [1, ∞), x ∈ [a, b], then the following
inequality holds:
(2.4)
Z b
f (x) − 1
f (u) du
b−a a
2 1− 1 3
1− 1 ( q
q
′ q
1
b−x
1 b−x
f (a)
≤ (b − a)
2
b−a
3 b−a
q 1
(b − x)2 (b − 3a + 2x) ′ b q
+m
f m 6(b − a)3
2 1− 1 q
1
(b − x)2 (3a − b − 2x) ′ q
x−a
+
+
f (a)
b−a
6
6(b − a)3
3 q 1 )
′ b q
1 x−a
f
+m
3 b−a
m for each x ∈ [a, b].
142
H. Kavurmacı, M. E. Özdemir, M. Avcı
Proof. By Lemma 2.1, and using the well known power mean inequality, we have
Z b
f (x) − 1
f (u) du
b−a a
≤ (b − a)
Z
b−x
b−a
tf ′ (ta + (1 − t)b) dt
0
+ (b − a)
Z
≤ (b − a)
b−x
b−a
0
+ (b − a)
≤ (b − a)
1
b−x
b−a
!1− 1
q
t dt
Z
Z
(1 − t)f ′ (ta + (1 − t)b) dt
Z
1
0
!1− 1
q
b−x
b−a
1− 1
q
1
2
b−x
b−a
(1 − t) dt
(
q
tf ′ (ta + (1 − t)b) dt
2 1− 1
q
Z
!1
q
1
b−x
b−a
q
(1 − t)f ′ (ta + (1 − t)b) dt
!1
q
3
′ q
1 b−x
f (a)
3 b−a
q 1
(b − x)2 (b − 3a + 2x) ′ b q
+m
f m 6(b − a)3
2 1− 1 q
(b − x)2 (3a − b − 2x) ′ q
x−a
1
+
f (a)
+
b−a
6
6(b − a)3
3 q 1 )
′ b q
1 x−a
f
+m
3 b−a
m b−x
b−a
where we use the facts that
2
Z b−x
b−a
1 b−x
t dt =
,
2 b−a
0
Z
b−x
b−a
0
q
tf ′ (ta + (1 − t)b) dt
3
2
′ q
b−x
f (a) + m (b − x) (b − 3a + 2x)
b−a
6(b − a)3
2
Z 1
1 x−a
,
(1 − t) dt =
b−x
2 b−a
≤
1
3
q
′ b f
,
m b−a
and
Z
1
b−x
b−a
q
(1 − t)f ′ (ta + (1 − t)b) dt
≤
3 q
′ b (b − x)2 (3a − 2x − b) ′ q
1 x−a
1
f
.
+
f
(a)
+
m
3
6
6(b − a)
3 b−a
m The proof is completed.
2.10. Remark. Suppose that all the assumptions of Theorem 2.9 are satisfied. If we
choose x = a+b
, we obtain
2
!
1
Z b
1− q
′ 1 a
+
b
1
3
f
≤ (b − a)
f (a) + m q f ′ b ,
−
f
(u)
du
2
b−a a
8
m Inequalities for m-Convex Functions
143
which is the inequality (1.2).
3. Applications to special means
Let us recall the following means for two positive numbers.
(AM) The arithmetic mean
a+b
A = A(a, b) =
; a, b > 0,
2
(p-LM) The p-logarithmic mean

a
if a = b
Lp = Lp (a, b) = h bp+1 −ap+1 i p1
; a, b > 0, p ∈ R \ {−1, 0},

if a 6= b
(p+1)(b−a)
(IM) The identric mean

a
1
I = I(a, b) = 1 bb b−a
 aa
e
if a = b
if a 6= b
; a, b > 0.
The following propositions hold:
3.1. Proposition. Let a, b ∈ [0, ∞), and a < b, n ≥ 2 with m ∈ (0, 1]. Then we have
|An (a, b) − Ln
n (a, b)|
b n−1 a n−1 b−a
≤n
min 2A an−1 , m
, 2A (b)n−1 , m
.
8
m
m
Proof. The proof follows by Remark 2.3 on choosing f : [0, ∞) → [0, ∞), f (x) = xn ,
n ∈ Z, n ≥ 2, which is m-convex on [0, ∞).
3.2. Proposition. Let a, b ∈ [0, ∞), with a < b, and m ∈ (0, 1]. Then we have
1
1
I(a + 1, b + 1) ln
≤ b − a ηq + ηq ,
1
2
A(a, b) + 1 4
where
1
q
η1 = min
1
η2q = min
q
q

1
2m
 b+1
+ m a+b+2m

2
q
q

1
2m
 a+1
+ m a+b+2m

2
,
,
2
a+b+2
q
+m
2
m
b+m
q 


q
q 
m
2

+ m a+m
a+b+2
2
,
.

Proof. The proof follows by Corollary 2.7 on choosing f : [0, ∞) → (−∞, 0], f (x) =
− ln(x + 1), which is m-convex on [0, ∞), p > 1.
4. Applications to the midpoint formula for 1-convex functions
Let d be a division a = x0 < x1 < · · · < xn−1 < xn = b of the interval [a, b], and
consider the quadrature formula
Z b
(4.1)
f (x) dx = M (f, d) + E(f, d),
a
where
M (f, d) =
n−1
X
i=1
(xi+1 − xi ) f
x
+ xi 2
i+1
144
H. Kavurmacı, M. E. Özdemir, M. Avcı
is the midpoint formula and E(f, d) denotes the associated approximation error (see [12]).
Here, we obtain some error estimates for the midpoint formula.
4.1. Proposition. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R be
q
a differentiable function on I such that f ′ ∈ L([a, b]), where 0 ≤ a < b < ∞. If |f ′ | is
1
1
1-convex on [a, b] and p > 1, p + q = 1, then in (4.1), for every division d of [a, b], the
midpoint error satisfies
1
n−1
1
1X
(xi+1 − xi )2 µ1q + µ2q ,
|E(f, d)| ≤
4 i=0
where
µ1 = min
=
q
x +x
q
|f ′ (xi )| + f ′ ( i 2 i+1 )
2
q
′ xi +xi+1 q
) + |f ′ (xi )|
f (
2
,
,
q
( ′
x +x
q
|f (xi+1 )| + f ′ ( i 2 i+1 )
′ xi +xi+1 q
q)
) + |f ′ (xi )|
f (
2
2
2
µ2 = min
=
(
2
q
′ xi +xi+1 q
) + |f ′ (xi+1 )|
f (
2
2
,
′ xi +xi+1 q
q)
) + |f ′ (xi+1 )|
f (
2
2
.
Proof. Applying Corollary 2.7 for m = 1 to the subinterval [xi , xi+1 ], (i = 0, 1, 2, . . . , n −
1) of the division, we have
1
Z xi+1
1
1
f xi+1 + xi −
≤ xi+1 − xi µ1q + µ2q ,
f
(x)
dx
2
xi+1 − xi xi
4
where
µ1 =
µ2 =
′ xi +xi+1 q
q
) + |f ′ (xi )|
f (
2
2
′ xi +xi+1 q
q
) + |f ′ (xi+1 )|
f (
2
2
Hence, in (4.1) we have
Z b
n−1 Z
X
f (x)dx − M (f, d) = a
which completes the proof.
.
xi+1
+ xi f (x) dx − (xi+1 − xi ) f
2
xi
i=0
n−1
x
X Z xi+1
i+1 + xi ≤
f (x) dx − (xi+1 − xi ) f
2
xi
i=0
1
n−1
1
1X
≤
(xi+1 − xi )2 µ1q + µ2q ,
4 i=0
x
i+1
4.2. Proposition. Let I be an open real interval such that [0, ∞) ⊂ I. Let f : I → R be
q
a differentiable function on I such that f ′ ∈ L([a, b]), where 0 ≤ a < b < ∞. If |f ′ | is
Inequalities for m-Convex Functions
145
1-convex on [a, b], and q ∈ [1, ∞), x ∈ [a, b], then in (4.1), for every division d of [a, b],
the midpoint error satisfies
! n−1
1− 1
3 q X
|E(f, d)| ≤
(xi+1 − xi )2 f ′ (xi ) + f ′ (xi+1 ) .
8
i=0
Proof. Similar to that of Proposition 4.1 on using Remark 2.10 with m = 1.
References
[1] Alomari, M. and Darus, M. Some Ostrowski’s type inequalities for convex functions with
applications, RGMIA Res. Rep. Coll. 13 (1), Article 3, 2010.
[2] Alomari, M, Darus, M, Dragomir, S. S. and Cerone, P. Ostrowski’s inequalities for functions
whose derivatives are s-convex in the second sense, RGMIA Res. Rep. Coll. 12, Supplement,
Article 15, 2009.
[3] Bakula, M. K., Özdemir, M. E. and Pečarić, J. Hadamard type inequalities for m-convex and
(α, m)-convex functions, J. Inequal. Pure & Appl. Math. 9, Article 96, 2008.
[4] Bakula, M. K., Pečarić, J. and Ribičić, M. Companion inequalities to Jensen’s inequality
for m-convex and (α, m)-convex functions, J. Inequal. Pure & Appl. Math. 7, Article 194,
2006.
[5] Barnett, N. S., Cerone, P., Dragomir, S. S., Pinheiro, M. R. and Sofo, A. Ostrowski type
inequalities for functions whose modulus of derivatives are convex and applications, RGMIA
Res. Rep. Coll. 5 (2), Article 1, 2002.
[6] Cerone, P., Dragomir, S. S. and Roumeliotis, J. An inequality of Ostrowski type for mappings
whose second derivatives are bounded and applications, RGMIA Res. Rep. Coll. 1 (1), Article
4, 1998.
[7] Dragomir, S. S., Cho, Y. J. and Kim, S. S. Inequalities of Hadamard’s type for Lipschitzian
mappings and their applications, J. Math. Anal. Appl. 245 (2), 489–501, 2000.
[8] Dragomir, S. S. and Sofo, A. Ostrowski type inequalities for functions whose derivatives
are convex, Proceedings of the 4th International Conference on Modelling and Simulation,
Victoria University, Melbourne, Australia, November 11–13, 2002, RGMIA Res. Rep. Coll.
5, Supplement, Article 30, 2002.
[9] Dragomir, S. S. and Toader, G. Some inequalities for m-convex functions, Studia Univ.
Babeş-Bolyai Math. 38 (1), 21–28, 1993.
[10] Dragomir, S. S. and Wang, S. Applications of Ostrowski’s inequality to the estimation of
error bounds for some special means and some numerical quadrature rules, Appl. Math.
Lett. 11, 105–109, 1998.
[11] Kirmaci, U. S. Inequalities for differentiable mappings and applications to special means of
real numbers and to midpoint formula, Appl. Math. Comput. 147, 137–146, 2004.
[12] Pearce, C. E. M. and Pečarić, J. Inequalities for differentiable mappings with application to
special means and quadrature formula, Appl. Math. Lett. 13, 51–55, 2000.
[13] Set, E., Özdemir, M. E. and Sarıkaya, M. Z. Inequalities of Hermite-Hadamard’s type for
functions whose derivatives absolute values are m-convex, RGMIA Res. Rep. Coll. 13, Supplement, Article 5, 2010.
[14] Toader, G. Some generalizations of the convexity (Proceedings of The Colloquium On Approximation and Optimization, Univ. Cluj-Napoca, Cluj-Napoca, 1984), 329-338.