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) [email protected] (M. E. Özdemir)
[email protected] (M. Avcı) [email protected]
†
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.
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