International Journal of Analysis and Applications
ISSN 2291-8639
Volume 3, Number 1 (2013), 35-46
http://www.etamaths.com

TWO-POINT FUZZY OSTROWSKI TYPE INEQUALITIES

MUHAMMAD AMER LATIF1 AND SABIR HUSSAIN2,∗

Abstract. Two-point fuzzy Ostrowski type inequalities are proved for fuzzy

Hölder and fuzzy differentiable functions. The two-point fuzzy Ostrowski type

inequality for M-lipshitzian mappings is also obtained. It is proved that only
the two-point fuzzy Ostrowski type inequality for M-lipshitzian mappings is

sharp and as a consequence generalize the two-point fuzzy Ostrowski type

inequalities obtained for fuzzy differentiable functions.

1. Introduction

In 1938, A. M. Ostrowski proved an interesting integral inequality, estimating
the absolute value of deviation of a differentiable function by its integral mean as:

Theorem 1. Let f : [a,b] → R be a continuous function on [a,b] and differentiable
on (a,b). If f′ is bounded on (a,b), that is

‖f′‖ := sup
t∈(a,b)

|f(t)| < ∞,

then

(1.1)

∣∣∣∣∣f(x) − 1b−a
∫ b
a

f(t)dt

∣∣∣∣∣ ≤
[

1

4
+

(
x− a+b

2

)2
(b−a)2

]
(b−a)‖f′‖∞

It is easy to observe that (1.1) can be rewritten in equivalent from as follow:

(1.2)

∣∣∣∣∣f(x) − 1b−a
∫ b
a

f(t)dt

∣∣∣∣∣ ≤ (x−a)
2

+ (b−x)2

2 (b−a)
‖f′‖∞ .

Since that time when A. Ostrowski proved this famous inequality, many mathe-
matician have been working on it and have been applying it in numerical analysis
and probability, etc.

N. S. Barnett and S. S. Dragomir [5], proved, as a generalization of (1.1), the
following result:

If f : [a,b] → R is absolutely continuous on [a,b] and if [c,d] ⊂ [a,b], then∣∣∣∣∣ 1b−a
∫ b
a

f(t)dt−
1

d− c

∫ d
c

f(s)ds

∣∣∣∣∣
≤

{
b−a

4
+
d− c

2
+

1

b−a

[∣∣∣∣c + d2 − a + b2
∣∣∣∣− d− c2

]2}
‖f′‖∞ .(1.3)

2000 Mathematics Subject Classification. 26D07, 26D15, 26E50.

Key words and phrases. Ostrowski inequality, Fuzzy inequalities, Fuzzy real analysis.

c©2013 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

35



36 LATIF AND HUSSAIN

It is to be noted that for c = d = x, one can assume 1
d−c

∫d
c
f(s)ds = f(x), as a

limit case, and hence (1.3) takes the from of (1.1).
In [9], M. Matić and J. Pečarić gave a two-point Ostrowski type inequality, as

a generalization of (1.3), by replacing the condition of differentiability of f and
boundedness of f′ on (a,b) by a weaker condition that f is M-Lipschitzian on
[a,b], that is

|f(t) −f(s)| ≤ M |t−s| , ∀t,s ∈ [a,b] , M > 0.

It was also proved that the two-point Ostrowski inequality established in [9] is
sharp and gives tighter estimate than those of (1.3). The main result from [9] is
the following one:

Theorem 2. Let a,b,c,d ∈ R be such that

a ≤ c < d ≤ b, c−a + b−d > 0.

(i) If f : [a,b] → R is M-Lipschitzian on [a,b], with some constant M > 0,
then

(1.4)

∣∣∣∣∣ 1b−a
∫ b
a

f(t)dt−
1

d− c

∫ d
c

f(s)ds

∣∣∣∣∣ ≤ (c−a)
2

+ (b−d)2

2 (c−a + b−d)
M.

(ii) If f0 : [a,b] → R defined as

f0 (t) = |t−s0| ,

where

s0 =
bc−ad

c−a + b−d
,

then f0 is 1-Lipshitzian on [a,b] and we havd∣∣∣∣∣ 1b−a
∫ b
a

f(t)dt−
1

d− c

∫ d
c

f(s)ds

∣∣∣∣∣ = (c−a)
2

+ (b−d)2

2 (c−a + b−d)
.

Since fuzziness is a natural reality different than randomness and determinis-
m, therefore an attempt has been made by George A. Anastassiou [2] to extend
(1.1) to fuzzy setting context in 2003. In fact, George A. Anastassiou [2] proved
the following important results for fuzzy Hölder and fuzzy differentiable functions
respectively:

Theorem 3. Let f ∈ C ([a,b],RF), the space of fuzzy continous functions, x ∈ [a,b]
be fixed. If f fulfills the Hölder condition

D (f(y),f(z)) ≤ Lf |y −z|
α

, 0 < α ≤ 1, for all y, z ∈ [a,b] ,

for some Lf > 0. Then

D

(
1

b−a
� (FR)

∫ b
a

f(y)dy,f(x)

)

≤ Lf

(
(x−a)α+1 + (b−x)α+1

(α + 1) (b−a)

)
.(1.5)



FUZZY OSTROWSKI TYPE INEQUALITIES 37

Theorem 4. Let f ∈ C1 ([a,b],RF), the space of one time continuously differen-
tiable functions in the fuzzy sense. Then for x ∈ [a,b],

D

(
1

b−a
� (FR)

∫ b
a

f(y)dy,f(x)

)

≤

(
sup
t∈[a,b]

D (f′(t), õ)

)(
(x−a)2 + (b−x)2

2 (b−a)

)
.(1.6)

The inequalities in (1.5) and (1.6) are sharp as equalities are attained by the
choice of simple fuzzy real number valued functions. For further details on these
inequalities we refer the interested readers to [2].

The main purpose of the present paper is to establish two-point fuzzy Ostrows-
ki type inequalties for fuzzy Hölder, fuzzy differentiable and fuzzy M-Lipshitzian
functions in Section 2, which actually generalize the inequalities (1.5) and (1.6).

2. Preliminaries

In this section we point out some basic definitions and results which would help
us in the sequel of the paper, we begin with:

Definition 1. [11] Let us denote by RF the class of fuzzy subsets of real axis R (
i.e. u : R −→ [0, 1]), satisfying the following properties:

(1) ∀u ∈ RF, u is normal i.e.with u(x) = 1.
(2) ∀u ∈ RF, u is convex fuzzy set i.e.

u(tx + (1 − t)y) ≥ min{u(x),u(y)} ,∀t ∈ [0, 1] .
(3) ∀ u ∈ RF, u is upper semi-continuous on R.
(4) {x ∈ R : u(x) > 0} is compact.

The set RF is called the space of fuzzy real numbers.

Remark 1. It is clear that R ⊂ RF, because any real number x0 ∈ R, can be
described as the fuzzy number whose value is 1 for x = x0 and zero otherwise.

We will collect some further definitions and notations needed in the sequel. For
0 < r ≤ 1 and u ∈ RF , we define

[u]
r

= {x ∈ R : u(x) ≥ r}
[u]

0
= {x ∈ R : u(x) > 0}

Now it is well known that for each r ∈ [0, 1] , [u]r, is bounded closed interval. For
u,v ∈ RF and λ ∈ R, we have the sum u⊕ v and the product λ�u are defined
by [u⊕v]r = [u]r + [v]r, [λ�u]r = λ [u]r , ∀r ∈ [0, 1], where [u]r + [v]r means the
usual addition of two intervals as subsets of R and λ [u]rmeans the usual product
between a scalar and a subset of R.

Now we define D : RF ×RF −→ R∪{0} by
D(u,v) = sup

r∈[0,1]

(
max

{∣∣ur− −vr−∣∣ , ∣∣ur+ + ur+∣∣}) ,
where [u]

r
=
[
ur−,u

r
+

]
, [v]

r
=
[
vr−,v

r
+

]
, then (D,RF) is a metric space and it

possesses the following properties:

(i) D(u⊕w,v ⊕w) = D(u,v), ∀u,v,w ∈ RF.
(ii) D(λ� u, λ�v) = λD(u,v),∀u,v ∈ RF,∀λ ∈ R.



38 LATIF AND HUSSAIN

(iii) D(u⊕v,w ⊕e) ≤ D(u,w) + D(v,e), ∀u,v,w,e ∈ RF
Moreover it is well known that (RF,D) is a complete metric space.
Also we have the following theorem:

Theorem 5. [11]

i If we denote õ = X{0} then õ ∈ RF is neutral element with respect to ⊕,
i.e. u⊕ õ = õ⊕u, for all u ∈ RF.

ii With respect to 0̃ none of u ∈ RF\R has opposite in RF with respect to ⊕.
iii For any a,b ∈ R with a,b ≥ 0 or a,b ≤ 0, any u ∈ RF, we have (a+b)�u =

a�u⊕ b� u.∀a,b ∈ R the above property does not hold.
iv For any λ ∈ R and any u,v ∈ RF, we have λ� (u⊕v) = λ�u⊕λ�v.
v For any λ,µ ∈ R and any u ∈ R, we have λ� (µ�v) = (λ.µ) �v.

vi If we denote‖u‖F = D(u,õ), ∀u ∈ RF then ‖.‖F has the properties of a
usual norm on RF, i.e. ‖u‖F = 0 if and only if u = õ, ‖λ�u‖F = |λ| .‖u‖F
and ‖u⊕v‖F ≤‖u‖F + ‖v‖F , |‖u‖F + ‖v‖F| ≤ D(u,v).

Remark 2. The propositions (ii) and (iii) in theorem show us that (RF,⊕,�) is
not a linear space over R and consequently (RF,‖.‖F) cannot be a normed space.
However, the properties of D and those in theorem (iv)-(vi), have as an effect that
most of the metric properties of a functions defined on R with values in a Banach
space, can be extended to functions f : R −→ RF, called fuzzy number valued
functions.

Definition 2. A function f : R −→ RF is said to be continuous at x0 ∈ R if for
every ε > 0 we can find δ > 0 such that D(f(x),f (x0)) < ε, whenever |x−x0| < δ.
f is said to be continuous on R if it is continuous at every x ∈ R.

Lemma 1. For any a, b ∈ R, a, b ≥ 0 and u ∈ RF, we have

D (a�u,b�u) ≤ |a− b|D(u,õ),

where õ ∈ RF is defined by õ := X{0}.

Definition 3. Let x, y ∈ RF. If there exists a z ∈ RF such that x = y ⊕ z, then
we call z the H-difference of x and y, denoted by z = x	y.

Definition 4. Let T := [x0,x0 + β] ⊂ R, with β > 0. A function f : T −→ RF
is H-differentiable at x ∈ T if there exists a f′(x) ∈ RF such that the limits (with
respect to the metric D)

lim
h→0+

f (x + h) 	f(x)
h

, lim
h→0+

f(x) 	f (x−h)
h

exist and are equal to f′(x). We call f′ the derivative or h-derivative of f at x. If
f is H-differentiable at any x ∈ T , we call f differentiable or H-differentiable and
it has H-derivative over T the function f′.

Definition 5. Let f : [a,b] −→ RF. We say that f is Fuzzy- Riemann integrable
to I ∈ RF, if for every ε > 0, there exsit δ > 0 such that for any division P =
{[u,v] ; ξ} of [a,b] with the norms ∆ (P) < δ, we have

D
(∑∗

(v −u) �f (ξ) ,I
)
< ε,



FUZZY OSTROWSKI TYPE INEQUALITIES 39

where
∑∗

denotes the fuzzy summation. We choose to write

I := (FR)

∫ b
a

f(x)dx.

We also call an f as above (FR)-integrable.

Corollary 1. If f ∈ C ([a,b] ,RF) then f is (FR)-integrable.

Lemma 2. If f,g : [a,b] ⊆ R −→ RF are fuzzy continuous (with respect to the met-
ric D), then the function F : [a,b] −→ R+ ∪{0} defined by F(x) := D (f(x),g(x))
is continuous on [a,b], and

D

(
(FR)

∫ b
a

f(u)du, (FR)

∫ b
a

g(u)du

)
≤
∫ b
a

D (f(x),g(x)) dx.

Lemma 3. Let f : [a,b] ⊆ R −→ RF be fuzzy continuous. Then

(FR)

∫ x
a

f(t)dt

is fuzzy continuous function in x ∈ [a,b].

Proposition 1. Let F(t) := tn �u, t ≥ 0, n ∈ N and u ∈ RF be fixed. The (the
H-derivative)

F ′(t) = ntn−1 �u.

In particular when n = 1 then F ′(t) = u.

Proposition 2. Let I be an open interval of R and let f : I −→ RF be H-fuzzy
differentiable, c ∈ R. Then (c�f)′ exist and (c�f (x))′ = c�f′ (x).

Theorem 6. Let f : [a,b] −→ RF be fuzzy differentiable function on [a,b] with
H-derivative f′ which is assumed to be fuzzy continuous. Then

D (f (d) ,f (c)) ≤ (d− c) sup
t∈[c,d]

D (f′(t), õ) ,

for any c, d ∈ [a,b] with d ≥ c.

Theorem 7. Let I be closed interval in R. Let g : I → ζ := g(I) ⊆ R be differen-
tiable, and f : g(I) → RF be H-differentiable. Assume that g is strictly increasing.
Then (f ◦g)′ (x) exists and

(f ◦g)′ (x) = f (g(x)) �g′(x), ∀x ∈ I.

3. Main Results

In this section we prove a two-point Ostrowski type inequalities for fuzzy Hölder
and fuzzy differentiable functions in a similar fashion as in [2].

We first prove a two point Ostrowski inequality like (1.3) but for fuzzy differe-
itaible functions in:



40 LATIF AND HUSSAIN

Theorem 8. Let f ∈ C1 ([a,b],RF), the space of one time continuously differen-
tiable functions in the fuzzy sense. if x ∈ [c,d] ⊂ [a,b], then

D

(
1

b−a
� (FR)

∫ b
a

f (t) dt,
1

d− c
� (FR)

∫ b
a

f (s) ds

)

≤

(
sup
t∈[a,b]

D (f′(t), õ)

){
b−a

4
+
d− c

2
+

1

b−a

[∣∣∣∣c + d2 − a + b2
∣∣∣∣− d− c2

]2}
.

(3.1)

Proof. By using the properties of the metric D and (1.5), we have

D

(
1

b−a
� (FR)

∫ b
a

f (t) dt,
1

d− c
� (FR)

∫ b
a

f (s) ds

)

= D

((
1

b−a
� (FR)

∫ b
a

f (t) dt

)
⊕f(x),

(
1

d− c
� (FR)

∫ b
a

f (s) ds

)
⊕f(x)

)

= D

((
1

b−a
� (FR)

∫ b
a

f (t) dt

)
⊕f(x),f(x) ⊕

(
1

d− c
� (FR)

∫ b
a

f (s) ds

))

≤ D

(
1

b−a
� (FR)

∫ b
a

f (t) dt,f(x)

)
+ D

(
1

d− c
� (FR)

∫ b
a

f (s) ds,f(x)

)

≤

(
sup
t∈[a,b]

D (f′(t), õ)

)

14 +

(
x− a+b

2

b−a

)2
 (b−a) +


14 +

(
x− c+d

2

d− c

)2
 (d− c)


 .

Since the rest of the proof is similar to that of (1.3), we therefore omit the detals. �

Theorem 9. Let a,b,c,d ∈ R be such that

a ≤ c < d ≤ b, c−a + b−d > 0.

f ∈ C ([a,b],RF), the space of fuzzy continuous functions. Suppose f fulfills the
Hölder condition, that is

D (f(y),f(z)) ≤ Lf |y −z|
α

, 0 < α ≤ 1, for all y, z ∈ [a,b] ,

for some Lf > 0. Then

D

(
1

b−a
� (FR)

∫ b
a

f (t) dt,
1

d− c
� (FR)

∫ d
c

f (s) ds

)

≤ Lf

(
(c−a)α+1 + (b−d)α+1

(α + 1) (c−a + b−d)

)
.(3.2)

Proof. By using the substitution

t =
b−a
d− c

s−
bc−ad
d− c

, s ∈ [c,d] ,

by (v) of Theorem 5 and Lemma 2, we have that

1

b−a
� (FR)

∫ b
a

f (t) dt =
1

d− c
� (FR)

∫ d
c

f

(
b−a
d− c

s−
bc−ad
d− c

)
ds.



FUZZY OSTROWSKI TYPE INEQUALITIES 41

Thus

D

(
1

b−a
� (FR)

∫ b
a

f (t) dt,
1

d− c
� (FR)

∫ d
c

f (s) ds

)

= D

(
1

d− c
� (FR)

∫ d
c

f

(
b−a
d− c

s−
bc−ad
d− c

)
ds,

1

d− c
� (FR)

∫ d
c

f (s) ds

)

=
1

d− c
D

(
(FR)

∫ d
c

f

(
b−a
d− c

s−
bc−ad
d− c

)
ds, (FR)

∫ d
c

f (s) ds

)

≤
1

d− c

∫ d
c

D

(
f

(
b−a
d− c

s−
bc−ad
d− c

)
,f (s)

)
ds

≤
Lf
d− c

∫ d
c

∣∣∣∣b−ad− cs− bc−add− c −s
∣∣∣∣α ds

=
Lf
d− c

∫ d
c

∣∣∣∣c−a + b−dd− c s− bc−add− c
∣∣∣∣α ds

=
Lf (c−a + b−d)

α

(d− c)α+1
∫ d
c

∣∣∣∣s− bc−adc−a + b−d
∣∣∣∣α ds

=
Lf (c−a + b−d)

α

(d− c)α+1
∫ d
c

|s−s0|
α
ds,

(3.3)

where s0 =
bc−ad

c−a+b−d.
We now observe that

s0 − c =
(d− c) (c−a)
c−a + b−d

≥ 0

and

d−s0 =
(d− c) (b−d)
c−a + b−d

≥ 0,

and hence s0 ∈ [c,d].
Therefore,∫ d

c

|s−s0|
α
ds =

∫ s0
c

(s0 −s)
α
ds +

∫ d
s0

(s−s0)
α
ds

=
1

α + 1

[
(s0 − c)

α+1
+ (s0 −d)

α+1
]

=
(d− c)α+1

(α + 1) (c−a + b−d)α+1
[
(c−a)α+1 + (b−d)α+1

]
(3.4)

Substitution of (3.4) in (3.3) gives (3.2).
This completes the proof. �

Remark 3. The inequalities (3.1) and (3.2) generalize the inequalitites (1.3) and
(1.6) respectively but are not sharp as the equality cannot be attained by a partic-
ular choice of the fuzzy real number valued functions, since if we choose f∗(t) =
|t−s0|

α � u, 0 < α ≤ 1, with u ∈ RF fixed, t ∈ [a,b] and s0 = bc−adc−a+b−d . Then



42 LATIF AND HUSSAIN

f∗ ∈ C ([a,b],RF), as for letting tn → t, tn ∈ [a,b] and using Lemma 1, we have

D (f∗ (tn) ,f
∗ (t)) = D (|tn −s0|

α �u, |t−s0|
α �u)

≤ ||tn −s0|
α −|t−s0|

α|D (u,õ) → 0, as n →∞.

Furthermore

D (f∗ (t) ,f∗ (s)) = D (|t−s0|
α �u, |s−s0|

α �u)
≤ ||t−s0|

α −|s−s0|
α|D (u,õ)

≤ |t−s|α D (u,õ) .

This shows that for Lf∗ = D (u,õ), we have

D (f∗ (t) ,f∗ (s)) ≤ Lf∗ |t−s|
α

, 0 < α ≤ 1, for any t,s ∈ [a,b] .

and therefore f∗ satisfies the Hölder condition.
Finally by the properties of (FR)-integrable functions and (iii) of Theorem 5, we
have

1

d− c
� (FR)

∫ d
c

f∗ (s) ds

=
(d− c)α

(α + 1) (c−a + b−d)α+1
[
(c−a)α+1 + (b−d)α+1

]
�u.(3.5)

Since

s0 −a =
(b−a) (c−a)
c−a + b−d

≥ 0

and

b−s0 =
(b−a) (b−d)
c−a + b−d

≥ 0

implies that s0 ∈ [a,b].
By similar arguments as in obtaining (3.4), we get that

1

b−a
� (FR)

∫ b
a

f∗ (t) dt

=
(b−a)α

(α + 1) (c−a + b−d)α+1
[
(c−a)α+1 + (b−d)α+1

]
�u(3.6)

Now it is evident from (3.5), (3.6) and Lemma 1 that

D

(
1

b−a
� (FR)

∫ b
a

f∗ (t) ,
1

d− c
� (FR)

∫ d
c

f∗ (s) ds

)

=

(
(c−a)α+1 + (b−d)α+1

(α + 1) (c−a + b−d)

)(
(b−a)α − (d− c)α

(c−a + b−d)α
)
D (u,õ) .

This shows that (1.6) is not sharp.

Our next result is about fuzzy differentiable functions and is stated as follow:

Theorem 10. Let a,b,c,d ∈ R be such that

a ≤ c < d ≤ b, c−a + b−d > 0.



FUZZY OSTROWSKI TYPE INEQUALITIES 43

f ∈ C1 ([a,b],RF), the space of fuzzy one time continuously differentiable functions.
Then

D

(
1

b−a
� (FR)

∫ b
a

f (t) dt,
1

d− c
� (FR)

∫ d
c

f (s) ds

)

≤

(
sup
t∈[a,b]

D (f′(t), õ)

)(
(c−a)2 + (b−d)2

2 (c−a + b−d)

)
.(3.7)

Proof. Again by using the substitution

t =
b−a
d− c

s−
bc−ad
d− c

, s ∈ [c,d] ,

and by (v) of Theorem 5, we have that

1

b−a
� (FR)

∫ b
a

f (t) dt =
1

d− c
� (FR)

∫ d
c

f

(
b−a
d− c

s−
bc−ad
d− c

)
ds.

Now by Lemma 2 and Theorem 6, we get that

D

(
1

b−a
� (FR)

∫ b
a

f (t) dt,
1

d− c
� (FR)

∫ d
c

f (s) ds

)

= D

(
1

d− c
� (FR)

∫ d
c

f

(
b−a
d− c

s−
bc−ad
d− c

)
ds,

1

d− c
� (FR)

∫ d
c

f (s) ds

)

=
1

d− c
D

(
(FR)

∫ d
c

f

(
b−a
d− c

s−
bc−ad
d− c

)
ds, (FR)

∫ d
c

f (s) ds

)

≤
1

d− c

∫ d
c

D

(
f

(
b−a
d− c

s−
bc−ad
d− c

)
,f (s)

)
ds

≤
sup
s∈[c,d]

D (f′(s), õ)

d− c

∫ d
c

∣∣∣∣b−ad− cs− bc−add− c −s
∣∣∣∣ds

=

sup
s∈[c,d]

D (f′(s), õ)

d− c

∫ d
c

∣∣∣∣c−a + b−dd− c s− bc−add− c
∣∣∣∣ds

≤
sup
t∈[a,b]

D (f′(t), õ) (c−a + b−d)

(d− c)2
∫ d
c

∣∣∣∣s− bc−adc−a + b−d
∣∣∣∣ds

=

sup
t∈[a,b]

D (f′(t), õ) (c−a + b−d)

(d− c)2
∫ d
c

|s−s0|ds,

(3.8)

where s0 =
bc−ad

c−a+b−d.
We now observe that

s0 − c =
(d− c) (c−a)
c−a + b−d

≥ 0

and

d−s0 =
(d− c) (b−d)
c−a + b−d

≥ 0,



44 LATIF AND HUSSAIN

and hence s0 ∈ [c,d].
Therefore, ∫ d

c

|s−s0|ds =
∫ s0
c

(s0 −s) ds +
∫ d
s0

(s−s0) ds

=
1

2

[
(s0 − c)

2
+ (s0 −d)

2
]

=
(d− c)2

2 (c−a + b−d)2
[
(c−a)2 + (b−d)2

]
(3.9)

Substituting (3.9) in (3.8), we get (3.7).
�

Remark 4. Inequality (3.7) in Theorem 9 is not sharp. Moreover, we note that if

c = d = x we can assume 1
d−c � (FR)

∫d
c
f (s) ds = f(x), as a limit case, so (3.2)

and (3.7) reduce to (1.5) and (1.6) respectively. This fact also reveals that although
our results are not sharp but generalize the inequalities (1.5) and (1.6).

Our last result is about fuzzy M-Lipshitzian mappings and is stated as follow:

Theorem 11. Let a,b,c,d ∈ R be such that

a ≤ c < d ≤ b, c−a + b−d > 0.

f ∈ C ([a,b],RF), the space of fuzzy continuous functions. Suppose f is M-
Lipshitzian, that is

D (f(y),f(z)) ≤ M |y −z| , for all y, z ∈ [a,b] ,

for some M > 0. Then

D

(
1

b−a
� (FR)

∫ b
a

f (t) dt,
1

d− c
� (FR)

∫ d
c

f (s) ds

)

≤ M

(
(c−a)2 + (b−d)2

2 (c−a + b−d)

)
.(3.10)

Inequality (3.10) is sharp, in fact attained by f∗(t) = |t−s0| � u, u ∈ RF being
fixed and s0 =

bc−ad
c−a+b−d .

Proof. The proof of (3.10) is similar to that of (3.2) we, therefore omit the detals
for the intrested reader.
It remains only to prove that (3.10) is sharp. It is clear that f∗ ∈ C ([a,b],RF),
since for letting tn → t, tn ∈ [a,b], then Lemma 1 we have

D (f∗(tn),f
∗(t)) = D (|tn −s0|�u, |t−s0|�u)

≤ ||tn −s0|− |t−s0||D (u,õ)
≤ |tn − t|D (u,õ) → 0, as n →∞.

Moreover, again by Lemma 1, we get that

D (f∗(t),f∗(s)) = D (|t−s0|�u, |s−s0|�u)
≤ ||t−s0|− |s−s0||D (u,õ)
≤ |t−s|D (u,õ) .



FUZZY OSTROWSKI TYPE INEQUALITIES 45

That is for M = D (u,õ), we get

D (f∗(t),f∗(s)) ≤ M |t−s| , ∀t,s ∈ [a,b] .

So f∗ is M-Lipshitzian function.
Now by the similar reasoning as in Remark 1, we have

1

d− c
� (FR)

∫ d
c

f∗ (s) ds

=
d− c

2 (c−a + b−d)2
[
(c−a)2 + (b−d)2

]
�u.(3.11)

Since

s0 −a =
(b−a) (c−a)
c−a + b−d

≥ 0

and

b−s0 =
(b−a) (b−d)
c−a + b−d

≥ 0

implies that s0 ∈ [a,b].
Similarly we also have

1

b−a
� (FR)

∫ b
a

f∗ (t) dt

=
b−a

(c−a + b−d)2
[
(c−a)2 + (b−d)2

]
�u(3.12)

Now it is aparent from (3.11), (3.12) and Lemma 1 that

D

(
1

b−a
� (FR)

∫ b
a

f∗ (t) ,
1

d− c
� (FR)

∫ d
c

f∗ (s) ds

)

=

(
(c−a)2 + (b−d)2

2 (c−a + b−d)

)
D (u,õ) .

Hence it is proved that (3.10) is sharp. �

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1College of Science, Department of Mathematics, University of Hail, Hail 2440,

Saudi Arabia

2Department of Mathematics, University of Engineering and Technology, Lahore,

Pakistan

∗Corresponding author