4FUZZY OSTROWSKIINEQUALITIES

We present optimal upper bounds for the deviation of a fuzzy continuousfunction from its fuzzy average over [a; b] � R, error is measured in the D-fuzzy metric. The established fuzzy Ostrowski type inequalities are sharp,in fact attained by simple fuzzy real number valued functions. These in-equalities are given for fuzzy Hölder and fuzzy di¤erentiable functions andthese facts are re�ected in their right-hand sides. This chapter relies on[13].

4.1 Introduction

Ostrowski inequality (see [92]) has as follows����� 1

b� a

Z b

a

f(y)dy � f(x)����� �

1

4+

�x� a+b

2

�2(b� a)2

!(b� a)kf 0k1;

where f 2 C1([a; b]), x 2 [a; b]. The last inequality is sharp, see [5].Since 1938 when A. Ostrowski proved his famous inequality, see [92],

many people have been working about and around it, in many di¤erentdirections and with a lot of applications in Numerical Analysis and Prob-ability, etc.One of the most notable works extending Ostrowski�s inequality is the

work of A.M. Fink, see [64]. The author in [5] continued that tradition.

G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 65–73.springerlink.com c© Springer-Verlag Berlin Heidelberg 2010

66 4. Fuzzy Ostrowski Inequalities

This chapter is mainly motivated by [5], [64], [92], [103] and extendsOstrowski type inequalities into the fuzzy setting, as fuzziness is a naturalreality genuine feature di¤erent than randomness and determinism.

4.2 Background

We use the Fuzzy Taylor formula.

Theorem 4.1 (Theorem 1 of [11]). Let T := [x0; x0 + �] � R, with � > 0.We assume that f (i) : T ! RF are H-di¤erentiable for all i = 0; 1; : : : ; n�1,for any x 2 T . (I.e., there exist in RF the H-di¤erences f (i)(x + h) �f (i)(x), f (i)(x)� f (i)(x� h), i = 0; 1; : : : ; n� 1 for all small h : 0 < h < �.Furthermore there exist f (i+1)(x) 2 RF such that the limits in D-distanceexist and

f (i+1)(x) = limh!0+

f (i)(x+ h)� f (i)(x)h

= limh!0+

f (i)(x)� f (i)(x� h)h

;

for all i = 0; 1; : : : ; n � 1.) Also we assume that f (n), is fuzzy continuouson T . Then for s � a, s; a 2 T we obtain

f(s) = f(a)� f 0(a)� (s� a)� f 00(a)� (s� a)2

2!

� � � � � f (n�1)(a)� (s� a)n�1

(n� 1)! �Rn(a; s);

where

Rn(a; s) := (FR)

Z s

a

�Z s1

a

� � ��Z sn�1

a

f (n)(sn)dsn

�dsn�1

�� � ��ds1:

Here Rn(a; s) is fuzzy continuous on T as a function of s.We use

Proposition 4.2 (Proposition 1 of [10]). Let F (t) := tn � u, t � 0, n 2 N,and u 2 RF be �xed. Then (the H-derivative)

F 0(t) = ntn�1 � u:

In particular when n = 1 then F 0(t) = u.We mention

Proposition 4.3 (Proposition 6 of [10]). Let I be an open interval of Rand let f : I ! RF be H-fuzzy di¤erentiable, c 2 R. Then

(c� f)0 exists and (c� f)0 = c� f 0(x):

We use the �Fuzzy Mean Value Theorem".

4.3 Results 67

Theorem 4.4 (Theorem 1 of [10]). Let f : [a; b]! RF be a fuzzy di¤eren-tiable function on [a; b] with H-fuzzy derivative f 0 which is assumed to befuzzy continuous. Then

D(f(d); f(c)) � (d� c) supt2[c;d]

D(f 0(t); ~o);

for any c; d 2 [a; b] with d � c.We �nally need the �Univariate Fuzzy Chain Rule".

Theorem 4.5 (Theorem 2 of [10]). Let I be a closed interval in R. Hereg : I ! � := g(I) � R is di¤erentiable, and f : � ! RF is H-fuzzy di¤er-entiable. Assume that g is strictly increasing. Then (f � g)0(x) exists and

(f � g)0(x) = f 0(g(x))� g0(x); 8x 2 I:

4.3 Results

We present the following

Theorem 4.6. Let f 2 C([a; b];RF ), the space of fuzzy continuous func-tions, x 2 [a; b] be �xed. We assume that f ful�lls the Hölder condition

D(f(y); f(z)) � Lf � jy � zj�; 0 < � � 1; 8y; z 2 [a; b];

for some Lf > 0. Then

D

1

b� a � (FR)Z b

a

f(y)dy; f(x)

!� Lf

�(x� a)�+1 + (b� x)�+1

(�+ 1)(b� a)

�:

(4.1)

Proof. We have that

D

1

b� a � (FR)Z b

a

f(y)dy; f(x)

!

= D

1

b� a � (FR)Z b

a

f(y)dy;1

b� a � (FR)Z b

a

f(x)dy

!

=1

b� aD Z b

a

f(y)dy;

Z b

a

f(x)dy

!(by Lemma 1.13)

� 1

b� a

Z b

a

D(f(y); f(x))dy

� Lfb� a

Z b

a

jy � xj�dy =�

Lfb� a

��(x� a)�+1 + (b� x)�+1

�+ 1

�:

68 4. Fuzzy Ostrowski Inequalities

�Optimality of (4.1) comes next.

Proposition 4.7. Inequality (4.1) is sharp, in fact, attained by f�(y) :=jy � xj� � u, 0 < � � 1, with u 2 RF �xed. Here x; y 2 [a; b].Proof. Clearly f� 2 C([a; b];RF ): for letting yn ! y, yn 2 [a; b], then

D(f�(yn); f�(y)) = D(jyn � xj� � u; jy � xj� � u)

(by Lemma 1.2)� jjyn � xj� � jy � xj�jD(u; ~o)! 0; as n! +1:

Furthermore

D(f�(y); f�(z)) = D(jy � xj� � u; jz � xj� � u)(by Lemma 1.2)

� jjy � xj� � jz � xj�jD(u; ~o)� jjy � xj � jz � xjj�D(u; ~o) � jy � zj�D(u; ~o):

That is, for Lf� := D(u; ~o) we get

D(f�(y); f�(z)) � Lf� jy � zj�; 0 < � � 1; any y; z 2 [a; b]:

So that f� is a Hölder function.Finally we have

D

1

b� a � (FR)Z b

a

f�(y)dy; f�(x)

!

= D

1

b� a � (FR)Z b

a

(jy � xj� � u)dy; ~o!

=1

b� a �D (FR)

Z b

a

(jy � xj� � u)dy; ~o!

=1

b� aD Z b

a

jy � xj�dy!� u; ~o

!

=1

b� aD��

(x� a)�+1 + (b� x)�+1�+ 1

�� u; ~o

�=

Lf�

b� a

�(x� a)�+1 + (b� x)�+1

�+ 1

�: �

Next comes the basic Ostrowski type fuzzy result in

Theorem 4.8 let f 2 C1([a; b];RF ), the space of one time continuouslydi¤erentiable functions in the fuzzy sense. Then for x 2 [a; b],

D

1

b� a � (FR)Z b

a

f(y)dy; f(x)

!� supt2[a;b]

D(f 0(t); ~o)

!�(x� a)2 + (b� x)2

2(b� a)

�:

(4.2)

4.3 Results 69

Inequality (4.2) is sharp at x = a, in fact attained by f�(y) := (y � a)(b�a)� u, u 2 RF being �xed.

Proof. We observe that

D

1

b� a � (FR)Z b

a

f(y)dy; f(x)

!

= D

1

b� a � (FR)Z b

a

f(y)dy;1

b� a � (FR)Z b

a

f(x)dy

!

=1

b� aD (FR)

Z b

a

f(y)dy; (FR)

Z b

a

f(x)dy

!(by Lemma 1.13)

� 1

b� a

Z b

a

D(f(y); f(x))dy

(by Theorem 4.4)� 1

b� a

Z b

a

jy � xj supt2[a;b]

D(f 0(t); ~o)

!dy

=

supt2[a;b]

D(f 0(t); ~o)

!b� a

�(x� a)2 + (b� x)2

2

�;

proving (4.2).By Propositions 4.2, 4.3 and Theorem 4.5 we get that f�0(y) = (b�a)�u.

We have that

L:H:S:(4:2) = D

1

b� a � (FR)Z b

a

((y � a)(b� a)� u)dy; ~o!

= D

(FR)

Z b

a

((y � a)� u)dy; ~o!

= D

Z b

a

(y � a)dy!� u; ~o

!

= D

�(b� a)22

� u; ~o�=(b� a)22

D(u; ~o):

And

R:H:S:(4:2) = supt2[a;b]

D((b� a)� u; ~o) (b� a)2

=(b� a)22

D(u; ~o):

That is equality in (4.2) is attained. �We conclude with the following Ostrowski type inequality fuzzy general-

ization in

70 4. Fuzzy Ostrowski Inequalities

Theorem 4.9. Let f 2 Cn+1([a; b];RF ), n 2 N, the space of (n+ 1) timescontinuously di¤erentiable functions on [a; b] in the fuzzy sense. Call

M :=nXi=1

(b� a)i(i+ 1)!

D(f (i)(a); ~o):

Then

D

1

b� a � (FR)Z b

a

f(x)dx; f(a)

!� (4.3)"

M +

supt2[a;b]

D(f (n+1)(t); ~o)

!(b� a)n+1(n+ 2)!

#:

If f (i)(a) = ~o, i = 1; : : : ; n. Then

D

1

b� a � (FR)Z b

a

f(x)dx; f(a)

!� supt2[a;b]

D(f (n+1)(t); ~o)

!(b� a)n+1(n+ 2)!

:

(4.4)Inequalities (4.3) and (4.4) are sharp, in fact attained by

f�(x) := (b� a)(x� a)n+1 � u; u 2 RF being �xed:

Corollary 4.10. Let f 2 C2([a; b];RF ). Then

D

1

b� a � (FR)Z b

a

f(x)dx; f(a)

!(4.5)

�"(b� a)2

D(f 0(a); ~o) +

supt2[a;b]

D(f 00(t); ~o)

!(b� a)26

#:

When f 0(a) = ~o, then

D

1

b� a � (FR)Z b

a

f(x)dx; f(a)

!� supt2[a;b]

D(f 00(t); ~o)

!(b� a)26

:

(4.6)

Proof of Theorem 4.9. Let x 2 [a; b], then by Theorem 4.1 we get

f(x) =

n�1X�

i=1

f (i)(a)� (x� a)i

i!�Rn(a; x);

where

Rn(a; x) := (FR)

Z x

a

�Z x1

a

� � ��Z xn�1

a

f (n)(xn)dxn

�dxn�1

�� � ��dx1

4.3 Results 71

(here we need x � a). We observe that

D

1

b� a � (FR)Z b

a

f(x)dx; f(a)

!

=1

b� aD (FR)

Z b

a

f(x)dx; (FR)

Z b

a

f(a)dx

!

=1

b� aD (FR)

Z b

a

n�1X�

i=0

f (i)(a)� (x� a)i

i!�Rn(a; x)

!dx;

(FR)

Z b

a

f(a)dx

!

=1

b� a �D (FR)

Z b

a

n�1X�

i=1

f (i)(a)� (x� a)i

i!�Rn(a; x)

!dx; ~o

!

=1

b� aD (FR)

Z b

a

nX�

i=1

f (i)(a)� (x� a)i

i!�Rn(a; x)

!dx;

(FR)

Z b

a

f (n)(a)� (x� a)n

n!dx

!

=1

b� aD

nX�

i=1

(FR)

Z b

a

f (i)(a)� (x� a)i

i!dx

� (FR)Z b

a

Rn(a; x)dx; (FR)

Z b

a

f (n)(a)� (x� a)n

n!dx

!

=1

b� aD

nX�

i=1

f (i)(a)� (b� a)i+1

(i+ 1)!� (FR)

Z b

a

Rn(a; x)dx;

(FR)

Z b

a

f (n)(a)� (x� a)n

n!dx

!

� 1

b� a

"nXi=1

(b� a)i+1(i+ 1)!

D(f (i)(a); ~o)

+ D

(FR)

Z b

a

Rn(a; x)dx; (FR)

Z b

a

f (n)(a)� (x� a)n

n!dx

!#

= M +1

b� aD (FR)

Z b

a

Rn(a; x)dx; (FR)

Z b

a

f (n)(a)� (x� a)n

n!dx

!

72 4. Fuzzy Ostrowski Inequalities

= M +1

b� aD�(FR)

Z b

a

�Z x

a

�Z x1

a

� � ��Z xn�1

a

f (n)(xn)dxn

�dxn�1

�� � ��dx1

�dx;

(FR)

Z b

a

�Z x

a

�Z x1

a

� � ��Z xn�1

a

f (n)(a)dxn

�dxn�1

�� � ��dx1

�dx

��

(by Lemmas 1.13, 1.15)� M +

1

b� a

�Z b

a

�Z x

a

�Z x1

a

� � ��Z xn�1

a

D(f (n)(xn); f(n)(a)

�dxn

�dxn�1

�� � ��dx1

�dx

�(by Theorem 4.4)

� M +1

b� a

�Z b

a

�Z x

a

�Z x1

a

� � ��Z xn�1

a

(xn � a)

� supt2[a;b]

D(f (n+1)(t); ~o)

!dxn

�dxn�1

�� � ��dx1

�dx

�

= M +

�supt2[a;b]

D(f (n+1)(t); ~o)�

b� a(b� a)n+2(n+ 2)!

= M +

supt2[a;b]

D(f (n+1)(t); ~o)

!(b� a)n+1(n+ 2)!

:

We have established inequalities (4.3) and (4.4).Consider g(x) := c(x � a)` � u, x 2 [a; b], c > 0, ` 2 Z+, u 2 RF �xed.

We prove that g is fuzzy continuous. Let xn 2 [a; b] such that xn ! x asn! +1. Then

D(g(xn); g(x)) = D(c(xn � a)` � u; c(x� a)` � u)� cj(xn � a)` � (x� a)`jD(u; ~o)! 0:

Hence by the last argument, Propositions 4.2, 4.3 and Theorem 4.5 weobtain that f� 2 Cn+1([a; b];RF ).We see that

f�(i)(a) = ~o; for i = 1; : : : ; n:

That is M = 0. Furthermore it holds

f�(n+1)(x) = (b� a)(n+ 1)!� u:

4.3 Results 73

Finally, we notice that

L:H:S:((4:3); (4:4)) = D

1

b� a � (FR)Z b

a

((b� a)(x� a)n+1 � u)dx; ~o!

= D

u�

Z b

a

(x� a)n+1dx; ~o!= D

�u� (b� a)

n+2

n+ 2; ~o

�=

(b� a)n+2n+ 2

D(u; ~o):

Also we �nd

R:H:S:((4:3); (4:4)) = (b�a)(n+1)!D(u; ~o) (b� a)n+1

(n+ 2)!=(b� a)n+2n+ 2

D(u; ~o):

Proving (4.3) and (4.4) sharp, in fact attained inequalities. �