Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 1, Issue 2, Article 11, 2000
OSTROWSKI TYPE INEQUALITIES FROM A LINEAR FUNCTIONAL
POINT OF VIEW
B. GAVREA AND I. GAVREA
U NIVERSITY BABE Ş -B OLYAI C LUJ -NAPOCA , D EPARTMENT OF M ATHEMATICS AND C OMPUTERS , S TR .
M IHAIL KOG ĂLNICEANU 1, 3400 C LUJ -NAPOCA , ROMANIA
gavreab@yahoo.com
T ECHNICAL U NIVERSITY C LUJ -NAPOCA , D EPARTMENT OF M ATHEMATICS , S TR . C. DAICOVICIU 15, 3400
C LUJ -NAPOCA , ROMANIA
Ioan.Gavrea@math.utcluj.ro
Received 13 Septemer, 1999; accepted 31 January, 2000
Communicated by P. Cerone
A BSTRACT. Inequalities are obtained using P0 -simple functionals. Applications to Lipschitzian
mappings are given.
Key words and phrases: P0 -simple functionals, inequalities, Lipschitz mappings.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
Let I be a bounded interval of the real axis. We denote by B(I) the set of all functions which
are bounded on [a, b].
Let A be a positive linear functional A : B(I) → R, such that A(e0 ) = 1, where ei : I → R,
ei (x) = xi , ∀ x ∈ I, i ∈ N.
The following inequality is known in literature as the Grüss inequality for the functional A.
Theorem 1.1. Let f, g : I → R be two bounded functions such that m1 ≤ f (x) ≤ M1 and
m2 ≤ g(x) ≤ M2 for all x ∈ I, m1 , M1 , m2 and M2 are constants. Then the inequality:
1
(1.1)
|A(f g) − A(f )A(g)| ≤ (M1 − m1 )(M2 − m2 )
4
holds.
In 1938 Ostrowski (cf. for example [7, p. 468]) proved the following result:
Theorem 1.2. Let f : I → R be continuous on (a, b) whose derivative f 0 : (a, b) → R is
bounded on (a, b), i.e.
kf 0 k∞ := sup |f 0 (t)| < ∞.
t∈(a,b)
ISSN (electronic): 1443-5756
c 2000 Victoria University. All rights reserved.
002-99
2
B. G AVREA AND I. G AVREA
Then
#
"
Z b
a+b 2
x
−
1
1
2
f (x) −
(b − a)kf 0 k∞
f (t)dt ≤
+
b−a a
4
(b − a)2
(1.2)
for all x ∈ [a, b]. The constant 41 is best.
In the recent paper [4] S.S. Dragomir and S. Wang proved the following version of Ostrowski’s inequality.
Theorem 1.3. Let f : I → R be a differentiable mapping in the interior of I and a, b ∈ int(I)
with a < b. If f 0 ∈ L1 [a, b] and γ ≤ f 0 (x) ≤ Γ for all x ∈ [a, b] then we have the following
inequality:
Z b
1
f
(b)
−
f
(a)
a
+
b
1
≤ (b − a)(Γ − γ)
f (x) −
f
(t)dt
−
x
−
(1.3)
4
b−a
b−a
2
a
for all x ∈ [a, b].
The following inequality for mappings with bounded variation can be found in [1]:
Theorem 1.4. Let f : I → R be a mapping of bounded variation. Then for all x ∈ [a, b] we
have the inequality
Z b
b
a + b _
1
f,
(1.4)
f (t)dt − f (x)(b − a) ≤ (b − a) + x −
2
2 a
a
where
b
W
f denotes the total variation of f .
a
The constant 21 is the best possible one.
In [2] S.S. Dragomir gave the following result for Lipschitzian mappings:
Theorem 1.5. Let f : [a, b] → R be an L-Lipschitzian mapping on [a, b], i.e.
|f (x) − f (y)| ≤ L|x − y|, for all x, y ∈ [a, b].
Then we have the inequality
"
#
Z b
a+b 2
x
−
1
2
(1.5)
f (t)dt − f (x)(b − a) ≤ L(b − a)2
+
2
4
(b
−
a)
a
for all x ∈ [a, b].
The constant 14 is the best possible one.
S.S. Dragomir, P. Cerone, J. Roumeliotis and S. Wang in [3] proved the following theorem:
Theorem 1.6. Let f, w : (a, b) ⊆ R → R be so that w(s) ≥ 0 on (a, b), w is integrable on
Rb
(a, b) and a w(s)ds > 0, f is of r-Hölder type, i.e.
(1.6)
|f (x) − f (y)| ≤ H|x − y|r , for all x, y ∈ (a, b)
where H > 0 and r ∈ (0, 1] are given. If w, f ∈ L1 (a, b), then we have the inequality:
Z b
Z b
1
1
(1.7)
w(s)f (s)ds ≤ H R b
|x − s|r w(s)ds
f (x) − R b
w(s)ds a
w(s)ds a
a
a
for all x ∈ (a, b).
The constant factor 1 in the right hand side cannot be replaced by a smaller one.
The aim of this paper is to improve the results from Theorems 1.1 – 1.6 using an unitary
method.
J. Ineq. Pure and Appl. Math., 1(2) Art. 11, 2000
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O STROWSKI T YPE I NEQUALITIES FROM A L INEAR F UNCTIONAL P OINT OF V IEW
3
2. AUXILIARY RESULTS
Let X = (X, d) be a compact metric space and C(X) the Banach lattice of real-valued
continuous functions on the compact metric space X = (X, d), endowed with the max norm
k · kX .
For a function f ∈ C(X), the modulus of continuity (with respect to the metric d) is defined
by:
ω(f ; t) = ωd (f ; t) = sup |f (x) − f (y)|, t ≥ 0.
d(x,y)≤t
The least concave majorant of this modulus with respect to the variable t is given by

;x)

sup (t−x)ω(f ;y)+(y−t)ω(f
for 0 ≤ t ≤ d(X);

y−x
 0≤x≤t≤y
x6=y
ω
e (f ; t) =



ω(f ; d(X))
for t > d(X),
where d(X) < ∞ is the diameter of the compact space X.
We denote by LipM α = LipM (α; X) the set of all Lipschitzian functions of order α, α ∈
[0, 1] having the same Lipschitz constant M . That is f ∈ LipM α iff for all x, y ∈ X
|f (x) − f (y)| ≤ M dα (x, y).
We see that
LipM (α; X) = {g ∈ C(X) : ω(g; t) ≤ M tα }.
Let I = [a, b] be a compact interval of the real axis, S a subspace of C(I), and A a linear
functional defined on S. The following definition was given by T. Popoviciu in [8].
Definition 2.1. The linear functional A defined on the subspace S which contains all polynomials is Pn -simple (n ≥ −1) if
(i) A(en+1 ) 6= 0
(ii) for every f ∈ S there are the distinct points t1 , t2 , . . . , tn+2 in [a, b] such that
A(f ) = A(en+1 )[t1 , t2 , . . . , tn+2 ; f ],
where [t1 , t2 , . . . , tn+2 ; f ] is the divided difference of the function f on the points t1 , t2 , . . . , tn+2 .
In [5] the following result is proved. The proof is reproduced here for completeness.
Theorem 2.1. Let A be a bounded linear functional, A : C(I) → R. If A is P0 -simple then for
all f ∈ C(I) we have
kAk
2|A(e1 )|
(2.1)
|A(f )| ≤
ω
e f;
.
2
kAk
Proof. For g ∈ C 1 (I) we have
|A(f )| = |A(f − g) + A(g)| ≤ kAkkf − gk + |A(g)|
≤ kAkkf − gk + |A(e1 )|kg 0 k.
From this inequality we obtain
|A(f )| ≤
inf (kAkkf − gk + |A(e1 )|kg 0 k)
g∈C 1 (I)
and using the following result (see [10])
t 0
1
inf1
kf − gk + kg k = ω
e (f ; t),
g∈C (I)
2
2
we obtain the relation (2.1).
J. Ineq. Pure and Appl. Math., 1(2) Art. 11, 2000
t≥0
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4
B. G AVREA AND I. G AVREA
The following result was proved by I. Raşa [9].
Theorem 2.2. Let k be a natural number such that 0 ≤ k ≤ n and A : C (k) [a, b] → R a
bounded linear functional, A 6= 0, A(ei ) = 0 for i = 0, 1, . . . , n such that for every f ∈
C (k) [a, b] Pn -nonconcave A(f ) ≥ 0. Then A is Pn -simple.
A function f ∈ C (k) [a, b] is called P0 -nonconcave if for any n + 2 points t1 , t2 , . . . , tn+2 ∈
[a, b] the inequality
[t1 , t2 , . . . , tn+2 ; f ] ≥ 0
holds.
Another criterion for Pn -simple functionals was given by A. Lupaş in [6]. He proved that
a bounded linear functional A : C[a, b] → R for which A(ek ) = 0, k = 0, 1, . . . , n and
A(en+1 ) 6= 0 is Pn -simple if and only if A is Pn -simple on C (n+1) [a, b].
Now we can prove the following result (see also [5]):
Theorem 2.3. Let A be a bounded linear functional, A : C(I) → R. If A(e1 ) 6= 0 and the
inequality (2.1) holds for any f ∈ C(I) then A is P0 -simple.
Proof. We can assume that A(e1 ) > 0. Combining the results of I. Raşa and A. Lupaş, it is
sufficient, for the proof of the theorem, to show that
A(f ) ≥ 0
(2.2)
for every nondecreasing differentiable function f defined on I.
For such a function we have
|A(f )| ≤ A(e1 )kf 0 k.
Let B be the linear functional defined by
B(f ) =
where
Z
F (t) =
A(F )
,
A(e1 )
t
f (u)du,
f ∈ C[0, 1].
0
The functional B is bounded and for any f ∈ C(I) we have
|B(f )| ≤ kf k
with B(e0 ) = 1.
Let f be a continuous function such that f ≥ 0, f 6= 0.
From the inequalities
f
0 ≤ e0 −
≤1
kf k
we obtain
B(f ) f 1−
≤ B e0 −
≤ 1.
kf k
kf k These inequalities imply that
(2.3)
B(f ) ≥ 0.
Further, let f be a differentiable function on I such that f 0 ≥ 0, then, from (2.3) we obtain
B(f 0 ) ≥ 0.
Since B(f 0 ) = A(f ), the inequality (2.2) is thus proved.
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O STROWSKI T YPE I NEQUALITIES FROM A L INEAR F UNCTIONAL P OINT OF V IEW
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3. A N I NTEGRAL I NEQUALITY OF O STROWSKI T YPE
The following inequality of Ostrowski type holds.
Theorem 3.1. Let f be a continuous function on [a, b] and w : (a, b) → R+ an integrable
Rb
function on (a, b) such that a ω(s)ds = 1. Then for any continuous function f the following
inequality:
Z x
Rx
Z b
w(t)(x
−
t)dt
a
Rx
(3.1) f (x) −
w(s)f (s)ds ≤
w(t)dt ω
e[a,x] f ;
w(t)
a
a
a
!
Rb
Z b
w(t)(t
−
x)dt
+
w(t)dt ω
e[x,b] f ; x R b
w(t)dt
x
x
holds, where x is a fixed point in (a, b).
Proof. From Theorem 2.3 we get that the linear functionals
A1 : C[a, x] → R,
defined by
A2 : C[x, b] → R
x
Z
x
Z
w(t)dt −
A1 (f ) = f (x)
f (t)w(t)dt
a
a
and
b
Z
b
Z
w(t)dt −
A2 (f ) = f (x)
f (t)w(t)dt
x
x
are P0 -simple.
It is easy to see that:
x
Z
kA1 k = 2
Z
w(t)dt
and
kA2 k = 2
a
From the inequality:
Z
Z b
f (x) −
w(s)f (s)ds ≤
w(t)dt.
x
x
a
a
b
!
Z b
|A1 (e1 )
A2 (e1 )
w(t)dt ω
e f; R x
+
w(t)dt ω
e f; R b
w(t)dt
w(t)dt
x
a
x
and from the results
Z
|A1 (e1 )| =
x
Z
w(t)(x − t)dt
and
|A2 (e1 )| =
a
b
w(t)(t − x)dt,
x
(3.1) follows.
Corollary 3.2. Let f be a continuous function on [a, b], such that f ∈ LipM1 (α, [a, x]) and
f ∈ LipM2 (β; [x, b]). Then
Z x
1−α Z x
α
Z b
(3.2) f (x) −
w(s)f (s)ds ≤ M1
w(t)dt
w(t)(x − t)dt
a
a
a
Z
+ M2
b
1−β Z b
β
w(t)dt
w(t)(t − x)dt .
x
x
Proof. The proof follows from the inequality (3.1) and the fact that
ω
e1 (g; t) ≤ M tr
for any function g, g ∈ LipM (α, [c, d]), where ω
e1 is taken on the interval [c, d].
Corollary 3.2 is an improvement of the result of Theorem 1.6.
J. Ineq. Pure and Appl. Math., 1(2) Art. 11, 2000
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6
B. G AVREA AND I. G AVREA
1
Remark 3.1. In the particular case when w(t) = b−a
the inequality (3.2) becomes:
R
b
f (s)ds (x − a)α+1
(b − x)β+1
1
a
(3.3)
+ M2
f (x) −
≤ M1
α
β
b−a 2
2
b−a
#
"
2
x − a+b
1
2
≤ max(M1 , M2 )
+
(b − a).
4
(b − a)2
Inequality (3.3) improves the inequality (1.5).
Corollary 3.3. Let f : [a, b] → R be continuous on (a, b), whose derivative f 0 : (a, b) → R is
bounded on (a, b) and w a function as in Theorem 3.1. Then we have the following inequality:
Z x
Z b
Z b
(3.4)
w(s)f (s)ds ≤
w(t)(x − t)dt +
w(t)(t − x)dt kf 0 k∞ .
f (x) −
a
a
x
Proof. The above inequality is a consequence of the inequality (3.1) and the fact that
ω
e (f ; t) ≤ kf 0 k∞ t.
The inequality of Ostrowski follows from (3.4) if we consider
1
w(t) =
, t ∈ [a, b].
b−a
Corollary 3.4. Let f : I → R be a mapping with bounded variation and w a function as in
Theorem 3.1. Then for all x ∈ [a, b] we have the inequalities
Z b
Z x
Z b
x
b
_
_
(3.5)
w(s)f (s)ds ≤
f
w(t)dt + f
w(t)dt
f (x) −
a
(3.6)
a
a

Z b
f (x) −
≤ 1 +
w(s)f
(s)ds
2
a
x
x
R

Rb
x
b
w(t)dt
−
w(t)dt
a
_
x
 f.
2
a
Proof. It is clear that
(3.7)
ω
e [a, x](f ; t) ≤
x
_
a
f
and
ω
e [x, b](f, t) ≤
b
_
f
x
for every positive number t.
Thus, inequality (3.5) follows from (3.7).
Rb
Rx
For the proof of the inequality (3.6) we note that, if we suppose a w(t)dt ≤ 12 then x w(t)dt ≥
1
and vice versa.
2
For definiteness we assume that
Z x
Z b
1
1
w(t)dt ≤
and
w(t)dt ≥ .
2
2
a
x
We then have
Z x
Z b
Z b
x
b
x
b
_
_
_
1_
f
w(t)dt + f
w(t)dt ≤
f+ f
w(t)dt
2 a
a
x
x
a
x
x
Z b
b
b
_
1_
1
f+ f
w(t)dt −
=
2 a
2
x
x
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O STROWSKI T YPE I NEQUALITIES FROM A L INEAR F UNCTIONAL P OINT OF V IEW
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and so
(3.8)
Z
x
_
f
a
x
Z b
b
_
w(t)dt + f
w(t)dt ≤
a
x
x
1
+
2
Rb
x
w(t)dt −
2
Rx
a
w(t)dt
!
b
_
f.
a
From the inequalities (3.5) and (3.8), the inequality (3.6) follows.
Remark 3.2. The inequality from Theorem 1.4 follows if we take in (3.6)
1
w(t) =
.
b−a
Theorem 3.5. Let g be a continuous differentiable function on [a, b] such that g(a) = g(b) = 0.
Then the inequality
Z b
3
3
g(x)
(x − a)2 + (b − x)2
1
2
(x
−
a)
+
(y
−
b)
0
(3.9)
g(t)dt ≤
ω
e g;
2 − b−a
8(b − a)
3 (x − a)2 + (y − b)2
a
holds, where x is an arbitrary point in (a, b).
Proof. The following functional A defined on C[a, b] by
Z b
a+b
1
t−
f (t)dt
A(f ) =
b−a a
2
is a linear bounded functional having its norm equal to b−a
. For every increasing function f we
4
have:
A(f ) ≥ 0.
Using Theorem 2.3, we deduce that the functional A is P0 -simple with
(b − a)2
.
12
From Theorem 2.1, we obtain the following inequality:
Z b
1
b−a
a+b
2
(3.10)
t−
f (t)dt ≤
ω
e f ; (b − a) .
b − a
2
8
3
A(e1 ) =
a
Inequality (3.10) holds for every continuous function f .
Let us suppose that f is differentiable on [a, b]. From the inequality (3.10) (written for f 0 ) we
obtain the following inequality:
Z b
1
b−a
f
(a)
+
f
(b)
2
0
≤
(3.11)
f (t)dt −
ω
e f ; (b − a) .
b − a
2
8
3
a
Now, we can prove the inequality (3.9). We have the following identity:
Z b
Z x
g(x)
1
x−a
1
g(a) + g(x)
(3.12) −
+
g(t)dt =
g(t)dt −
2
b−a a
b−a x−a a
2
Z b
b−x
1
g(b) + g(x)
+
g(t)dt −
.
b−a b−x a
2
Using the relations (3.11) and (3.12) we obtain
Z b
g(x)
1
(3.13) −
g(t)dt
2
b−a a
(x − a)2
(b − x)2
0 2
0 2
≤
ω
e g ; (x − a) +
ω
e g ; (b − x) .
8(b − a)
3
8(b − a)
3
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8
B. G AVREA AND I. G AVREA
As the function ω
e (g 0 ; ·), is concave, then from (3.13) and using Jensen’s inequality, we obtain
the inequality (3.9).
Corollary 3.6. Let g be a continuous differentiable function on [a, b] such that g(a) = g(b) = 0,
then the following inequality
#
"
Z b
a+b 2
g(x)
x
−
1
1
2
(3.14)
(b − a)kg 0 k∞
g(t)dt ≤
+
2 − b−a
8
2(b − a)
a
is valid for all x ∈ [a, b].
Proof. It is well known that
(3.15)
ω
e (g 0 ; t) ≤ 2kg 0 k∞ ,
for every positive number t.
The inequality (3.15) then readily follows from the inequality (3.14).
Remark 3.3. The result from the Theorem 1.3 can be written in terms of ω
e using the inequality
(3.13) for the function
x−a
b−x
f (b) −
f (a).
g(x) = f (x) −
b−a
b−a
In [5] the following result was proved:
Let A be a linear positive functional A : C[0, 1] → R, A(e0 ) = 1 and ϕ, ϕ : [0, 1] → R a
continuous increasing function such that A(e1 ϕ) − A(e1 )A(ϕ) > 0. Then the following Grüss
type inequality
A(|ϕ − A(ϕ)|)
2(A(e1 ϕ) − A(e1 )A(ϕ))
(3.16)
|A(ϕf ) − A(ϕ)A(f )| ≤
ω
e f;
2
A(|ϕ − A(ϕ)|)
holds.
We are interested in the following open problem:
Open problem. Let A be a linear positive functional defined on C[0, 1] and f, g be two
continuous functions. Do positive numbers δ1 = δ1 (f ) < 1 and δ2 = δ2 (f ) < 1 exist such that
1
|A(f g) − A(f )A(g)| ≤ ω
e (f ; δ1 )e
ω (f, δ2 )?
4
R EFERENCES
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applications, Math. Inequal. Appl. (in press).
[2] S.S. DRAGOMIR, On the Ostrowski integral inequality for Lipschitzian mappings and applications, Comput. Math. Appl., 38 (1999), 33–37.
[3] S.S. DRAGOMIR, P. CERONE, J. ROUMELIOTIS AND S. WANG, A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math.
Soc. Sc. Math. Roumanie, 42(90)(4), (1999), 301–314.
[4] S.S. DRAGOMIR AND S. WANG, An Inequality of Ostrowski-Grüss’ type and its applications to
the estimation of error bounds for some special means and for some numerical quadrature rules,
Comput. Math. Appl., 33(11) (1997), 15–20.
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O STROWSKI T YPE I NEQUALITIES FROM A L INEAR F UNCTIONAL P OINT OF V IEW
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[7] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers.
[8] T. POPOVICIU, Sur le reste dans certains formules lineaires d’approximation de l’analyse, Mathematica, Cluj., 1(24), 95–142.
[9] I. RAŞA, Sur les fonctionnelles de la forme simple au sens de T. Popoviciu, L’Anal. Num. et la
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J. Ineq. Pure and Appl. Math., 1(2) Art. 11, 2000
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