Volume 8 (2007), Issue 4, Article 117, 13 pp.
SHARP GRÜSS-TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES
ARE OF BOUNDED VARIATION
SEVER S. DRAGOMIR
S CHOOL OF C OMPUTER S CIENCE AND M ATHEMATICS
V ICTORIA U NIVERSITY
PO B OX 14428, M ELBOURNE C ITY
VIC 8001, AUSTRALIA .
sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/dragomir
Received 05 June, 2007; accepted 31 October, 2007
Communicated by G. Milovanović
A BSTRACT. Sharp Grüss-type inequalities for functions whose derivatives are of bounded variation (Lipschitzian or monotonic) are given. Applications in relation with the well-known Čebyšev, Grüss, Ostrowski and Lupaş inequalities are provided as well.
Key words and phrases: Riemann-Stieltjes integral, Functions of bounded variation, Lipschitzian functions, Integral inequalities, Čebyšev, Grüss, Ostrowski and Lupaş type inequalities.
2000 Mathematics Subject Classification. 26D15, 26D10, 41A55.
1. I NTRODUCTION
In 1998, S.S. Dragomir and I. Fedotov [10] introduced the following Grüss type error functional
Z b
Z b
1
D (f ; u) :=
f (t) du (t) − [u (a) − u (b)] ·
f (t) dt
b−a a
a
Rb
in order to approximate the Riemann-Stieltjes integral a f (t) du (t) by the simpler quantity
Z b
1
[u (a) − u (b)] ·
f (t) dt.
b−a a
In the same paper the authors have shown that
1
(1.1)
|D (f ; u)| ≤ · L (M − m) (b − a) ,
2
provided that u is L−Lipschitzian, i.e., |u (t) − u (s)| ≤ L |t − x| for any t, s ∈ [a, b] and f is
Riemann integrable and satisfies the condition
−∞ < m ≤ f (t) ≤ M < ∞
The constant
quantity.
188-07
1
2
for any t ∈ [a, b] .
is best possible in (1.1) in the sense that it cannot be replaced by a smaller
2
S EVER S. D RAGOMIR
In [11], the same authors established another result for D (f ; u) , namely
b
_
1
|D (f ; u)| ≤ K (b − a) (u) ,
2
a
(1.2)
W
provided that u is of bounded variation on [a, b] with the total variation ba (u) and f is
K−Lipschitzian. Here 12 is also best possible.
In [8], by introducing the kernel Φu : [a, b] → R given by
1
(1.3)
Φu (t) :=
[(t − a) u (b) + (b − t) u (a)] − u (t) , t ∈ [a, b] ,
b−a
the author has obtained the following integral representation
Z b
(1.4)
D (f ; u) =
Φu (t) df (t) ,
a
Rb
where u, f : [a, b] → R are bounded functions such that the Riemann-Stieltjes integral a f (t) du (t)
Rb
and the Riemann integral a f (t) dt exist. By the use of this representation he also obtained the
following bounds for D (f ; u) ,
(1.5)
|D (f ; u)|

W

sup |Φu (t)| · ba (f ) if u is continuous and f is of bounded variation;



t∈[a,b]



Rb
≤
L a |Φu (t)| dt if u is Riemann integrable and f is L-Lipschitzian;






 Rb
|Φu (t)| dt if u is continuous and f is monotonic nondecreasing.
a
If u is monotonic nondecreasing and K (u) is defined by
Z b
a+b
4
u (t) dt (≥ 0) ,
K (u) :=
t−
2
(b − a)2 a
then
1
1
|D (f ; u)| ≤ L (b − a) [u (b) − u (a) − K (u)] ≤ L (b − a) [u (b) − u (a)] ,
2
2
provided that f is L−Lipschitzian on [a, b] .
Here 21 is best possible in both inequalities.
Also, for u monotonic nondecreasing on [a, b] and by defining Q (u) as
Z b
1
a+b
Q (u) :=
u (t) sgn t −
dt (≥ 0) ,
b−a a
2
(1.6)
we have
(1.7)
|D (f ; u)| ≤ [u (b) − u (a) − Q (u)] ·
b
_
(f ) ≤ [u (b) − u (a)] ·
a
b
_
(f ) ,
a
provided that f is of bounded variation on [a, b] . The first inequality in (1.7) is sharp.
Finally, the case when u is convex and f is of bounded variation produces the bound
(1.8)
b
_
1 0
0
|D (f ; u)| ≤
u (b) − u+ (a) (b − a) (f ) ,
4 −
a
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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G RÜSS - TYPE I NEQUALITIES
3
with 14 the best constant (when u0− (b) and u0+ (a) are finite) and if f is monotonic nodecreasing
and u is convex on [a, b] , then
(1.9)
0 ≤ D (f ; u)
Z b
u0− (b) − u0+ (a)
a+b
≤2·
·
t−
f (t) dt
b−a
2
a
 1 0
u− (b) − u0+ (a) max {|f (a)| , |f (b)|} (b − a)

2




0
1/q
1
0
if p > 1,
≤
1/q u− (b) − u+ (a) kf kp (b − a)
(q+1)




 0
u− (b) − u0+ (a) kf k1 ,
1
p
+
1
q
= 1;
are sharp constants (when u0− (b) and u0+ (a) are finite) and k·kp are the usual
R
p1
b
Lebesgue norms, i.e., kf kp := a |f (t)|p dt , p ≥ 1.
The main aim of the present paper is to provide sharp upper bounds for the absolute value of
D (f ; u) under various conditions for u0 , the derivative of an absolutely continuous function u,
and f of bounded variation (Lipschitzian or monotonic). Natural applications for the Čebyšev
functional that complement the classical results due to Čebyšev, Grüss, Ostrowski and Lupaş
are also given.
where 2 and
1
2
2. P RELIMINARY R ESULTS
We have the following integral representation of Φu .
Lemma 2.1. Assume that u : [a, b] → R is absolutely continuous on [a, b] and such that the
derivative u0 exists on [a, b] (eventually except at a finite number of points). If u0 is Riemann
integrable on [a, b] , then
Z b
1
(2.1)
Φu (t) :=
K (t, s) du0 (s) , t ∈ [a, b] ,
b−a a
where the kernel K : [a, b]2 → R is given by
(
(b − t) (s − a) if s ∈ [a, t] ,
(2.2)
K (t, s) :=
(t − a) (b − s) if s ∈ (t, b].
Proof. We give, for simplicity, a proof only in the case when u0 is defined on the entire interval,
and for which we have used the usual convention that u0 (a) := u0+ (a) , u0 (b) := u0− (b) and the
lateral derivatives are finite.
Since u0 is assumed to be Riemann integrable on [a, b] , it follows that the Riemann-Stieltjes
Rt
Rb
integrals a (s − a) du0 (s) and t (b − s) du0 (s) exist for each t ∈ [a, b] . Now, integrating by
parts in the Riemann-Stieltjes integral, we have succesively
Z b
Z t
Z b
0
0
K (t, s) du (s) = (b − t)
(s − a) du (s) + (t − a)
(b − s) du0 (s)
a
t
a
t Z t
0
0
= (b − t) (s − a) u (s) −
u (s) ds
a
a
b Z b
0
0
+ (t − a) (b − s) u (s) −
u (s) ds
t
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
t
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4
S EVER S. D RAGOMIR
= (b − t) [(t − a) u0 (t) − (u (t) − u (a))]
+ (t − a) [− (b − t) u0 (t) + u (b) − u (t)]
= (t − a) [u (b) − u (t)] − (b − t) [u (t) − u (a)]
= (b − a) Φu (t) ,
for any t ∈ [a, b] , and the representation (2.1) is proved.
The following result provides a sharp bound for |Φu | in the case when u0 is of bounded
variation.
Theorem 2.2. Assume that u : [a, b] → R is as in Lemma 2.1. If u0 is of bounded variation on
[a, b] , then
b
b
_
(t − a) (b − t) _ 0
1
|Φu (t)| ≤
(u ) ≤ (b − a) (u0 ) ,
b−a
4
a
a
(2.3)
W
where ba (u0 ) denotes the total variation of u0 on [a, b] .
The inequalities are sharp and the constant 14 is best possible.
Proof. It is well known that, if p : [α, β] → R is continuous and v : [α, β] → R is of bounded
Rβ
variation, then the Riemann-Stieltjes integral α p (s) dv (s) exists and
Z β
β
_
p (s) dv (s) ≤ sup |p (s)| (v) .
α
s∈[α,β]
α
Now, utilising the representation (2.1) we have successively:
(2.4)
|Φu (t)|
Z t
Z b
1
0
0
≤
(b − t) (s − a) du (s) + (t − a) (b − s) du (s)
b−a
a
t
"
#
t
b
_
_
1
(b − t) sup (s − a) · (u0 ) + (t − a) sup (b − s) · (u0 )
≤
b−a
s∈[a,t]
s∈[t,b]
a
t
" t
#
b
b
_
(t − a) (b − t) _ 0
(t − a) (b − t) _ 0
(u ) + (u0 ) =
(u ) .
=
b−a
b
−
a
a
t
a
The second inequality is obvious by the fact that (t − a) (b − t) ≤ 41 (b − a)2 , t ∈ [a, b] .
For the sharpness of the inequalities, assume that there exist A, B > 0 so that
b
(2.5)
b
_
(t − a) (b − t) _ 0
|Φu (t)| ≤ A ·
(u ) ≤ B (b − a) (u0 ) ,
b−a
a
a
with u as in the assumption of the theorem. Then, for t = a+b
, we get from (2.5) that
2
b
b
_
_
u (a) + u (b)
1
a
+
b
0
(2.6)
−u
≤ A (b − a) (u ) ≤ B (b − a) (u0 ) .
2
2
4
a
a
. This function is absolutely conConsider the function u : [a, b] → R, u (t) = t − a+b
2 W
tinuous, u0 (t) = sgn t − a+b
, t ∈ [a, b] \ a+b
and ba (u0 ) = 2. Then (2.6) becomes
2
2
b−a
≤ 12 A (b − a) ≤ 2B (b − a) , which implies that A ≥ 1 and B ≥ 14 .
2
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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G RÜSS - TYPE I NEQUALITIES
5
Corollary 2.3. With the assumptions of Theorem 2.2, we have
b
_
u (a) + u (b)
a + b 1
0
(2.7)
−u
≤ 4 (b − a) (u ) .
2
2
a
The constant
1
4
is best possible.
The Lipschitzian case is incorporated in the following result.
Theorem 2.4. Assume that u : [a, b] → R is absolutely continuous on [a, b] with the property
that u0 is K−Lipschitzian on (a, b) . Then
1
1
(t − a) (b − t) K ≤ (b − a)2 K.
2
8
are best possible.
|Φu (t)| ≤
(2.8)
The constants
1
2
and
1
8
Proof. We utilise the fact that, for an L−Lipschitzian function p : [α, β] → R and a Riemann
Rβ
integrable function v : [α, β] → R, the Riemann-Stieltjes integral α p (s) dv (s) exists and
Z β
Z β
p (s) dv (s) ≤ L
|p (s)| ds.
α
α
Then, by (2.1), we have that
Z t
Z b
1
0
0
(2.9)
|Φu (t)| ≤
(b − t) (s − a) du (s) + (t − a) (b − s) du (s)
b−a
a
t
1
1
1
2
2
K (b − t) (t − a) + K (t − a) (b − t)
≤
b−a 2
2
1
= (t − a) (b − t) K,
2
which proves the first part of (2.8). The second part is obvious.
Now, for the sharpness of the constants, assume that there exist the constants C, D > 0 such
that
|Φu (t)| ≤ C (b − t) (t − a) K ≤ D (b − a)2 K,
(2.10)
provided that u is as in the hypothesis of the theorem. For t = a+b
, we get from (2.10) that
2
u (a) + u (b)
a + b 1
(2.11)
−
u
≤ CK (b − a)2 ≤ D (b − a)2 K.
2
2
4
2 . Then u0 (t) = t − a+b is Lipschitzian with the
Consider u : [a, b] → R, u (t) = 12 t − a+b
2
2
constant K = 1 and (2.11) becomes
1
1
(b − a)2 ≤ C (b − a)2 ≤ D (b − a)2 ,
8
4
which implies that C ≥ 12 and D ≥ 18 .
Corollary 2.5. With the assumptions of Theorem 2.4, we have
u (a) + u (b)
1
a
+
b
≤ (b − a)2 K.
−u
(2.12)
8
2
2
The constant
1
8
is best possible.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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6
S EVER S. D RAGOMIR
Remark 2.6. If u0 is absolutely continuous and ku00 k∞ := ess supt∈[a,b] |u00 (t)| < ∞, then we
can take K = ku00 k∞ , and we have from (2.8) that
1
1
(2.13)
|Φu (t)| ≤ (t − a) (b − t) ku00 k∞ ≤ (b − a)2 ku00 k∞ .
2
8
1
1
The constants 2 and 8 are best possible in (2.13).
From (2.12) we also get
1
u (a) + u (b)
a
+
b
≤ (b − a)2 ku00 k ,
(2.14)
−u
∞
8
2
2
in which
1
8
is the best possible constant.
3. B OUNDS IN THE C ASE WHEN u0
IS OF
B OUNDED VARIATION
We can start with the following result:
Theorem 3.1. Assume that u : [a, b] → R is as in Lemma 2.1. If u0 and f are of bounded
variation on [a, b] , then
b
|D (f ; u)| ≤
(3.1)
and the constant
1
4
b
_
_
1
(b − a) (u0 ) · (f ) ,
4
a
a
is best possible in (3.1).
Proof. We use the following representation of the functional D (f ; u) obtained in [8] (see also
[9] or [6]):
Z b
(3.2)
D (f ; u) =
Φu (t) df (t) .
a
Then we have the bound
Z b
b
_
|D (f ; u)| = Φu (t) df (t) ≤ sup |Φu (t)| (f )
t∈[a,b]
a
a
b
≤
b
_
1 _ 0
(u ) sup [(t − a) (b − t)] · (f )
b−a a
t∈[a,b]
a
b
b
_
_
1
= (b − a) (u0 ) · (f ) ,
4
a
a
where, for the last inequality we have used (2.3).
To prove the sharpness of the constant 14 , assume that there is a constant E > 0 such that
(3.3)
|D (f ; u)| ≤ E (b − a)
b
_
a
(u0 ) ·
b
_
(f ) .
a
. Then u0 (t) = sgn t − a+b , t ∈ [a, b] \ a+b .
Consider u : [a, b] → R, u (t) = t − a+b
2
2
2
The total variation on [a, b] is 2 and
Z b
Z a+b
Z b
2
a+b
f (t) dt +
f (t) dt =
sgn t −
f (t) dt.
D (f ; u) = −
a+b
2
a
a
2
a+b
Now, if we choose f (t) = sgn t − 2 , then we obtain from (3.1) b − a ≤ 4E (b − a) , which
implies that E ≥ 14 .
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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G RÜSS - TYPE I NEQUALITIES
7
The following result can be stated as well:
Theorem 3.2. Assume that u : [a, b] → R is as in Lemma 2.1. If the derivative u0 is of bounded
variation on [a, b] while f is L−Lipschitzian on [a, b] , then
b
(3.4)
_
1
|D (f ; u)| ≤ L (b − a)2 (u0 ) .
6
a
Proof. We have
Z b
Z b
|D (f ; u)| = Φu (t) df (t) ≤ L
|Φu (t)| dt
a
a
Z b
b
L _ 0
≤
(u )
(t − a) (b − t) dt
b−a a
a
b
_
1
= L (b − a)2 (u0 ) ,
6
a
where for the second inequality we have used the inequality (2.3).
1
6
is the best possible constant
Remark 3.3. It is an open problem whether or not the constant
in (3.4).
When the integrand f is monotonic, we can state the following result as well:
Theorem 3.4. Assume that u is as in Theorem 3.1. If f is monotonic nondecreasing on [a, b] ,
then
Wb 0 Z b a
+
b
a (u )
t −
f (t) dt
(3.5)
|D (f ; u)| ≤ 2 ·
·
b−a
2 a
 Wb
1
0


a (u ) max {|f (a)| , |f (b)|} (b − a) ;
2




Wb 0
1/q
1
if p > 1, p1 + 1q = 1;
≤
1/q
a (u ) kf kp (b − a)
(q+1)




 Wb 0

a (u ) kf k1 ,
R
p1
b
where kf kp := a |f (t)|p dt , p ≥ 1 are the Lebesgue norms. The constants 2 and
best possible in (3.5).
1
2
are
Proof. It is well known that, if p : [α, β] → R is continuous and v : [α, β] →
R R is monotonic
Rβ
β
nondecreasing, then the Riemann-Stieltjes integral α p (t) dv (t) exists and α p (t) dv (t) ≤
Rβ
Rb
|p
(t)|
dv
(t)
.
Then,
on
applying
this
property
for
the
integral
Φu (t) df (t) , we have
α
a
Z b
Z b
(3.6)
|D (f ; u)| = Φu (t) df (t) ≤
|Φu (t)| df (t)
a
Wb
≤
a
0
(u )
·
b−a
a
Z
b
(t − a) (b − t) df (t) ,
a
where for the last inequality we used (2.3).
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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8
S EVER S. D RAGOMIR
Integrating by parts in the Riemann-Stieltjes integral, we have
Z b
b Z b
(t − a) (b − t) df (t) = f (t) (b − t) (t − a) −
[−2t + (a + b)] f (t) dt
a
a
a
Z b
a+b
=2
t−
f (t) dt,
2
a
which together with (3.6) produces the first part of (3.5).
Rb
The second part is obvious by the Hölder inequality applied for the integral a t − a+b
f (t) dt
2
and the details are omitted.
and f (t) = sgn t − a+b ,
For the sharpness of the constants we use as examples u (t) = t − a+b
2
2
t ∈ [a, b] . The details are omitted.
4. B OUNDS IN THE C ASE
WHEN
u0
L IPSCHITZIAN
IS
The following result can be stated as well:
Theorem 4.1. Let u : [a, b] → R be absolutely continuous on [a, b] with the property that u0 is
K−Lipschitzian on (a, b) . If f is of bounded variation, then
b
_
1
|D (f ; u)| ≤ (b − a)2 K
(f ) .
8
a
(4.1)
The constant
1
8
is best possible in (4.1).
Proof. Utilising (2.8), we have successively:
Z b
b
_
|D (f ; u)| = Φu (t) df (t) ≤ sup |Φu (t)| (f )
t∈[a,b]
a
1
≤ K sup [(b − t) (t − a)]
2 t∈[a,b]
=
1
(b − a)2 K
8
b
_
a
b
_
(f )
a
(f ) ,
a
and the inequality (4.1) is proved.
Now, for the sharpness of the constant, assume that the inequality holds with a constant
G > 0, i.e.,
(4.2)
2
|D (f ; u)| ≤ G (b − a) K
b
_
(f ) .
a
for u and f as in the statement of
the theorem.
a+b 2
1
and f (t) = sgn t − a+b
, t ∈ [a, b] . Then u0 (t) = t − a+b
Consider u (t) := 2 t − 2
2
2
is K−Lipschitzian with the constant K = 1 and
Z b
a+b
a+b
(b − a)2
D (f ; u) =
sgn t −
· t−
dt =
.
2
2
4
a
2
W
≤ 2G (b − a)2 , which implies that G ≥
Since ba (f ) = 2, hence from (4.2) we get (b−a)
4
1
.
8
The following result may be stated as well:
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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G RÜSS - TYPE I NEQUALITIES
9
Theorem 4.2. Let v : [a, b] → R be as in Theorem 4.1. If f is L−Lipschitzian on [a, b] , then
1
(b − a)3 KL.
(4.3)
|D (f ; u)| ≤
12
1
The constant 12
is best possible in (4.3).
Proof. We have by (2.8), that:
Z b
|D (f ; u)| = Φu (t) df (t)
a
Z b
≤L
|Φu (t)| dt
a
Z b
1
1
(b − t) (t − a) dt = KL (b − a)3 ,
≤ LK
2
12
a
and the inequality is proved.
For the sharpness, assume that (4.3) holds with a constant F > 0. Then
|D (f ; u)| ≤ F (b − a)3 KL,
(4.4)
provided f and u are as in the hypothesis of the theorem.
2
Consider f (t) = t − a+b
and u (t) = 12 t − a+b
. Then u0 is Lipschitzian with the constant
2
2
K = 1 and f is Lipschitzian with the constant L = 1. Also,
2
Z b
a+b
(b − a)3
D (f ; u) =
t−
dt =
,
2
12
a
and by (4.4) we get
(b−a)3
12
≤ F (b − a)3 which implies that F ≥ 12 .
Finally, the case of monotonic integrands is enclosed in the following result.
Theorem 4.3. Let u : [a, b] → R be as in Theorem 4.1. If f is monotonic nondecreasing, then
Z b
a+b
(4.5)
|D (f ; u)| ≤ K
t−
f (t) dt
2
a
 1
K max {|f (a)| , |f (b)|} (b − a)2 ;

4




1+1/q
1
if p > 1, p1 + 1q = 1;
≤
1/q K kf kp (b − a)
2(q+1)




 1
(b − a) K kf k1 .
2
The first inequality is sharp. The constant
1
4
is best possible.
Proof. We have
Z
b
|D (f ; u)| ≤
|Φu (t)| df (t)
Z b
1
≤ K
(b − t) (t − a) df (t)
2
a
Z b
a+b
f (t) dt
=K
t−
2
a
and the first inequality is proved. The second part follows by the Hölder inequality.
sharpness
of the first inequality
and of the constant 14 follows by choosing u (t) =
The
a+b
a+b
t −
and f (t) = sgn t −
. The details are omitted.
2
2
a
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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10
S EVER S. D RAGOMIR
5. A PPLICATIONS FOR THE Č EBYŠEV F UNCTIONAL
The above result can naturally be applied in obtaining various sharp upper bounds for the
absolute value of the Čebyšev functional C (f, g) defined by
Z b
Z b
Z b
1
1
1
(5.1)
C (f, g) :=
f (t) g (t) dt −
f (t) dt ·
g (t) dt,
b−a a
b−a a
b−a a
where f, g : [a, b] → R are Lebesgue integrable functions such that f g is also Lebesgue integrable.
There are various sharp upper bounds for |C (f, g)| and in the following we will recall just a
few of them.
In 1934, Grüss [13] showed that
1
(M − m) (N − n)
4
under the assumptions that f and g satisfy the bounds
(5.2)
|C (f, g)| ≤
(5.3)
−∞ < m ≤ f (t) ≤ M < ∞
and
− ∞ < n ≤ g (t) ≤ N < ∞
for almost every t ∈ [a, b] , where m, M, n, N are real numbers. The constant 41 is best possible
in the sense that it cannot be replaced by a smaller quantity.
Another less known result, even though it was established by Čebyšev in 1882 [1], states that
1
kf 0 k∞ kg 0 k∞ (b − a)2 ,
12
0 0
provided that f , g exist and are continuous in [a, b] and kf 0 k∞ = supt∈[a,b] |f 0 (t)| . The con1
stant 12
cannot be replaced by a smaller quantity. The Čebyšev inequality also holds if f, g are
absolutely continuous on [a, b] , f 0 , g 0 ∈ L∞ [a, b] and k·k∞ is replaced by the ess sup norm
kf 0 k∞ = ess supt∈[a,b] |f 0 (t)| .
In 1970, A. Ostrowski [16] considered a mixture between Grüss and Čebyšev inequalities by
proving that
(5.4)
|C (f, g)| ≤
1
(b − a) (M − m) kg 0 k∞ ,
8
provided that f satisfies (5.3) and g is absolutely continuous and g 0 ∈ L∞ [a, b] .
Three years after Ostrowski, A. Lupaş [14] obtained another bound for C (f, g) in terms of
the Euclidean norms of the derivatives. Namely, he proved that
(5.5)
(5.6)
|C (f, g)| ≤
|C (f, g)| ≤
1
(b − a) kf 0 k2 kg 0 k2 ,
π2
provided that f and g are absolutely continuous and f 0 , g 0 ∈ L2 [a, b] . Here π12 is also best
possible.
Recently, Cerone and Dragomir [2], proved the following result:
Z b
Z b
1
1
f (t) −
dt,
(5.7)
|C (f, g)| ≤ inf kg − γk∞ ·
f
(s)
ds
γ∈R
b−a a b−a a
provided f ∈ L [a, b] and g ∈ C [a, b] .
As particular cases of (5.7), we can state the results:
Z b
Z b
1
1
dt
(5.8)
|C (f, g)| ≤ kgk∞
f
(t)
−
f
(s)
ds
b−a a b−a a
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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G RÜSS - TYPE I NEQUALITIES
11
if g ∈ C [a, b] and f ∈ L [a, b] and
(5.9)
1
1
|C (f, g)| ≤ (M − m)
2
b−a
Z b
Z b
1
f (t) −
dt,
f
(s)
ds
b−a a
a
where m ≤ g (x) ≤ M for x ∈ [a, b] . The constants 1 in (5.8) and 12 in (5.9) are best possible.
The inequality (5.9) has been obtained before in a different way in [5].
For generalisations in abstract Lebesgue spaces, best constants and discrete versions, see [3].
For other results on the Čebyšev functional, see [6], [7] and [12].
R t Now, assume that g : [a, b] → R is Lebesgue integrable on [a, b] . Then the function u (t) :=
g (s) ds is absolutely continuous on [a, b] and we can consider the function
a
Z t
Z
t−a b
(5.10)
Φ̃g (t) := Φu (t) =
g (s) ds −
g (s) ds, t ∈ [a, b] .
b−a a
a
Utilising Lemma 2.1, we can state the following representation result.
Lemma 5.1. If g is absolutely continuous, then
Z b
1
(5.11)
Φ̃g (t) =
K (t, s) dg (s) ,
b−a a
t ∈ [a, b] ,
where K is given by (2.2).
As a consequence of Theorems 2.2 and 2.4, we also have the inequalities:
Proposition 5.2. Assume that g is Lebesgue integrable on [a, b] .
(i) If g is of bounded variation on [a, b] , then
b
b
(t − a) (b − t) _
_
1
(g) ≤ (b − a) (g) .
Φ̃g (t) ≤
b−a
4
a
a
(5.12)
The inequalities are sharp and 14 is best possible.
(ii) If g is K−Lipschitzian on [a, b] , then
1
1
2
(5.13)
Φ̃g (t) ≤ (b − t) (t − a) K ≤ (b − a) K.
2
8
The constants
1
2
and
1
8
are best possible.
We notice that the functions g1 : [a, b] → R, g1 (t) = sgn t − a+b
and g2 : [a, b] → R,
2
a+b
g (t) = t − 2 realise equality in (5.12) and (5.13), respectively.
Rt
Now, we observe that for u (t) = a g (s) ds, s ∈ [a, b] , we have the identity:
D (f, u) = (b − a) C (f, g) .
(5.14)
Utilising this identity and Theorems 3.1 and 3.4, we can state the following result.
Proposition 5.3. Assume that g is of bounded variation on [a, b] .
(i) If f is of bounded variation on [a, b] , then
b
|C (f, g)| ≤
(5.15)
The constant
1
4
b
_
1_
(g) · (f ) .
4 a
a
is best possible in (5.15).
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 117, 13 pp.
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12
S EVER S. D RAGOMIR
(ii) If f is monotonic nondecreasing, then
(5.16)
b
_
Z b
a+b
1
|C (f, g)| ≤ 2 (g) ·
t−
f (t) dt
2
(b − a)2 a
a

W
1

· ba (g) max {|f (a)| , |f (b)|} ;

2




Wb
−1/p
1
if p > 1,
≤
1/q
a (g) kf kp (b − a)
(q+1)





 1 Wb (g) kf k .
a
1
b−a
The multiplicative constants 2 and
1
2
1
p
+
1
q
= 1;
are best possible in (5.16).
Finally, by Theorems 4.1 – 4.3 we also have the following sharp bounds for the Čebyšev
functional C (f, g) .
Proposition 5.4. Assume that g is K−Lipschitzian on [a, b] .
(i) If f is of bounded variation, then
b
(5.17)
_
1
|C (f, g)| ≤ · (b − a) K
(f ) .
8
a
The constant 18 is best possible.
(ii) If f is L−Lipschitzian, then
1
(b − a)2 KL.
12
1
The constant 12
is best possible in (5.18).
(iii) If f is monotonic nondecreasing, then
Z b
1
a+b
(5.19)
|C (f, g)| ≤ K ·
t−
f (t) dt
b−a a
2
 1
K (b − a) max {|f (a)| , |f (b)|} ;

4




1
K (b − a)1/q kf kp if p > 1,
≤
2(q+1)1/q




 1
K kf k1 .
2
(5.18)
|C (f, g)| ≤
The first inequality is sharp. The constant
1
4
1
p
+
1
q
= 1;
is best possible.
Remark 5.5. The inequalities (5.15) and (5.17) were obtained by P. Cerone and S.S. Dragomir
in [4, Corollary 3.5]. However, the sharpnes of the constants 14 and 18 were not discussed there.
Inequality (5.18) is similar to the Čebyšev inequality (5.4).
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