Journal of Inequalities in Pure and
Applied Mathematics
ON SOME GENERALIZATIONS OF STEFFENSEN’S INEQUALITY
AND RELATED RESULTS
volume 2, issue 3, article 28,
2001.
P. CERONE
School of Communications and Informatics
Victoria University of Technology
PO Box 14428
Melbourne City MC
8001 Victoria, Australia
EMail: pc@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/cerone
Received 8 January, 2001;
accepted 19 March, 2001.
Communicated by: H. Gauchman
Abstract
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2000
Victoria University
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Abstract
Steffensen’s inequality is generalised to allow bounds involving any two subintervals rather than restricting them to include the end points. Further results are
obtained involving an identity related to the generalised Chebychev functional
in which the difference of the mean of the product of functions and the product
of means of functions over different intervals is utilised. Bounds involving one
subinterval are also presented.
On Some Generalisations of
Steffensen’s Inequality and
Related Results
2000 Mathematics Subject Classification: 26D15, 26D10, 26D99.
Key words: Steffensen’s Inequality, Chebychev functional.
The author would like to express his sincere appreciation to the referee for the thorough and helpful comments that have aided significantly in improving the paper.
P. Cerone
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Contents
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Steffensen Type Results for General Subintervals . . . . . . . . . . 7
3
Steffensen and the Generalised Chebyshev Functional . . . . . . 17
References
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1.
Introduction
For two measurable functions f, g : [a, b] → R, define the functional, which is
known in the literature as Chebychev’s functional
(1.1)
T (f, g; a, b) := M (f g, a, b) − M (g; a, b) M (f ; a, b) ,
where the integral mean is given by
1
M (f ; a, b) =
b−a
(1.2)
Z
b
f (x) dx,
a
On Some Generalisations of
Steffensen’s Inequality and
Related Results
provided that the involved integrals exist.
The following inequality is well known in the literature as the Grüss inequality [10]
P. Cerone
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1
|T (f, g; a, b)| ≤ (M − m) (N − n) ,
4
(1.3)
provided that m ≤ f ≤ M and n ≤ g ≤ N a.e. on [a, b], where m, M, n, N are
real numbers. The constant 41 in (1.3) is the best possible.
Another inequality of this type is due to Chebychev (see for example [14, p.
207]). Namely, if f, g are absolutely continuous on [a, b] and f 0 , g 0 ∈ L∞ [a, b]
with kf 0 k∞ := ess sup |f 0 (t)| , then
t∈[a,b]
(1.4)
and the constant
|T (f, g; a, b)| ≤
1
12
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1
kf 0 k∞ kg 0 k∞ (b − a)2
12
is the best possible.
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Finally, let us recall a result by Lupaş ([11], see also [14, p. 210]), which
states that:
(1.5)
|T (f, g; a, b)| ≤
1
kf 0 k2 kg 0 k2 (b − a) ,
2
π
provided f, g are absolutely continuous and f 0 , g 0 ∈ L2 [a, b]. The constant π12
is the best possible here.
For other Grüss type inequalities, see the books [13] and [14], and the papers
[4]-[10], where further references are given.
Recently, Cerone and Dragomir [3] have pointed out generalisations of the
above results for integrals defined on two different intervals [a, b] and [c, d].
They defined a generalised Chebychev functional involving the mean of the
product of two functions, and the product of the means of each of the functions,
where one is over a different interval by
(1.6)
T (f, g; a, b, c, d) := M (f g, a, b) − M (g; a, b) M (f ; c, d) ,
with M (·, ·, ·) as defined in (1.2). They proved the following theorem.
Theorem 1.1. Let f, g : I ⊆ R → R be measurable on I and the intervals
[a, b], [c, d] ⊂ I. In addition, let m1 ≤ f ≤ M1 and n1 ≤ g ≤ N1 a.e. on [a, b]
with n2 ≤ f ≤ N2 a.e. on [c, d]. Then the following inequalities hold
(1.7)
|T (f, g; a, b, c, d)|
1
≤ T (g; a, b) + M 2 (g; a, b) 2
1
× T (f ; a, b) + T (f ; c, d) + (M (f ; a, b) − M (f ; c, d))2 2
On Some Generalisations of
Steffensen’s Inequality and
Related Results
P. Cerone
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# 12
2
N1 − n 1
≤
+ M2 (g; a, b)
2
"
# 21
2 2
M1 − m 1
M2 − m 2
×
+
+ (M (f ; a, b) − M (f ; c, d))2 ,
2
2
"
where T (f ; a, b) ≡ T (f, f ; a, b) which is as given by (1.1) and M (f ; a, b) by
(1.2).
Proof. The proof was based on the identity
Z bZ d
1
(1.8) T (f, g; a, b, c, d) =
g (x) (f (x) − f (y)) dydx,
(b − a) (d − c) a c
and the Cauchy-Buniakowski-Schwartz inequality for double integrals to give
On Some Generalisations of
Steffensen’s Inequality and
Related Results
P. Cerone
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2
|T (f, g; a, b, c, d)|
2
Z bZ d
1
=
g (x) (f (x) − f (y)) dydx
(b − a) (d − c) a c
Z b Z d
Z b
1
2
2
≤
g (x) dx
(f (x) − f (y)) dydx
b−a a
a
c
= M2 (g; a, b) T (f, f, a, b, c, d) .
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They noted that equivalent results to the second inequality in (1.7) could be
obtained if (1.4) and (1.5) relating to the Chebyshev and Lupaş inequalities were
used in the first inequality in (1.7).
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The following inequality is due to Steffensen ([15], see also [14, p. 181]).
Theorem 1.2. Let f, g : [a, b] → R be integrable mappings on [a, b] such that
f is nonincreasing and 0 ≤ g (t) ≤ 1 for t ∈ [a, b]. Then
Z
b
Z
f (t) dt ≤
(1.9)
b−λ
b
Z
f (t) g (t) dt ≤
a
a+λ
f (t) dt,
a
where
Z
(1.10)
λ=
b
g (t) dt.
On Some Generalisations of
Steffensen’s Inequality and
Related Results
a
P. Cerone
Hayashi obtains a similar result [14, p. 182] which may ostensibly be obtained from Theorem 1.2 be replacing g (t) by g(t)
where A is some positive
A
constant.
For Steffensen type inequalities with integrals over a measure space, see the
work of Gauchman [9].
It may be noted that both the generalised Chebyshev functional (1.6) and
Steffensen’s inequality (1.9) – (1.10) involve integrals of functions and of products of functions. The current article aims at investigating the relationship further.
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2.
Steffensen Type Results for General Subintervals
The following lemma will be useful for the results that follow.
Lemma 2.1. Let f, g : [a, b] → R be integrable mappings on [a, b]. Further, let
Rb
[c, d] ⊆ [a, b] with λ = d − c = a g (t) dt. Then the following identities hold.
Namely,
Z d
Z b
(2.1)
f (t) dt −
f (t) g (t) dt
c
a
Z c
Z d
=
(f (d) − f (t)) g (t) dt +
(f (t) − f (d)) (1 − g (t)) dt
a
c
Z b
+
(f (d) − f (t)) g (t) dt
On Some Generalisations of
Steffensen’s Inequality and
Related Results
P. Cerone
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d
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and
Z
(2.2)
a
b
Z
d
f (t) g (t) dt −
f (t) dt
c
Z c
Z d
=
(f (t) − f (c)) g (t) dt +
(f (c) − f (t)) (1 − g (t)) dt
a
c
Z b
+
(f (t) − f (c)) g (t) dt.
d
Proof. Let
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Z
(2.3)
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d
Z
f (t) dt −
S (c, d; a, b) =
c
b
f (t) g (t) dt, a ≤ c < d ≤ b,
a
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then
Z
d
Z
c
Z
(1 − g (t)) f (t) dt −
S (c, d; a, b) =
b
f (t) g (t) dt
f (t) g (t) dt +
d
Z d
Z d
=
(1 − g (t)) (f (t) − f (d)) dt + f (d)
(1 − g (t)) dt
c
Z c
Z cc
+
(f (d) − f (t)) g (t) dt − f (d)
g (t) dt
a
a
Z b
Z b
+
(f (d) − f (t)) g (t) dt − f (d)
g (t) dt.
c
a
d
d
The identity (2.1) is readily obtained on noting that
Z d
Z b
f (d)
dt −
g (t) dt = 0.
c
a
Identity (2.2) follows immediately from (2.1) and (2.3) on realising that (2.2) is
S (d, c; b, a) or, equivalently, −S (c, d; a, b).
Remark 2.1. If c = a in (2.1) and d = b in (2.2) then the identities obtained by
Mitrinović [12] using an idea of Apéry, are recaptured.
Theorem 2.2. Let f, g : [a, b] → R be integrable mappings on [a, b] and let f
Rb
be nonincreasing. Further, let 0 ≤ g (t) ≤ 1 and λ = a g (t) dt = di − ci ,
where [ci , di ] ⊂ [a, b] for i = 1, 2 and d1 ≤ d2 .
Then the result
Z d2
Z b
Z d1
(2.4)
f (t) dt − r (c2 , d2 ) ≤
f (t) g (t) dt ≤
f (t) dt + R (c1 , d1 ) ,
c2
a
c1
On Some Generalisations of
Steffensen’s Inequality and
Related Results
P. Cerone
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holds where,
Z
b
(f (c2 ) − f (t)) g (t) dt ≥ 0
r (c2 , d2 ) =
d2
and
Z
c1
(f (t) − f (d1 )) g (t) dt ≥ 0.
R (c1 , d1 ) =
a
Proof. From (2.1) and (2.3) of Lemma 2.1
Z c1
S (c1 , d1 ; a, b) +
(f (t) − f (d1 )) g (t) dt
a
Z d1
Z b
=
(f (t) − f (d1 )) (1 − g (t)) dt +
(f (d1 ) − f (t)) g (t) dt ≥ 0
c1
c1
a
a
and thus the right inequality is valid.
Now, from (2.2) and (2.3) of Lemma 2.1
Z
b
− S (c2 , d2 ; a, b) +
(f (c2 ) − f (t)) g (t) dt
d2
c2
Z
a
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d2
(f (t) − f (c2 )) g (t) dt +
=
P. Cerone
d1
by the assumptions of the theorem.
Hence, from (2.3)
Z d1
Z c1
Z b
f (t) dt +
(f (t) − f (d1 )) g (t) dt −
f (t) g (t) dt ≥ 0
Z
On Some Generalisations of
Steffensen’s Inequality and
Related Results
(f (c2 ) − f (t)) (1 − g (t)) dt ≥ 0
c2
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from the assumptions.
Thus, from (2.3)
Z
Z b
f (t) g (t) dt −
a
d2
b
Z
f (t) dt −
c2
(f (c2 ) − f (t)) g (t) dt ≥ 0,
d2
giving the left inequality.
Both r (c2 , d2 ) and R (c1 , d1 ) are nonnegative since f is nonincreasing and g
is nonnegative. The theorem is now completely proved.
Remark 2.2. If in Theorem 2.2 we take c1 = a and so d1 = a + λ, then
R (a, a + λ) = 0. Further, taking d2 = b so that c2 = b − λ gives r (b − λ, b) =
0. The Steffensen inequality (1.9) is thus recaptured. Since (1.10) holds, then
c2 ≥ a and d1 ≤ b giving [ci , di ] ⊂ [a, b]. Theorem 2.2 may thus be viewed as a
generalisation of the Steffensen inequality as given in Theorem 1.2, to allow for
two equal length subintervals that are not necessarily at the ends of [a, b].
It may be advantageous at times to gain coarser bounds that may be more
easily evaluated. The following corollary examines this aspect.
Corollary 2.3. Let the conditions of Theorem 2.2 hold. Then
Z b
Z b
(2.5)
f (t) dt − (b − d2 ) f (c2 ) ≤
f (t) g (t) dt
c2
≤
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a
Z
On Some Generalisations of
Steffensen’s Inequality and
Related Results
d1
f (t) dt − (c1 − a) f (d1 ) .
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Proof. From Theorem 2.2 on using the fact that 0 ≤ g (t) ≤ 1, gives
Z b
0 ≤ r (c2 , d2 ) =
(f (c2 ) − f (t)) g (t) dt
d2
Z
b
≤
Z
b
(f (c2 ) − f (t)) dt = (b − d2 ) f (c2 ) −
d2
and so
Z d2
Z
d2
f (t) dt − r (c2 , d2 ) ≥
c2
f (t) dt
d2
Z
b
f (t) dt − (b − d2 ) f (c2 ) +
c2
f (t) dt.
d2
Combining the two integrals produces the left inequality of (2.5). Similarly,
Z c1
Z c1
0 ≤ R (c1 , d1 ) =
(f (t) − f (d1 )) g (t) dt ≤
f (t) dt − (c1 − a) f (d1 ) ,
a
On Some Generalisations of
Steffensen’s Inequality and
Related Results
P. Cerone
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a
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producing
Z
d1
Z
f (t) dt + R (c1 , d1 ) ≤
c1
d1
f (t) dt − (c1 − a) f (d1 )
a
giving the right inequality.
Remark 2.3. If we take c1 = a and so d1 = a+λ and d2 = b such that c2 = b−λ
then (2.5) again recaptures Steffensen’s inequality as given in Theorem 1.2.
The following lemma produces alternative identities to those obtained in
Lemma 2.1. The current identities involve the integral mean of f (·) over the
subinterval [c, d].
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Lemma 2.4. Let f, g : [a, b] → R be integrable mappings on [a, b]. Define
Z x
G (x) =
g (t) dt
a
and
λ = G (b) = d − c
where [c, d] ⊂ [a, b].
The following identities hold
Z
b
f (x) g (x) dx −
(2.6)
On Some Generalisations of
Steffensen’s Inequality and
Related Results
d
Z
f (y) dy
a
P. Cerone
c
Z
= λ [f (b) − M (f ; c, d)] −
b
G (x) df (x)
a
and
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Z
d
Z
c
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b
f (y) dy −
(2.7)
f (x) g (x) dx
a
Z
b
[λ − G (x)] df (x) ,
= λ [M (f ; c, d) − f (a)] −
a
where M (f ; c, d) is the integral mean of f (·) over [c, d].
Proof. Consider
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Z
b
Z
f (x) g (x) dx −
L :=
a
d
f (y) dy
c
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then from the postulates G(b)
= 1 giving
d−c
Z b
Z b
Z d
1
L=
g (x) dx
f (x) g (x) dx −
f (y) dy.
d−c a
a
c
Combining the integrals gives
Z b
(2.8)
L=
g (x) [f (x) − M (f ; c, d)] dx,
a
where M (f ; c, d) is the integral mean of f over [c, d] as given by (1.2).
Integration by parts from (2.8) gives
b Z b
L = G (x) [f (x) − M (f ; c, d)] −
G (x) df (x)
a
a
and so
Z
Contents
G (x) df (x)
a
since G (b) = λ and G (a) = 0.
Now for the second identity.
Let
Z
d
c
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Z
f (y) dy −
U :=
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b
L = λ [f (b) − M (f ; c, d)] −
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a
then, from (2.6),
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Z
U = −L = −λ [f (b) − M (f ; c, d)] +
b
G (x) df (x) .
a
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Hence
Z
U = λ [M (f ; c, d) − f (a)] − λ [f (b) − f (a)] +
b
G (x) df (x)
a
and so combining the last two terms gives (2.7).
Theorem 2.5. Let f, g : [a, b] → R be integrable mappings
R x on [a, b] and let
f be nonincreasing. Further, let g (t) ≥ 0 and G (x) = a g (t) dt with λ =
G (b) = di − ci where [ci, di ] ⊂ [a, b] for i = 1, 2 and d1 < d2 . Then
Z
d2
On Some Generalisations of
Steffensen’s Inequality and
Related Results
f (y) dy − λ [M (f ; c2 , d2 ) − f (b)]
(2.9)
P. Cerone
c2
Z
≤
b
Z
d1
f (x) g (x) dx ≤
a
f (y) dy + λ [f (a) − M (f ; c1 , d1 )]
c1
Contents
where d2 > d1 .
Proof. From (2.6) together with the facts that f is nonincreasing and g (t) ≥ 0
then
Z
b
−
G (x) df (x) ≥ 0
a
gives
Z
b
Z
d2
f (x) g (x) dx −
a
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f (y) dy + λ [f (b) − M (f ; c2 , d2 )] ≥ 0
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c2
and so the left inequality is obtained.
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Similarly, from (2.7) and the postulates we have
Z b
−
[λ − G (x)] df (x) ≥ 0,
a
which gives
Z d1
Z b
f (y) dy + λ [M (f ; c1 , d1 ) − f (a)] −
f (x) g (x) dx ≥ 0.
c1
a
On Some Generalisations of
Steffensen’s Inequality and
Related Results
Remark 2.4. The lower and upper inequalities in (2.9) may be simplified to
λf (b) and λf (a) respectively since
Z
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d
f (y) dy = λM (f ; c, d) .
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c
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That is,
b
Z
f (x) g (x) dx ≤ λf (a) .
λf (b) ≤
(2.10)
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a
The result should not be overly surprising since it may be obtained directly from
the postulates since
Z b
Z b
Z b
inf f (x)
g (x) dx ≤
f (x) g (x) dx ≤ sup f (x)
g (x) dx.
x∈[a,b]
a
a
x∈[a,b]
a
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As a referee suggested, the result (2.10) readily follows on noting that
Z b
g (x) [f (x) − f (b)] dx ≥ 0
a
and
Z
b
f (x) [f (a) − f (x)] dx ≥ 0.
a
The motivation behind Lemma 2.4 and Theorem 2.5 was to obtain a Steffensen like inequality and it was not predictable in advance that the result would
reduce to (2.10).
On Some Generalisations of
Steffensen’s Inequality and
Related Results
P. Cerone
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3.
Steffensen and the Generalised Chebyshev Functional
Bounds will be obtained for the difference between the integral of the product
of two functions from the integral over a subinterval of one of the functions.
Theorem 3.1. Let f, g : [a, b] → R be integrable mappings on [a, b] such that
f is nonincreasing and 0 ≤ g (t) ≤ 1 for t ∈ [a, b]. Further, let [c, d] ⊆ [a, b]
Rb
with λ = d − c = a g (t) dt, then the following inequality holds. Namely,
Z b
Z d
(3.1)
|S| := f (x) g (x) dx −
f (y) dy a
c
"
≤ (b − a)
+
On Some Generalisations of
Steffensen’s Inequality and
Related Results
1
+
4
λ
b−a
f (c) − f (d)
2
2 # 12
"
×
f (a) − f (b)
2
Title Page
#
2
2
+ (M (f ; a, b) − M (f ; c, d))
where M (f ; a, b) is the integral mean.
Rb
Proof. Since d − c = a g (t) dt, then
Z b
Z b
Z d
1
S=
f (x) g (x) dx −
g (x) dx
f (y) dy
d−c a
a
c
that is
Z b
Z b
(3.2)
S=
f (x) g (x) dx − M (f ; c, d)
g (x) dx,
a
a
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2
1
2
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where M (f ; c, d) is as defined by (1.2).
Thus, from (3.2), S may be expressed in terms of the generalised Chebyshev
functional as defined in (1.6), namely
S = (b − a) T (f, g; a, b, c, d)
(3.3)
and so, from (1.7)
(3.4)
|S| = (b − a) |T (f, g; a, b, c, d)|
"
≤ (b − a) T (g; a, b) +
λ
b−a
2 # 12
On Some Generalisations of
Steffensen’s Inequality and
Related Results
1
× T (f ; a, b) + T (f ; c, d) + (M (f ; a, b) − M (f ; c, d))2 2 ,
where
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T (f ; a, b) = M f 2 ; a, b − (M (f ; a, b))2
λ
and M (g; a, b) = b−a
.
Hence, using the second inequality from (1.7) and (3.4) produces the stated
result (3.1) upon using (1.3) and the facts that f is nonincreasing and 0 ≤
g (t) ≤ 1 .
0
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0
Remark 3.1. If we had more stringent conditions on f and g such that the
Chebyshev and Lupaş results (1.4) and (1.5) could be utilised in (3.4), then
bounds in terms of the k·k∞ and k·k2 norms of the derivatives would result.
This will not be pursued further here however.
The following theorem expresses S as a double integral over a rectangular
region to obtain bounds for the Steffensen functional.
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Theorem 3.2. Let the conditions of Theorem 3.1 hold. The following inequality
is then valid. Namely,
Z b
Z d
(3.5)
|S| := f (x) g (x) dx −
f (y) dy a
c
4
≤ (a + b + c + d) M (f ; c, d) −
µ (f ; c, d)
d−c
Z c
Z b
+
f (x) dx −
f (x) dx
a
d
≤ (c − a) f (a) − (b − d) f (b) + (a + b + c + d) f (c)
− 2 (d + c) f (d) ,
where M (f ; c, d) is the integral mean and
Z
µ (f ; c, d) =
(3.6)
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d
xf (x) dx.
c
Proof. From (3.3) and (1.8)
(3.7)
On Some Generalisations of
Steffensen’s Inequality and
Related Results
Z Z
1 b d
g (x) (f (x) − f (y)) dydx
|S| =
d−c a c
Z Z
kgk∞ b d
≤
|f (x) − f (y)| dydx,
d−c a c
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where kgk∞ := ess sup |g (x)| = 1, from the postulates.
x∈[a,b]
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Thus,
1
|S| ≤
d−c
(3.8)
Z bZ
d
|f (x) − f (y)| dydx := I.
a
c
Now, using the fact that f is nonincreasing and that [c, d] ⊆ [a, b], we have
Z cZ
(d − c) I =
d
Z
d
(f (x) − f (y)) dydx +
a
Z
c
dZ d
Z
x
(f (y) − f (x)) dydx
c
Z
c
bZ d
(f (x) − f (y)) dydx +
(f (y) − f (x)) dydx
d
c
Z c
Z d
= (d − c)
f (x) dx − (c − a)
f (y) dy
a
c
Z dZ x
Z d
+
f (y) dydx −
(x − c) f (x) dx
c
c
c
Z d
Z dZ d
+
(d − x) f (x) dx −
f (y) dydx
c
c
x
Z d
Z b
+ (b − d)
f (y) dy − (d − c)
f (x) dx
c
d
Z c
Z b
= (d − c)
f (x) dx −
f (x) dx
a
d
Z d
+
[a + b + c + d − 4x] f (x) dx.
+
c
On Some Generalisations of
Steffensen’s Inequality and
Related Results
x
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c
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Here we have used the facts that
Z dZ x
Z
f (y) dydx =
c
and
Z
c
d
Z
d
(d − x) f (x) dx
c
d
d
Z
(x − c) f (x) dx.
f (y) dydx =
c
x
c
Some elementary simplification gives
Z
c
Z
f (x) dx −
(3.9) I =
a
On Some Generalisations of
Steffensen’s Inequality and
Related Results
b
f (x) dx
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d
+
4
d−c
Z
c
d
a+b+c+d
− x f (x) dx
4
and hence the first inequality results.
The coarser inequality is obtained using the fact that f is nonincreasing,
giving, from (3.9)
I ≤ (c − a) f (a) − (b − d) f (b)
+ (a + b + c + d) f (c) −
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4
f (d)
d−c
Z
d
xdx,
c
which upon simplification gives the second inequality in (3.5). The theorem is
thus completely proved.
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Corollary 3.3. Let the conditions of Theorem 3.1 hold. Then
Z b
(3.10)
−2cM (f ; c, d) − φ (c, d) ≤
f (x) g (x) dx
a
≤ 2dM (f ; c, d) + φ (c, d) ,
where
(3.11) φ (c, d) = (a + b) M (f ; c, d)
Z c
Z b
+
f (x) dx −
f (x) dx −
a
d
4
µ (f ; c, d)
d−c
with M (f ; c, d) being the integral mean and µ (f ; c, d) the mean of f over the
subinterval [c, d] given by (1.2) and (3.6) respectively.
Proof. From (3.5) and (3.8) we have that
Z b
Z
−I ≤
f (x) g (x) dx −
a
On Some Generalisations of
Steffensen’s Inequality and
Related Results
P. Cerone
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d
f (y) dy ≤ I
c
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so that from (3.9)
Z
I = φ (c, d) −
d
f (x) dx
c
giving the result as stated after some minor algebra.
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Rb
Remark 3.2. Equation (3.10) gives bounds for a f (x) g (x) dx in terms of
information known over the subinterval [c, d]. Let [c1 , d1 ] and [c2 , d2 ] be two
such subintervals with d1 < d2 and di − ci = λ. Then
Z b
(3.12)
m≤
f (x) g (x) dx ≤ M,
a
where
M = min {2d1 M (f ; c1 , d1 ) + φ (c1 , d1 ) , 2d2 M (f ; c2 , d2 ) + φ (c2 , d2 )}
On Some Generalisations of
Steffensen’s Inequality and
Related Results
and
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m = max {−2c1 M (f ; c1 , d1 ) − φ (c1 , d1 ) , −2c2 M (f ; c2 , d2 ) − φ (c2 , d2 )} ,
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with φ (·, ·) being as given in (3.11).
Particularising the result (3.12) on taking d2 = b and hence c2 = b − λ,
c1 = a and so d1 = a + λ, produces bounds in terms of subintervals at the ends
of [a, b]. That is,
Z
(3.13)
me ≤
b
f (x) g (x) dx ≤ Me ,
a
where
(3.14)
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Me = min{2 (a + λ) M (f ; a, a + λ) + φ (a, a + λ) ,
2bM (f ; b − λ, b) + φ (b − λ, b)}
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and
me = max{−2aM (f ; a, a + λ) − φ (a, a + λ) ,
−2 (b − λ) M (f ; b − λ, b) − φ (b − λ, b)}
with φ (·, ·) defined in (3.11).
For (3.14) on using (3.11), (1.2) and (3.6) gives
Z
Z b
4 a+λ
φ (a, a + λ) = (a + b) M (f ; a, a + λ) −
f (x) dx −
xf (x) dx
λ a
a+λ
On Some Generalisations of
Steffensen’s Inequality and
Related Results
and
P. Cerone
Z
φ (b − λ, b) = (a + b) M (f ; b − λ, b) +
b−λ
f (x) dx −
a
4
λ
Z
b
xf (x) dx.
b−λ
It may be possible that Me can either be tighter or coarser than the
Rb
bound in (1.9) and similarly with me and b−λ f (x) dx.
R a+λ
a
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f (x) dx
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References
[1] P. CERONE AND S.S. DRAGOMIR, On some inequalities for the expectation and variance, Korean J. Comp. & Appl. Math., 8(2) (2001), 357–380.
[2] P. CERONE AND S.S. DRAGOMIR, On some inequalities arising from
Montgomery’s identity, J. Compt. Anal. Applics., 6(2) (2001), 117–128.
[3] P. CERONE AND S.S. DRAGOMIR, Generalisations of the Grüss, Chebychev and Lupaş inequalities for integrals over different intervals, Inter. J.
Appl. Math., (accepted).
On Some Generalisations of
Steffensen’s Inequality and
Related Results
[4] S.S. DRAGOMIR, Grüss inequality in inner product spaces, The Australian Math Soc. Gazette, 26(2) (1999), 66–70.
[5] S.S. DRAGOMIR, A generalization of Grüss’ inequality in inner product
spaces and applications, J. Math. Anal. Appl., 237 (1999), 74–82.
[6] S.S. DRAGOMIR, A Grüss type integral inequality for mappings of rHölder’s type and applications for trapezoid formula, Tamkang J. of Math.,
31(1) (2000), 43–48.
P. Cerone
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[7] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. of
Pure and Appl. Math., 31(4) (2000), 397-415.
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[8] S.S. DRAGOMIR AND I. FEDOTOV, An inequality of Grüss’ type for
Riemann-Stieltjes integral and applications for special means, Tamkang J.
of Math., 29(4) (1998), 286–292.
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[9] H. GAUCHMAN, A Steffensen type inequality, J. Ineq. Pure and Appl.
Math., 1(1) (2000), Art. 3.
http://jipam.vu.edu.au/v1n1/009_99.html
[10] G. RGRÜSS, Über das Maximum
absoluten Betrages von
Rb
R des
b
b
1
1
f (x)g(x)dx − (b−a)2 a f (x)dx a g(x)dx, Math. Z. , 39(1935),
b−a a
215–226.
[11] A. LUPAŞ, The best constant in an integral inequality, Mathematica,
15(38) (2) (1973), 219–222.
[12] D.S. MITRINOVIĆ, The Steffensen inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 247-273 (1969), 1–14.
[13] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
[14] J. PEČARIĆ, F. PROCHAN AND Y. TONG, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, San Diego,
1992.
[15] J.F. STEFFENSEN, On certain inequalities between mean values, and
their applications to actuarial problems, Strand. Aktuarietids, (1918), 82–
97.
On Some Generalisations of
Steffensen’s Inequality and
Related Results
P. Cerone
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