Journal of Inequalities in Pure and
Applied Mathematics
NEW ČEBYŠEV TYPE INEQUALITIES VIA
TRAPEZOIDAL-LIKE RULES
volume 7, issue 1, article 31,
2006.
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001 (Maharashtra)
India.
Received 02 October, 2005;
accepted 16 January, 2006.
Communicated by: N.S. Barnett
EMail: bgpachpatte@gmail.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
018-06
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Abstract
In this paper we establish new inequalities similar to the Čebyšev integral inequality involving functions and their derivatives via certain Trapezoidal like
rules.
2000 Mathematics Subject Classification: 26D15, 26D20.
Key words: Čebyšev type inequalities, Trapezoid-like rules, Absolutely continuous
functions, Differentiable functions, Identities.
The author would like to express his sincere thanks to the referee and Professor Neil
Barnett for their valuable suggestions which improved the presentation of our results.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Proofs of Theorems 2.1 and 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 6
4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
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1.
Introduction
In 1882, P.L. Čebyšev [2] proved the following classical integral inequality (see
also [10, p. 207]):
(1.1)
|T (f, g)| ≤
1
(b − a)2 kf 0 k∞ kg 0 k∞ ,
12
where f, g : [a, b] → R are absolutely continuous functions, whose first derivatives f 0 , g 0 are bounded and
1
(1.2) T (f, g) =
b−a
Z
a
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
b
f (x) g (x) dx
Z b
Z b
1
1
−
f (x) dx
g (x) dx ,
b−a a
b−a a
provided the integrals in (1.2) exist.
The inequality (1.1) has received considerable attention and a number of
papers have appeared in the literature which deal with various generalizations,
extensions and variants, see [5] – [10]. The aim of this paper is to establish new
inequalities similar to (1.1) involving first and second order derivatives of the
functions f, g. The analysis used in the proofs is based on certain trapezoidal
like rules proved in [1, 3, 4].
B.G. Pachpatte
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2.
Statement of Results
In what follows R and 0 denote respectively the set of real numbers and the
derivative of a function. Let [a, b] ⊂ R; a < b. We use the following notations
to simplify the detail of presentation. For suitable functions f, g, m : [a, b] → R,
and the constants α, β ∈ R, we set:
Z bZ b
1
L (f ; a, b) =
(f 0 (t) − f 0 (s)) (t − s) dtds,
2
2 (b − a) a a
M (f ; a, b) =
1
2 (b − a)2
Z bZ
a
1
N (f , f ; a, b) =
2 (b − a)
0
00
1
P (α, β, f, g) = αβ −
b−a
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
b
(f 0 (t) − f 0 (s)) (m (t) − m (s)) dtds,
B.G. Pachpatte
(t − a) (b − t) {[f 0 ; a, b] − f 00 (t)} dt,
Title Page
a
Z
b
a
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Z b
Z b
α
g (t) dt + β
f (t) dt
a
a
Z b
Z b
1
1
+
f (t) dt
g (t) dt ,
b−a a
b−a a
f (b) − f (a)
,
b−a
g (a) + g (b)
a+b
G=
, A=f
,
2
2
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[f ; a, b] =
f (a) + f (b)
F =
,
2
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B=g
a+b
2
,
J. Ineq. Pure and Appl. Math. 7(1) Art. 31, 2006
http://jipam.vu.edu.au
f (a) + f (b) (b − a)2 0
−
[f ; a, b] ,
2
12
and define
F̄ =
Ḡ =
g (a) + g (b) (b − a)2 0
−
[g ; a, b] ,
2
12
Z
kf k∞ = sup |f (t)| < ∞,
t∈[a,b]
kf kp =
b
p1
p
|f 0 (t)| dt
< ∞,
a
for 1 ≤ p < ∞.
Theorem 2.1. Let f, g : [a, b] → R be absolutely continuous functions on [a, b]
with f 0 , g 0 ∈ L2 [a, b] , then,
12
1
(b − a)2
2
0 2
(2.1)
|P (F, G, f, g)| ≤
kf k2 − ([f ; a, b])
12
b−a
21
1
2
0 2
×
kg k2 − ([g; a, b])
,
b−a
(2.2)
12
(b − a)2
1
2
0 2
|P (A, B, f, g)| ≤
kf k2 − ([f ; a, b])
12
b−a
21
1
2
2
×
kg 0 k2 − ([g; a, b])
.
b−a
Theorem 2.2. Let f, g : [a, b] → R be differentiable functions so that f 0 , g 0 are
absolutely continuous on [a, b] , then,
(2.3)
4
P F̄ , Ḡ, f, g ≤ (b − a) kf 00 − [f 0 ; a, b]k kg 00 − [g 0 ; a, b]k .
∞
∞
144
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
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3.
Proofs of Theorems 2.1 and 2.2
From the hypotheses of Theorem 2.1, we have the following identities (see [3,
p. 654]):
Z b
1
(3.1)
F−
f (t) dt = L (f ; a, b) ,
b−a a
1
G−
b−a
(3.2)
Z
b
g (t) dt = L (g; a, b) .
a
Multiplying the left sides and right sides of (3.1) and (3.2) we get
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
P (F, G, f, g) = L (f ; a, b) L (g; a, b) .
(3.3)
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From (3.3) we have
(3.4)
|P (F, G, f, g)| = |L (f ; a, b)| |L (g; a, b)| .
Using the Cauchy-Schwarz inequality for double integrals,
Z bZ b
1
|L (f ; a, b)| ≤
(3.5)
|(f 0 (t) − f 0 (s)) (t − s)| dtds
2
2 (b − a) a a
12
Z bZ b
1
2
0
0
≤
(f (t) − f (s))
2 (b − a)2 a a
12
Z bZ b
1
×
(t − s)2 .
2 (b − a)2 a a
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By simple computation,
(3.6)
1
2 (b − a)2
Z bZ
a
b
2
(f 0 (t) − f 0 (s)) dtds
a
1
=
b−a
b
Z
2
0
(f (t)) dt −
a
1
b−a
Z
b
2
f (t) dt ,
0
a
and
(3.7)
1
2 (b − a)2
Z bZ
a
a
b
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
(b − a)2
(t − s) dtds =
.
12
2
B.G. Pachpatte
Using (3.6), (3.7) in (3.5),
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(3.8)
|L (f ; a, b)| ≤
1
b−a
2
√
kf 0 k2 − ([f ; a, b])2
b
−
a
2 3
12
.
Similarly,
(3.9)
12
1
b−a
2
0 2
|L (g; a, b)| ≤ √
kg k2 − ([g; a, b])
.
2 3 b−a
Using (3.8) and (3.9) in (3.4), we obtain (2.1).
From the hypotheses of Theorem 2.1, we have (see [4, p. 238]):
Z b
1
(3.10)
A−
f (t) dt = M (f ; a, b) ,
b−a a
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(3.11)
1
B−
b−a
Z
b
g (t) dt = M (g; a, b) ,
a
where m(t) involved in the notation M (·; a, b) is given by
(
t − a if t ∈ a, a+b
2
m (t) =
t − b if t ∈ a+b
,
b
.
2
Multiplying the left sides and right sides of (3.10) and (3.11), we get
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
(3.12)
P (A, B, f, g) = M (f ; a, b) M (g; a, b) .
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From (3.12),
(3.13)
|P (A, B, f, g)| = |M (f ; a, b)| |M (g; a, b)| .
Again using the Cauchy-Schwarz inequality for double integrals, we have,
Z bZ b
1
|M (f ; a, b)| ≤
|(f 0 (t) − f 0 (s)) (m (t) − m (s))| dtds
2
2 (b − a) a a
21
Z bZ b
1
2
0
0
≤
(f (t) − f (s)) dtds
2 (b − a)2 a a
12
Z bZ b
1
×
(m (t) − m (s))2 dtds .
(3.14)
2 (b − a)2 a a
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By simple computation,
(3.15)
1
2 (b − a)2
Z bZ
a
b
2
(f 0 (t) − f 0 (s)) dtds
a
1
=
b−a
Z
b
0
2
(f (t)) −
a
1
b−a
Z
b
2
f (t) dt ,
0
a
and
(3.16)
1
2 (b − a)2
Z bZ
a
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
b
(m (t) − m (s))2 dtds
B.G. Pachpatte
a
1
=
b−a
Z
b
2
(m (t)) −
a
1
b−a
Z
b
2
m (t) dt .
a
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It is easy to observe that
Z
b
m (t) dt = 0,
a
and
1
b−a
Z
a
b
m2 (t) dt =
(b − a)2
.
12
Using (3.15), (3.16) and the above observations in (3.14) we get
(3.17)
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21
b−a
1
2
2
|M (f ; a, b)| ≤ √
kf 0 k2 − ([f ; a, b])
.
2 3 b−a
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Similarly ,
12
1
b−a
2
0 2
.
(3.18)
|M (g; a, b)| ≤ √
kg k2 − ([g; a, b])
2 3 b−a
Using (3.17) and (3.18) in (3.13) we get (2.2).
From the hypotheses of Theorem 2.2, we have the following identities (see
[1, p. 197]):
Z b
1
(3.19)
f (t) dt − F̄ = N (f 0 , f 00 ; a, b) ,
b−a a
Z b
1
(3.20)
g (t) dt − Ḡ = N (g 0 , g 00 ; a, b) .
b−a a
Multiplying the left sides and right sides of (3.19) and (3.20), we get
(3.21)
P F̄ , Ḡ, f, g = N (f 0 , f 00 ; a, b) N (g 0 , g 00 ; a, b) .
From (3.21),
(3.22)
P F̄ , Ḡ, f, g = |N (f 0 , f 00 ; a, b)| |N (g 0 , g 00 ; a, b)| .
By simple computation, we have,
Z b
1
0
00
|N (f , f ; a, b)| ≤
(t − a) (b − t) |[f 0 ; a, b] − f 00 (t)|dt
2 (b − a) a
Z b
1
00
0
kf (t) − [f ; a, b]k∞
(t − a) (b − t)dt
≤
2 (b − a)
a
(b − a)2 00
(3.23)
=
kf (t) − [f 0 ; a, b]k∞ .
12
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
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Similarly,
(3.24)
(b − a)2 00
|N (g , g ; a, b)| ≤
kg (t) − [g 0 ; a, b]k∞ .
12
0
00
Using (3.23) and (3.24) in (3.22), we get the required inequality in (2.3).
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
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4.
Applications
In this section we present applications of the inequalities established in Theorem
2.1, to obtain results which are of independent interest.
Let X be a continuous random variable having
R bthe probability density function (p.d.f.) h : [a, b] ⊂ R → R+ and E (x) = a th (t) dt its expectation
and
Rx
the cumulative density function H : [a, b] → [0, 1], i.e. H (x) = a h (t) dt,
Rb
x ∈ [a, b] . Then H(a) = 0, H(b) = 1 and H(a)+H(b)
= 12 , a H (x) dx
2
= b − E (X).
Let f = g = h and choose in (2.1) H instead of f and g and 12 instead of F
and G . By simple computation, we have,
1 1
1
1
b − E (X)
P
, , H, H = −
(b − E (X)) 1 −
,
2 2
4 b−a
b−a
and the right hand side in (2.1) is equal to
1 (b − a) khk22 − 1 ,
12
and hence the following inequality holds:
1
− 1 (b − E (X)) 1 − b − E (X) ≤ 1 (b − a) khk2 − 1 .
2
4 b − a
b−a
12
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
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Let a, b >
consider
the function f : (0, ∞) → R defined by f (x) = x1 ,
0 anda+b
a+b
2
then f 2 = g 2 = a+b .
J. Ineq. Pure and Appl. Math. 7(1) Art. 31, 2006
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Let g = f and choose in (2.2) x1 instead of f and g and
B. By simple computation, we have,
P
2
2 1 1
,
, ,
a+b a+b x x
=
2
a+b
instead of A and
2
log b − log a
−
a+b
b−a
2
,
0 2 2
1
1
(b − a)2
1 −
; a, b
=
.
b − a x 2
x
3a3 b3
Using the above facts in (2.2), the following inequality holds:
2
log b − log a
−
a+b
b−a
2
≤
(b − a)4
.
36a3 b3
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
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References
[1] N.S. BARNETT AND S.S. DRAGOMIR, On the perturbed trapezoid formula, Tamkang J. Math., 33(2) (2002), 119–128.
[2] P.L. ČEBYŠEV , Sur les expressions approximatives des limites, Proc.
Math. Soc. Charkov, 2 (1882), 93–98.
[3] S.S. DRAGOMIR AND S. MABIZELA, Some error estimates in the trapezoidal quadrature rule, RGMIA Res. Rep.Coll., 2(5) (1999), 653–663.
[ONLINE: http://rgmia.vu.edu.au/v2n5.html].
[4] S.S. DRAGOMIR, J. ŠUNDE AND C. BUŞE, Some new inequalities
for Jeffreys divergence measure in information theory, RGMIA Res. Rep.
Coll., 3(2) (2000), 235–243. [ONLINE: http://rgmia.vu.edu.
au/v3n2.html].
New Čebyšev Type Inequalities
via Trapezoidal-like Rules
B.G. Pachpatte
Title Page
Contents
[5] H.P. HEINIG AND L. MALIGRANDA, Chebyshev inequality in function
spaces, Real Analysis and Exchange, 17 (1991-92), 211–247.
[6] D.S. MITRINOVIĆ , J.E. PEČARIĆ AND A.M. FINK , Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
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[7] B.G. PACHPATTE, On trapezoid and Grüss like integral inequalities,
Tamkang J. Math., 34(4) (2003) , 365–369.
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[8] B.G. PACHPATTE, New weighted multivariate Grüss type inequalities, J.
Inequal. Pure and Appl. Math., 4(5) (2003), Art. 108. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=349].
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[9] B.G. PACHPATTE, A note on Chebychev-Grüss type inequalities for differentiable functions, Tamusi Oxford J. Math. Sci., to appear.
[10] J.E. PEČARIĆ, F. PORCHAN AND Y. TONG, Convex Functions, Partial
Orderings and Statistical Applications, Academic Press, San Diego, 1992.
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via Trapezoidal-like Rules
B.G. Pachpatte
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