OSTROWSKI TYPE INEQUALITIES OVER BALLS
AND SHELLS VIA A TAYLOR-WIDDER FORMULA
Ostrowski Type Inequalities via a
Taylor-Widder Formula
GEORGE A. ANASTASSIOU
Department of Mathematical Sciences
The University of Memphis
Memphis, TN 38152, U.S.A.
EMail: ganastss@memphis.edu
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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Contents
Received:
06 August, 2007
Accepted:
27 November, 2007
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
26D10, 26D15.
Key words:
Ostrowski inequality, multivariate inequality, ball and shell, Taylor-Widder formula, extended complete Tschebyshev system.
Abstract:
The classical Ostrowski inequality for functions on intervals estimates the value
of the function minus its average in terms of the maximum of its first derivative.
This result is extended to higher order over shells and balls of RN , N ≥ 1, with
respect to an extended complete Tschebyshev system and the generalized radial
derivatives of Widder type. We treat radial and non-radial functions.
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Contents
1
Introduction
3
2
Background
4
3
Results on the Shell
8
4
Results on the Sphere
14
5
Addendum
23
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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1.
Introduction
The classical Ostrowski inequality (of 1938, see [12]) is
!
Z b
a+b 2
1
x
−
1
2
+
f (y)dy − f (x) ≤
(b − a)kf 0 k∞ ,
b − a
2
4
(b
−
a)
a
1
for f ∈ C ([a, b]), x ∈ [a, b], and it is a sharp inequality. This was extended to
RN , N ≥ 1, over balls and shells in [5], [6], [4]. Earlier this extension was done
over boxes and rectangles, see [2, p. 507-520], and [3], see also [1]. The produced
Ostrowski type inequalities, in the above mentioned references, were mostly sharp
and they involved the first and higher order derivatives of the engaged function f .
Here we derive a set of very general higher order Ostrowski type inequalities
over shells and balls with respect to an extended complete Tschebyshev system (see
[10]) and generalized derivatives of Widder type (see [15]). The proofs are based on
the polar method and the general Taylor-Widder formula (see [15], 1928). Our results generalize the higher order Ostrowski type inequalities established in the above
mentioned references.
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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2.
Background
The following are taken from [15]. Let f, u0 , u1 , . . . , un ∈ C n+1 ([a, b]),n ≥ 0, and
the Wronskians
Wi (x) := W [u0 (x), u1 (x), . . . , ui (x)]
u0 (x) u1 (x) . . . ui (x)
0
u0 (x) u01 (x) . . . u0i (x)
:= ..
.
(i)
(i)
u0 (x) u(i)
ui (x)
1 (x) . . .
,
Ostrowski Type Inequalities via a
Taylor-Widder Formula
i = 0, 1, . . . , n.
vol. 8, iss. 4, art. 106, 2007
Assume Wi (x) > 0 over [a, b]. Clearly then
φ0 (x) := W0 (x) = u0 (x),
W1 (x)
φ1 (x) :=
,...,
(W0 (x))2
Wi (x)Wi−2 (x)
,
φi (x) :=
(Wi−1 (x))2
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i = 2, 3, . . . , n
W [u0 (x), u1 (x), . . . , ui−1 (x), f (x)]
,
Wi−1 (x)
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i = 1, . . . , n + 1;
L0 f (x) := f (x), ∀ x ∈ [a, b].
Then for i = 1, . . . , n + 1 we have
Li f (x) = φ0 (x)φ1 (x) · · · φi−1 (x)
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are positive on [a, b].
For i ≥ 0, the linear differentiable operator of order i:
Li f (x) :=
George A. Anastassiou
d
1
d
1
d
d 1 d f (x)
···
.
dx φi−1 (x) dx φi−2 (x) dx
dx φ1 (x) dx φ0 (x)
Close
Consider also
u0 (t)
u1 (t)
u00 (t)
u01 (t)
1 ···
···
gi (x, t) :=
Wi (t) (i−1)
(i−1)
u0 (t) u1 (t)
u0 (x)
u1 (x)
···
ui (t)
···
u0i (t)
···
···
(i−1)
· · · ui (t)
···
ui (x)
,
i = 1, 2, . . . , n; g0 (x, t) := uu00(x)
, ∀x, t ∈ [a, b].
(t)
Note that gi (x, t) as a function of x is a linear combination of u0 (x), u1 (x), . . . , ui (x)
and it holds
Z x
Z x1
Z xi−2
Z xi−1
φ0 (x)
gi (x, t) =
φ1 (x)
···
φi−1 (xi−1 )
φi (xi )dxi dxi−1 . . . dx1
φ0 (t) · · · φi (t) t
t
t
t
Z x
1
φ0 (s) · · · φi (s)gi−1 (x, s)ds,
i = 1, 2, . . . , n.
=
φ0 (t) · · · φi (t) t
Example 2.1 ([15]). The sets
{1, x, x2 , . . . , xn },
{1, sin x, − cos x, − sin 2x, cos 2x, . . . , (−1)n−1 sin nx, (−1)n cos nx}
fulfill the above theory.
We mention
Theorem 2.1 (Karlin and Studden (1966), see [10, p. 376]). Let u0 , u1 , . . . , un ∈
C n ([a, b]), n ≥ 0. Then {ui }ni=0 is an extended complete Tschebyshev system on
[a, b] iff Wi (x) > 0 on [a, b], i = 0, 1, . . . , n.
We also mention
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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Theorem 2.2 (D. Widder, [15, p. 138]). Let the functions
f (x), u0 (x), u1 (x), . . . , un (x) ∈ C n+1 ([a, b]),
and the Wronskians W0 (x), W1 (x), . . . , Wn (x) > 0 on [a, b], x ∈ [a, b]. Then for
t ∈ [a, b] we have
f (x) = f (t)
u0 (x)
+ L1 f (t)g1 (x, t) + · · · + Ln f (t)gn (x, t) + Rn (x),
u0 (t)
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
where
Z
Rn (x) :=
vol. 8, iss. 4, art. 106, 2007
x
gn (x, s)Ln+1 f (s)ds.
t
For example, one could take u0 (x) = c > 0. If ui (x) = xi , i = 0, 1, . . . , n,
defined on [a, b], then
(x − t)i
,
i!
So under the assumptions of Theorem 2.2, we have
Li f (t) = f (i) (t) and
(2.1) f (x) = f (y)
u0 (x)
+
u0 (y)
n
X
gi (x, t) =
t ∈ [a, b].
gn (x, t)Ln+1 f (t)dt,
∀ x, y ∈ [a, b].
If u0 (x) = c > 0, then
(2.2) f (x) = f (y)+
i=1
Z
Li f (y)gi (x, y)+
x
gn (x, t)Ln+1 f (t)dt,
y
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x
y
n
X
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+
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Li f (y)gi (x, y)
Z
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∀ x, y ∈ [a, b].
Close
We call Li the generalized Widder-type derivative.
We need
Notation 1. Let A be a spherical shell ⊆ RN , N ≥ 1, i.e. A := B(0, R2 )−B(0, R1 ),
0 < R1 < R2 .
Here the ball B(0, R) := {x ∈ RN : |x| < R}, R > 0, where | · | is the Euclidean
norm, also S N −1 := {x ∈ RN : |x| = 1} is the unit sphere in RN with surface area
2π N/2
ωN := Γ(N/2)
. For x ∈ RN − {0} one can write uniquely x = rω, where r > 0,
ω ∈ S N −1 .
Let f ∈ C n+1 (A), n ≥ 0. If f is radial, i.e., f (x) = g(r), where r = |x|,
R1 ≤ r ≤ R2 , then g ∈ C n+1 ([R1 , R2 ]).
For radial f define
(2.3)
θi f (x) := Li g(r),
Here
g (i) (r) =
∂ i f (x)
,
∂ri
For F ∈ C(A) we have
Z
Z
(2.4)
F (x)dx =
A
i = 1, . . . , n + 1,
all
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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∀ x ∈ A.
i = 1, . . . , n + 1.
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Z
S N −1
R2
F (rω)rN −1 dr dω.
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We notice that
(2.5)
Ostrowski Type Inequalities via a
Taylor-Widder Formula
N
R2N − R1N
and
V ol(A) =
Z
R2
Close
sN −1 ds = 1,
R1
ωN (R2N − R1N )
.
N
3.
Results on the Shell
Remark 1. Here, let u0 , u1 , . . . , un ∈ C n+1 ([R1 , R2 ]), and W0 , W1 , . . . , Wn > 0 on
[R1 , R2 ], 0 < R1 < R2 , n ≥ 0 integer, with u0 (r) = c > 0.
Let also f ∈ C n+1 (A). We assume first that f is radial, i.e. there exists g such
that f (x) = g(r), r = |x|, R1 ≤ r ≤ R2 . Clearly g ∈ C n+1 ([R1 , R2 ]).
Let x ∈ A. Then by using the polar method (2.4) we obtain
R
f
(y)dy
A
E(x) := f (x) −
V ol(A) R
R
R2
N −1
N S N −1 R1 f (sω)s
ds dω (3.1)
= f (x) −
ωN (R2N − R1N )
R
RR
N S N −1 R12 g(s)sN −1 ds dω (3.2)
= g(r) −
N
N
ω
(R
−
R
)
N
2
1
Z R2
N
(3.3)
= g(r) −
g(s)sN −1 ds
N
N
R2 − R1
R
Z R2 1
Z R2
N
N
−1
N
−1
(3.4)
= g(r)s
ds
−
g(s)s
ds
R2N − R1N
R1
R1
Z R2
N
(3.5)
=
(g(r) − g(s))sN −1 ds =: (∗).
N
N
R2 − R1
R1
Let s, r ∈ [R1 , R2 ], then by generalized Taylor’s formula (2.2) we get
(3.6)
g(s) − g(r) =
n
X
i=1
Li g(r)gi (s, r) + Rn (r, s),
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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where
Z
(3.7)
Rn (r, s) :=
s
gn (s, t)Ln+1 g(t)dt.
r
But it holds
(3.8)
Z s
|Rn (r, s)| ≤ |gn (s, t)|dt kLn+1 gk∞,[R1 ,R2 ] .
r
By calling
(3.9)
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
Z s
Nn (r, s) := |gn (s, t)|dt ,
∀ s, r ∈ [R1 , R2 ],
vol. 8, iss. 4, art. 106, 2007
r
we get
(3.10)
|Rn (r, s)| ≤ Nn (r, s)kLn+1 gk∞,[R1 ,R2 ] ,
∀ s, r ∈ [R1 , R2 ].
Therefore by (3.5), (3.6), we have
#
Z R2 "X
n
N
N −1
(3.11)
(∗) =
L
g(r)g
(s,
r)
+
R
(r,
s)
s
ds
i
i
n
R2N − R1N R1 i=1
"
X
n Z R2
N
N
−1
≤
Li g(r)gi (s, r)s
ds
N
N
R2 − R1
R1
i=1
Z R2
N −1
+
|Rn (r, s)|s
ds
R1
Z R2
"X
n
by (3.10) N
N −1
≤
|L
g(r)|
g
(s,
r)s
ds
i
i
R2N − R1N
R1
i=1
Z R2
N −1
(3.12)
+(kLn+1 gk∞,[R1 ,R2 ] )
Nn (r, s)s
ds .
R1
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We have established the following result.
Theorem 3.1. Let u0 , u1 , . . . , un ∈ C n+1 ([R1 , R2 ]), W0 , W1 , . . . , Wn > 0 on [R1 , R2 ],
0 < R1 < R2 , n ≥ 0 integer, with u0 (r) = c > 0. Let f ∈ C n+1 (A) be radial, i.e.
there exists g such that f (x) = g(r), r = |x|, R1 ≤ r ≤ R2 , x ∈ A.
Then
R
f (y)dy A
E(x) := f (x) −
V ol(A) Z R2
N
N
−1
= g(r) −
g(s)s
ds
N
N
R2 − R1
R1
Z R2
"X
n
N
N
−1
≤
|L
g(r)|
g
(s,
r)s
ds
i
i
R2N − R1N
R
1
i=1
Z R2
N −1
+(kLn+1 gk∞,[R1 ,R2 ] )
(3.13)
Nn (r, s)s
ds
R1
Z R2
"X
n
N
N −1
=
|θi f (x)| gi (s, |x|)s
ds
N
N
R2 − R1
R1
i=1
Z R2
N −1
+(kθn+1 f k∞,A )
Nn (|x|, s)s
ds .
R1
Corollary 3.2. Let the conditions of Theorem 3.1 hold. Assume further that Li g(r0 ) =
0, i = 1, . . . , n, for a fixed r0 ∈ [R1 , R2 ]. For all x0 = r0 ω ∈ A, ω ∈ S N −1 , we have
R
f (y)dy A
E(x0 ) = f (x0 ) −
V ol(A) Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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(3.14)
Z R2
N
N −1
= g(r0 ) −
g(s)s
ds
R2N − R1N
R1
Z R 2
N
N −1
≤
(kLn+1 gk∞,[R1 ,R2 ] )
Nn (r0 , s)s
ds
R2N − R1N
R1
Z R 2
N
N −1
=
(kθn+1 f k∞,A )
Nn (|x0 |, s)s
ds .‘
R2N − R1N
R1
Interesting cases also arise when r0 = R1 or R2 .
We continue Remark 1 with
Remark 2. Now let f ∈ C n+1 (A), n ≥ 0, x ∈ A, x = rω, r > 0. Clearly for
fixed ω ∈ S N −1 , since the function f (rω), r ∈ [R1 , R2 ] is radial, it also belongs to
C n+1 ([R1 , R2 ]).
By applying the internal inequality (3.13) we get
Z R2
N
N −1
(3.15) f (rω) −
f
(sω)s
ds
R2N − R1N
R1
"
Z R2
X
n
N
N −1
≤
|(L
f
(·ω))(r)|
g
(s,
r)s
ds
i
i
R2N − R1N
R1
i=1
Z R2
N −1
+(kLn+1 f (·ω)k∞,[R1 ,R2 ] )
Nn (r, s)s
ds .
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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For non-radial f we define again
(3.16)
θi f (x) = θi f (rω) := (Li f (·ω))(r),
all
i = 1, . . . , n + 1, ∀ x ∈ A.
Here the involved
∂ i f (rω)
∂ i f (x)
=
,
∂ri
∂ri
i = 1, . . . , n + 1,
are the radial derivatives. In a sense θi is a generalized radial derivative of Widdertype.
Hence
"X
n
N
(3.17) R.H.S.(3.15) ≤
|θi f (rω)|
R2N − R1N
i=1
Z R2
Z R2
N −1
N −1
Nn (|x|, s)s
ds .
gi (s, r)s
ds + kθn+1 f k∞,A
R1
R1
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
Therefore, by (3.15) and (3.17) we have
Z
Γ N Z
1
2
(3.18) N/2
f (rω)dω −
f (y)dy 2π
V ol(A) A
S N −1
"
(
)
Z R2
n Z
N
Γ(N/2) X
N −1
|θ
f
(rω)|dω
g
(s,
r)s
ds
≤
i
i
2π N/2
R2N − R1N
N
−1
S
R1
i=1
Z R2
N −1
+kθn+1 f k∞,A
Nn (|x|, s)s
ds .
R1
We have established the following.
Theorem 3.3. Let u0 , u1 , . . . , un ∈ C n+1 ([R1 , R2 ]); W0 , W1 , . . . , Wn > 0 on [R1 , R2 ],
0 < R1 < R2 , n ≥ 0 an integer, with u0 (r) = c > 0. Let f ∈ C n+1 (A), x ∈ A;
x = rω, r ∈ [R1 , R2 ], ω ∈ S N −1 .
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Then
R
R
N
f
(rω)dω
Γ
f
(y)dy
2
sN −1
≤ f (x) −
(3.19) E(x) = f (x) − A
V ol(A) 2π N/2
(
"
)
Z R2
n Z
X
Γ N2
N
N −1
+
|θ
f
(rω)|dω
g
(s,
r)s
ds
i
i
N
N
2π N/2 i=1
R2 − R1
sN −1
R1
Z R2
N −1
Nn (|x|, s)s
ds , ∀ x ∈ A.
+kθn+1 f k∞,A
R1
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
Corollary 3.4. Let the conditions of Theorem 3.3 hold. Assume that
θi f (r0 ω) = 0, i = 1, . . . , n, for a fixed r0 ∈ [R1 , R2 ],
∀ ω ∈ S N −1 ;
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also consider all x0 = r0 ω ∈ A for any ω ∈ S N −1 . Then
R
f
(y)dy
A
E(x0 ) = f (x0 ) −
V ol(A) R
N
Γ
)
f
(r
ω)dω
0
2
S N −1
≤ f (x0 ) −
N/2
2π
Z R 2
N
N −1
+
kθn+1 f k∞,A
(3.20)
Nn (|x0 |, s)s
ds .
R2N − R1N
R1
When r0 = R1 or R2 is of special interest.
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4.
Results on the Sphere
Notation 2. Let N ≥ 1, B(0, R) := {x ∈ RN : |x| < R} be the ball in RN centered
N
at the origin and of radius R > 0. Note that V ol(B(0, R)) = ωNNR .
Let f from B(0, R) into R and consider f to be radial, i.e. f (x) = g(r), where
r = |x|, 0 ≤ r ≤ R. We assume g ∈ C n+1 ([0, R]), n ≥ 0. Clearly then f ∈
C(B(0, R)).
For F ∈ C(B(0, R)) we have
Z R
Z
Z
N −1
(4.1)
F (x)dx =
F (rω)r
dr dω.
S N −1
B(0,R)
0
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
Title Page
We notice that
(4.2)
Contents
N
RN
Z
R
sN −1 ds = 1.
0
The operator θi in the radial case, is as defined in (2.3), now ∀x ∈ B(0, R).
In the non-radial case, for f ∈ C n+1 (B(0, R)), θi is defined as in (3.16), ∀ x ∈
B(0, R) − {0}.
We make the following remark.
Remark 3. Here, let u0 , u1 , . . . , un ∈ C n+1 ([0, R]) and W0 , W1 , . . . , Wn > 0 on
[0, R], R > 0, n ≥ 0 an integer, with u0 (r) = c > 0.
We again first assume that f is radial on B(0, R), i.e. there exists g such that
f (x) = g(r), r = |x|, 0 ≤ r ≤ R. Assume further that g ∈ C n+1 ([0, R]). Let
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x ∈ B(0, R). Then by using the polar method (4.1) we obtain
R
f (y)dy B(0,R)
(4.3)
E(x) := f (x) −
V ol(B(0, R)) R
R
R
n−1
N
g(s)s
ds
dω S N −1
0
= g(r) −
N
ω
R
N
Z R
N
= g(r) − N
g(s)sn−1 ds
R
Z R 0
by (4.2) N N −1
=: (∗).
(4.4)
=
(g(r)
−
g(s))s
ds
RN 0
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
Title Page
Let s, r ∈ [0, R], then by the generalized Taylor’s formula (2.2) we get
(4.5)
g(s) − g(r) =
n
X
Li g(r)gi (s, r) + Rn (r, s),
i=1
where
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Z
(4.6)
Contents
Rn (r, s) :=
s
gn (s, t)Ln+1 g(t)dt.
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r
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But it holds that
(4.7)
Z s
|Rn (r, s)| ≤ |gn (s, t)|dt kLn+1 gk∞,[0,R] .
r
By calling
(4.8)
Z s
Nn (r, s) := |gn (s, t)|dt ,
r
∀ s, r ∈ [0, R],
Close
we get
(4.9)
|Rn (r, s)| ≤ Nn (r, s)kLn+1 gk∞,[0,R] ,
∀ s, r ∈ [0, R].
Therefore by (4.4) and (4.5), we have
Z "
#
n
N R X
(∗) = N Li g(r)gi (s, r) + Rn (r, s) sN −1 ds
R 0
" n Zi=1
#
Z R
N X R
(4.10)
Li g(r)gi (s, r)sN −1 ds +
|Rn (r, s)|sN −1 ds
≤ N
R
0
0
i=1
" n
Z
by (4.9) N X
R
N −1
≤
|Li g(r)| gi (s, r)s
ds
N
R
0
i=1
Z R
N −1
(4.11)
+(kLn+1 gk∞,[0,R] )
Nn (r, s)s
ds .
0
We have established the next result.
Theorem 4.1. Let u0 , u1 , . . . , un ∈ C n+1 ([0, R]), W0 , W1 , . . . , Wn > 0 on [0, R],
R > 0, n ≥ 0 an integer, with u0 (r) = c > 0. Let f from B(0, R) into R be radial,
i.e. there exists g such that f (x) = g(r), r = |x|, 0 ≤ r ≤ R, ∀ x ∈ B(0, R); further
assume that g ∈ C n+1 ([0, R]).
Then
R
f
(y)dy
B(0,R)
E(x) := f (x) −
V ol(B(0, R)) Z R
N
= g(r) − N
g(s)sn−1 ds
R
0
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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(4.12)
" n
Z R
N X
N
−1
≤ N
|Li g(r)| gi (s, r)s
ds
R
0
i=1
Z R
N −1
+(kLn+1 gk∞,[0,R] )
Nn (r, s)s
ds
0
" n
Z R
N X
N −1
|θi f (x)| gi (s, |x|)s
ds
= N
R
0
i=1
Z R
N −1
+(kθn+1 f k∞,B(0,R) )
Nn (|x|, s)s
ds .
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
0
The following corollary holds.
Corollary 4.2. Let the conditions of Theorem 4.1 hold. Assume further that Li g(r0 ) =
0, i = 1, . . . , n, for a fixed r0 ∈ [0, R]; consider all x0 = r0 ω ∈ B(0, R) and any
ω ∈ S N −1 .
Then
R
f (y)dy B(0,R)
E(x0 ) := f (x0 ) −
V ol(B(0, R)) Z R
N
n−1
= g(r0 ) − N
g(s)s ds
R
0
Z R
N
N −1
≤ N (kLn+1 gk∞,[0,R] )
Nn (r0 , s)s
ds
R
0
Z R
N
N −1
(4.13)
Nn (|x0 |, s)s
ds .
= N (kθn+1 f k∞,B(0,R) )
R
0
Interesting cases especially arise when r0 = 0 or R.
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We continue Remark 3 with
Remark 4. Let f be non-radial. Here assume that f ∈ C n+1 (B(0, R)). Consider
x ∈ B(0, R) − {0}, which is written uniquely as x = rω, r ∈ (0, R], ω ∈ S N −1 .
Then by using again the polar method (4.1) we obtain
R
R
f (y)dy B(0,R)
S N −1 f (rω)dω
(4.14)
−
ωN
V ol(B(0, R)) R
R
R
R
N −1
N
f
(sω)s
ds
dω N
−1
S N −1 f (rω)dω
S
0
(4.15)
=
−
N
ω
ω
R
N
N
Z
Z R
by (4.2) N
= f (rω)sN −1 ds dω
ωN RN
S N −1
0
Z R
Z
N
N −1
−
f
(sω)s
ds
dω
ωN RN
N
−1
S
0
Z
Z R
N N −1
=: (∗).
(4.16)
=
(f
(rω)
−
f
(sω))s
ds
dω
ωN RN S N −1
0
Clearly here f (·ω) ∈ C n+1 ((0, R]).
Let ρ ∈ (0, R], then by (2.2) we get
(4.17)
f (ρω) − f (rω) =
n
X
((Li (f (·ω)))(r))gi (ρ, r) + Rn (r, ρ),
i=1
where
Z
(4.18)
Rn (r, ρ) :=
ρ
gn (ρ, t)(Ln+1 (f (·ω)))(t)dt.
r
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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That is
(4.19)
n
X
f (ρω) − f (rω) =
(θi f (rω))gi (ρ, r) + Rn (r, ρ),
i=1
with
Z
ρ
Rn (r, ρ) =
(4.20)
gn (ρ, t)θn+1 f (tω)dt.
r
We further assume that
George A. Anastassiou
θ := kθn+1 f k∞,B(0,R)−{0} < +∞.
(4.21)
Therefore we obtain
(4.22)
Z
|Rn (r, ρ)| ≤ r
ρ
|gn (ρ, t)|dt θ.
By calling
(4.23)
Ostrowski Type Inequalities via a
Taylor-Widder Formula
Z
Nn (r, ρ) := r
ρ
|gn (ρ, t)|dt ,
∀ ρ, r ∈ [0, R],
vol. 8, iss. 4, art. 106, 2007
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Page 19 of 25
we get
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|Rn (r, ρ)| ≤ Nn (r, ρ)θ.
(4.24)
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Hence by (4.19) we obtain
|f (ρω) − f (rω)| ≤
(4.25)
≤
n
X
i=1
n
X
i=1
Close
|θi f (rω)||gi (ρ, r)| + |Rn (r, ρ)|
|θi f (rω)||gi (ρ, r)| + Nn (r, ρ)θ.
By the continuity of f and gi , i = 1, . . . , n,and by taking the limit as ρ → 0 in the
external inequality (4.25), we obtain
(4.26)
|f (0) − f (rω)| ≤
n
X
|θi f (rω)||gi (0, r)| + Nn (r, 0)θ.
i=1
Notice here that gn (ρ, t) is jointly continuous in (ρ, t) ∈ [0, R]2 , hence Nn (r, ρ)
is continuous in ρ ∈ [0, R].
That is, ∀ s ∈ [0, R] we get
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
(4.27)
|f (sω) − f (rω)| ≤
n
X
|θi f (rω)||gi (s, r)| + Nn (r, s)θ.
i=1
Consequently by (4.16) and (4.27) we find
Z R
Z
N
N −1
(4.28)
(∗) ≤
|f (sω) − f (rω)|s
ds dω
ωN RN S N −1
0
" n Z
Z R
X
N
N −1
(4.29)
≤
|θi f (rω)||gi (s, r)|s
ds dω
ωN RN i=1
S N −1
0
Z R
Z
N −1
+θ
Nn (r, s)s
ds dω
S N −1
0
R
Pn R
R
N −1
|θ
f
(rω)||g
(s,
r)|s
ds
dω
i
i
N
−1
i=1
S
0
=
V ol(B(0, R))
R
R
θN 0 Nn (r, s)sN −1 ds
(4.30)
+
RN
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=
N
X
 R
S N −1

i=1
(4.31)
+
|θi f (rω)|dω
R R
0
|gi (s, r)|s
V ol(B(0, R))
R
R
N −1
θN 0 Nn (r, s)s
ds
RN
N −1
ds


.
Ostrowski Type Inequalities via a
Taylor-Widder Formula
That is
(4.32)
R
R
f
(y)dy
f
(rω)dω
B(0,R)
S N −1
∆ := −
ωN
V ol(B(0, R)) 
 R
R R
N −1
n
|θ
f
(rω)|dω
|g
(s,
r)|s
ds
X
i
i
N
−1
0

 S
≤
V
ol(V
(0,
R))
i=1
R
R
θN 0 Nn (r, s)sN −1 ds
.
+
RN
The following theorem holds.
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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Page 21 of 25
Theorem 4.3. Let u0 , u1 , . . . , un ∈ C n+1 ([0, R]), W0 , W1 , . . . , Wn > 0 on [0, R],
R > 0, n ≥ 0 an integer, with u0 (r) = c > 0. Let f ∈ C n+1 (B(0, R)). Assume that
Full Screen
θ := kθn+1 f k∞,B(0,R)−{0} < +∞.
Close
(4.33)
Let x ∈ B(0, R) − {0}, which is written uniquely as x = rω, r ∈ (0, R], ω ∈ S N −1 .
Go Back
Then
R
f (y)dy B(0,R)
E(x) := f (x) −
V ol(B(0, R)) Z
Γ(N/2)
0
0
f (rω )dω ≤ f (x) −
N/2
2π
S N −1

 R
R R
0
0
N −1
n
|θ
f
(rω
)dω
|g
(s,
r)|s
ds
X
i
i
N −1
0
 S

+
V
ol(B(0,
R))
i=1
R
R
N −1
θN 0 Nn (r, s)s
ds
+
(4.34)
.
RN
We finally give the following corollary.
Corollary 4.4. Let the conditions of Theorem 4.3 hold. Assume further that θi f (r0 ω 0 ) =
0, ∀ ω 0N −1 , for some r0 ∈ (0, R], for all i = 1, . . . , n. Consider all x0 = r0 ω ∈
B(0, R) − {0}, and any ω ∈ S N −1 .
Then
R
f (y)dy B(0,r)
E(x0 ) := f (x0 ) −
V ol(B(0, R)) Z
Γ(N/2)
0
0
≤ f (x0 ) −
f (|x0 |ω )dω N/2
2π
N −1
R S
R
θN 0 Nn (|x0 |, s)sN −1 ds
(4.35)
.
+
RN
An interesting case is when r0 = R.
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
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5.
Addendum
We give
Proposition 5.1. Let f ∈ C 1 (B(0, R)) such that f is radial, i.e. f (x) = g(r),
r = |x|, 0 ≤ r ≤ R, ∀ x ∈ B(0, R). Then
(i) ∃ g 0 ∈ C((0, R]),
(ii) ∃ g 0 (0) = 0.
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
Proof. (i) is obvious.
(ii) Let u be a unit vector, h > 0, then
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g(h) − g(0)
f (hu) − f (0)
=
h
h
∇f (0) · hu + o(h)
(by first order multivariate Taylor’s formula) =
h
∇f (0) · hu o(h)
o(h)
=
+
= ∇f (0) · u +
h
h
h
o(h)
h→0
−→ ∇f (0) · u, by
→ 0.
h
The last is true for any unit vector u. Thus ∇f (0) = 0, proving the claim.
By Proposition 5.1, we see that, it may well be that g 0 is discontinuous at zero, if
only f ∈ C 1 (B(0, R)).
Therefore the assumption that g ∈ C n+1 ([0, R]) in Theorem 4.1 seems to be the
best.
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References
[1] G.A. ANASTASSIOU, Ostrowski type inequalities, Proc. AMS, 123 (1995),
3775–3781.
[2] G.A. ANASTASSIOU, Quantitative Approximations, Chapman & Hall/CRC,
Boca Raton, New York, 2001.
[3] G.A. ANASTASSIOU, Multivariate Ostrowski type inequalities, Acta Math.
Hungarica, 76(4) (1997), 267–278.
Ostrowski Type Inequalities via a
Taylor-Widder Formula
George A. Anastassiou
vol. 8, iss. 4, art. 106, 2007
[4] G.A. ANASTASSIOU, Ostrowski type inequalities over spherical shells, accepted in Serdica.
Title Page
[5] G.A. ANASTASSIOU AND J.A. GOLDSTEIN, Ostrowski type inequalities
over Euclidean domains, Rend. Lincei Mat. Appl., 18 (2007), 305–310.
[6] G.A. ANASTASSIOU AND J.A. GOLDSTEIN, Higher order Ostrowski type
inequalities over Euclidean domains, J. Math. Anal. and Applics., 337 (2008),
962–968.
[7] N.S. BARNETT AND S.S. DRAGOMIR, An Ostrowski type inequality for
double integrals and applications for cubature formulae, Soochow J. Math.,
27(1) (2001), 1–10.
[8] P. CERONE, Approximate multidimensional integration through dimension
reduction via the Ostrowski functional, Nonlinear Funct. Anal. Appl., 8(3)
(2003), 313–333.
[9] S.S. DRAGOMIR, N.S. BARNETT AND P. CERONE, An n-dimensional version of Ostrowski’s inequality for mappings of the Hölder type, Kyungpook
Math. J., 40(1) (2000), 65–75.
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[10] S. KARLIN AND W.J. STUDDEN, Tchebycheff Systems: with Applications in
Analysis and Statistics, Interscience, New York (1966).
[11] M. MATIC, J. PEČARIĆ AND N. UJEVIĆ, Weighted version of multivariate
Ostrowski type inequalities, Rocky Mountain J. Math., 31(2) (2001), 511–538.
[12] A. OSTROWSKI, Über die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938), 226–227.
Ostrowski Type Inequalities via a
Taylor-Widder Formula
[13] B.G. PACHPATTE, On an inequality of Ostrowski type in three independent
variables, J. Math. Anal. Appl., 249(2) (2000), 583–591.
vol. 8, iss. 4, art. 106, 2007
[14] B.G. PACHPATTE, An inequality of Ostrowski type in n independent variables, Facta Univ. Ser. Math. Inform., 16 (2001), 21–24.
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[15] D.V. WIDDER, A generalization of Taylor’s series, Transactions of AMS, 30(1)
(1928), 126–154.
George A. Anastassiou
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