Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 4, Article 85, 2004
A NEW SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS AND A
CORRESPONDING SUBCLASS OF STARLIKE FUNCTIONS WITH FIXED
SECOND COEFFICIENT
G. MURUGUSUNDARAMOORTHY AND N. MAGESH
D EPARTMENT OF M ATHEMATICS
V ELLORE I NSTITUTE OF T ECHNOLOGY,
D EEMED U NIVERSITY
V ELLORE - 632014, I NDIA .
gmsmoorthy@yahoo.com
D EPARTMENT OF M ATHEMATICS
A DHIYAMAAN C OLLEGE OF E NGINEERING
H OSUR - 635109, I NDIA .
nmagi_2000@yahoo.co.in
Received 22 June, 2004; accepted 25 August, 2004
Communicated by A. Sofo
A BSTRACT. Making use of Linear operator theory, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients. The
main object of this paper is to obtain coefficient estimates distortion bounds, closure theorems
and extreme points for functions belonging to this new class. The results are generalized to
families with fixed finitely many coefficients.
Key words and phrases: Univalent, Convex, Starlike, Uniformly convex, Uniformly starlike, Linear operator.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION
Denoted by S the class of functions of the form
∞
X
(1.1)
f (z) = z +
an z n
n=2
that are analytic and univalent in the unit disc 4 = {z : |z| < 1} and by ST and CV the
subclasses of S that are respectively, starlike and convex. Goodman [2, 3] introduced and
defined the following subclasses of CV and ST.
A function f (z) is uniformly convex (uniformly starlike) in 4 if f (z) is in CV (ST ) and has
the property that for every circular arc γ contained in 4, with center ξ also in 4, the arc f (γ)
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
The authors would like to thank the referee for his insightful suggestions.
125-04
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G. M URUGUSUNDARAMOORTHY
AND
N. M AGESH
is convex (starlike) with respect to f (ξ). The class of uniformly convex functions is denoted
by U CV and the class of uniformly starlike functions by U ST (for details see [2]). It is well
known from [4, 5] that
00 00
zf (z) zf (z)
f ∈ U CV ⇔ 0
≤ Re 1 + 0
.
f (z) f (z)
In [5], Rønning introduced a new class of starlike functions related to U CV defined as
0
0 zf (z)
zf (z)
f ∈ Sp ⇔ − 1 ≤ Re
.
f (z)
f (z)
Note that f (z) ∈ U CV ⇔ zf 0 (z) ∈ Sp . Further, Rønning generalized the class Sp by introducing a parameter α, −1 ≤ α < 1,
0
0
zf (z)
zf (z)
− 1 ≤ Re
−α .
f ∈ Sp (α) ⇔ f (z)
f (z)
Now we define the function φ(a, c; z) by
(1.2)
φ(a, c; z) = z +
∞
X
(a)n−1
n=2
(c)n−1
zn,
for c 6= 0, −1, −2, . . . , a 6= −1; z ∈ ∆ where (λ)n is the Pochhammer symbol defined by
(
)
1;
n=0
Γ(n + λ)
(1.3) (λ)n =
=
.
Γ(λ)
λ(λ + 1)(λ + 2) . . . (λ + n − 1), n ∈ N = {1, 2, . . . }
Carlson and Shaffer [1] introduced a linear operator L(a, c), by
(1.4)
L(a, c)f (z) = φ(a, c; z) ∗ f (z)
∞
X
(a)n−1
=z+
an z n , z ∈ 4,
(c)
n−1
n=2
where ∗ stands for the Hadamard product or convolution product of two power series
∞
∞
X
X
n
ϕn z and ψ(z) =
ψn z n
ϕ(z) =
n=1
n=1
defined by
(ϕ ∗ ψ)(z) = ϕ(z) ∗ ψ(z) =
∞
X
ϕn ψn z n .
n=1
We note that L(a, a)f (z) = f (z), L(2, 1)f (z) = zf 0 (z), L(m + 1, 1)f (z) = Dm f (z), where
Dm f (z) is the Ruscheweyh derivative of f (z) defined by Ruscheweyh [6] as
z
(1.5)
Dm f (z) =
∗ f (z), m > −1.
(1 − z)m+1
Which is equivalently,
z dm m−1
Dm f (z) =
{z
f (z)}.
m! dz m
For β ≥ 0 and −1 ≤ α < 1, we let S(α, β) denote the subclass of S consisting of functions
f (z) of the form (1.1) and satisfying the analytic criterion
z(L(a, c)f (z))0
z(L(a, c)f (z))0
(1.6)
Re
−α >β
− 1 , z ∈ 4.
L(a, c)f (z)
L(a, c)f (z)
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A N EW S UBCLASS OF U NIFORMLY C ONVEX F UNCTIONS AND A C ORRESPONDING ...
We also let T S(α, β) = S(α, β)
the form
T
f (z) = z −
(1.7)
3
T where T, the subclass of S consisting of functions of
∞
X
an z n , an ≥ 0,
∀ n ≥ 2,
n=2
was introduced and studied by Silverman [7].
The main object of this paper is to obtain necessary and sufficient conditions for the functions f (z) ∈ T S(α, β). Furthermore we obtain extreme points, distortion bounds and closure
properties for f (z) ∈ T S(α, β) by fixing the second coefficient.
2. T HE C LASS S(α, β)
In this section we obtain necessary and sufficient conditions for functions f (z) in the classes
T S(α, β).
Theorem 2.1. A function f (z) of the form (1.1) is in S(α, β) if
∞
X
(2.1)
[n(1 + β) − (α + β)]
n=2
(a)n−1
|an | ≤ 1 − α,
(c)n−1
−1 ≤ α < 1, β ≥ 0.
Proof. It suffices to show that
z(L(a, c)f (z))0
z(L(a, c)f (z))0
β
− 1 − Re
− 1 ≤ 1 − α.
L(a, c)f (z)
L(a, c)f (z)
We have
0
z(L(a, c)f (z))0
z(L(a,
c)f
(z))
β − 1 − Re
−1
L(a, c)f (z)
L(a, c)f (z)
z(L(a, c)f (z))0
≤ (1 + β) − 1
L(a, c)f (z)
P∞
n−1
(1 + β) n=2 (n − 1) (a)
|a |
(c)n−1 n
≤
.
P∞ (a)n−1
1 − n=2 (c)n−1 |an |
This last expression is bounded above by (1 − α) if
∞
X
[n(1 + β) − (α + β)]
n=2
(a)n−1
|an | ≤ 1 − α,
(c)n−1
and hence the proof is complete.
Theorem 2.2. A necessary and sufficient condition for f (z) of the form (1.7) to be in the class
T S(α, β), −1 ≤ α < 1, β ≥ 0 is that
(2.2)
∞
X
[n(1 + β) − (α + β)]
n=2
(a)n−1
an ≤ 1 − α.
(c)n−1
Proof. In view of Theorem 2.1, we need only to prove the necessity. If f (z) ∈ T S(α, β) and z
is real then
P∞
P
(a)n−1
(a)n−1
n−1
n−1 1− ∞
n
a
z
a
z
(n
−
1)
n
n
n=2 (c)n−1
(c)n−1
n=2
−α≥β
.
P∞ (a)n−1
P∞ (a)n−1
n−1
n−1
1−
1−
an z
an z
n=2 (c)n−1
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n=2 (c)n−1
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G. M URUGUSUNDARAMOORTHY
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Letting z → 1 along the real axis, we obtain the desired inequality
∞
X
(a)n−1
[n(1 + β) − (α + β)]
an ≤ 1 − α, −1 ≤ α < 1, β ≥ 0.
(c)
n−1
n=2
Corollary 2.3. Let the function f (z) defined by (1.7) be in the class T S(α, β). Then
(1 − α)(c)n−1
an ≤
, n ≥ 2, −1 ≤ α ≤ 1, β ≥ 0.
[n(1 + β) − (α + β)](a)n−1
Remark 2.4. In view of Theorem 2.2, we can see that if f (z) is of the form (1.7) and is in the
class T S(α, β) then
(1 − α)(c)
.
(2.3)
a2 =
(2 + β − α)(a)
By fixing the second coefficient, we introduce a new subclass T Sb (α, β) of T S(α, β) and
obtain the following theorems.
Let T Sb (α, β) denote the class of functions f (z) in T S(α, β) and be of the form
∞
b(1 − α)(c) 2 X
(2.4)
f (z) = z −
z −
an z n (an ≥ 0), 0 ≤ b ≤ 1.
(2 + β − α)(a)
n=3
Theorem 2.5. Let function f (z) be defined by (2.4). Then f (z) ∈ T Sb (α, β) if and only if
∞
X
(a)n−1
(2.5)
[n(1 + β) − (α + β)]
an ≤ (1 − b)(1 − α),
(c)
n−1
n=3
−1 ≤ α < 1, β ≥ 0.
Proof. Substituting
b(1 − α) (c)
, 0 ≤ b ≤ 1.
(2 + β − α) (a)
in (2.2) and simple computation leads to the desired result.
a2 =
Corollary 2.6. Let the function f (z) defined by (2.4) be in the class T Sb (α, β). Then
(1 − b)(1 − α)(c)n−1
(2.6)
an ≤
, n ≥ 3, −1 ≤ α ≤ 1, β ≥ 0.
[n(1 + β) − (α + β)](a)n−1
Theorem 2.7. The class T Sb (α, β) is closed under convex linear combination.
Proof. Let the function f (z) be defined by (2.4) and g(z) defined by
∞
b(1 − α) (c) 2 X
(2.7)
g(z) = z −
z −
dn z n ,
(2 + β − α) (a)
n=3
where dn ≥ 0 and 0 ≤ b ≤ 1.
Assuming that f (z) and g(z) are in the class T Sb (α, β), it is sufficient to prove that the
function H(z) defined by
(2.8)
H(z) = λf (z) + (1 − λ)g(z), (0 ≤ λ ≤ 1)
is also in the class T Sb (α, β).
Since
(2.9)
∞
b(1 − α)(c) 2 X
H(z) = z −
z −
{λan + (1 − λ)dn }z n ,
(2 + β − α)(a)
n=3
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A N EW S UBCLASS OF U NIFORMLY C ONVEX F UNCTIONS AND A C ORRESPONDING ...
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an ≥ 0, dn ≥ 0, 0 ≤ b ≤ 1, we observe that
(2.10)
∞
X
[n(1 + β) − (α + β)]
n=3
(a)n−1
(λan + (1 − λ)dn ) ≤ (1 − b)(1 − α)
(c)n−1
which is, in view of Theorem 2.5, again, implies that H(z) ∈ T Sb (α, β) which completes the
proof of the theorem.
Theorem 2.8. Let the functions
fj (z) = z −
(2.11)
∞
b(1 − α)(c) 2 X
z −
an, j z n , an,j ≥ 0
(2 + β − α)(a)
n=3
be in the class T Sb (α, β) for every j (j = 1, 2, . . . , m). Then the function F (z) defined by
m
X
F (z) =
(2.12)
µj fj (z),
j=1
is also in the class T Sb (α, β), where
m
X
(2.13)
µj = 1.
j=1
Proof. Combining the definitions (2.11) and (2.12), further by (2.13) we have
!
∞
m
b(1 − α)(c) 2 X X
(2.14)
F (z) = z −
z −
µj an,j z n .
(2 + β − α)(a)
n=3
j=1
Since fj (z) ∈ T Sb (α, β) for every j = 1, 2, . . . , m, Theorem 2.5 yields
∞
X
(2.15)
[n(1 + β) − (α + β)]
n=3
(a)n−1
an,j ≤ (1 − b)(1 − α),
(c)n−1
for j = 1, 2, . . . , m. Thus we obtain
∞
X
(a)n−1
[n(1 + β) − (α + β)]
(c)n−1
n=3
=
m
X
µj
j=1
∞
X
m
X
!
µj an,j
j=1
[n(1 + β) − (α + β)]
n=3
(a)n−1
an,j
(c)n−1
!
≤ (1 − b)(1 − α)
in view of Theorem 2.5. So, F (z) ∈ T Sb (α, β).
Theorem 2.9. Let
f2 (z) = z −
(2.16)
b(1 − α)(c) 2
z
(2 + β − α)(a)
and
(2.17)
fn (z) = z −
b(1 − α)(c) 2
(1 − b)(1 − α)(c)n−1
z −
zn
(2 + β − α)(a)
[n(1 + β) − (α + β)](a)n−1
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G. M URUGUSUNDARAMOORTHY
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for n = 3, 4, . . . . Then f (z) is in the class T Sb (α, β) if and only if it can be expressed in the
form
f (z) =
(2.18)
∞
X
λn fn (z),
n=2
where λn ≥ 0 and
∞
P
λn = 1.
n=2
Proof. We suppose that f (z) can be expressed in the form (2.18). Then we have
∞
(1 − b)(1 − α)(c)n−1
b(1 − α)(c) 2 X
z −
λn
zn
f (z) = z −
(2 + β − α)(a)
[n(1
+
β)
−
(α
+
β)](a)
n−1
n=3
=z−
(2.19)
∞
X
An z n ,
n=2
where
A2 =
(2.20)
b(1 − α)(c)
(2 + β − α)(a)
and
An =
(2.21)
λn (1 − b)(1 − α)(c)n−1
, n = 3, 4, . . . .
[n(1 + β) − (α + β)](a)n−1
Therefore
∞
X
(a)n−1
[n(1 + β) − (α + β)]
An = b(1 − α) +
λn (1 − b)(1 − α)
(c)n−1
n=3
n=2
∞
X
= (1 − α)[b + (1 − λ2 )(1 − b)]
≤ (1 − α),
(2.22)
it follows from Theorem 2.2 and Theorem 2.5 that f (z) is in the class T Sb (α, β). Conversely,
we suppose that f (z) defined by (2.4) is in the class T Sb (α, β). Then by using (2.6), we get
(2.23)
an ≤
(1 − b)(1 − α)(c)n−1
, (n ≥ 3).
[n(1 + β) − (α + β)](a)n−1
Setting
(2.24)
λn =
[n(1 + β) − (α + β)](a)n−1
an , (n ≥ 3)
(1 − b)(1 − α)(c)n−1
and
(2.25)
λ2 = 1 −
∞
X
λn ,
n=3
we have (2.18). This completes the proof of Theorem 2.9.
Corollary 2.10. The extreme points of the class T Sb (α, β) are functions fn (z), n ≥ 2 given by
Theorem 2.9.
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3. D ISTORTION T HEOREMS
In order to obtain distortion bounds for the function f ∈ T Sb (α, β), we first prove the following lemmas.
Lemma 3.1. Let the function f3 (z) be defined by
b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3
z −
z .
(2 + β − α)(a)
(3 + 2β − α)(a)2
Then, for 0 ≤ r < 1 and 0 ≤ b ≤ 1,
b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3
(3.2)
|f3 (reiθ )| ≥ r −
r −
r
(2 + β − α)(a)
(3 + 2β − α)(a)2
with equality for θ = 0. For either 0 ≤ b < b0 and 0 ≤ r ≤ r0 or b0 ≤ b ≤ 1,
b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3
(3.3)
|f3 (reiθ )| ≤ r +
r −
r
(2 + β − α)(a)
(3 + 2β − α)(a)2
with equality for θ = π, where
(3.1)
(3.4) b0 =
f3 (z) = z −
1
2(1 − α)(c)(c)2
× {−[(3 + 2β − α)(a)2 (c) + 4(2 + β − α)(a)(c)2 − (1 − α)(c)(c)2 ]
+ [((3 + 2β − α)(a)2 (c) + 4(2 + β − α)(a)(c)2 − (1 − α)(c)(c)22
+ 16(2 + β − α)(1 − α)(a)(c)(c)22 ]1/2 }
and
(3.5) r0 =
1
{−2(1 − b)(2 + β − α)(a)(c + 1)
b(1 − b)(1 − α)(c)2
+ [4(1 − b)2 (2 + β − α)2 (a)2 (c + 1)2
+ b2 (1 − b)(3 + 2β − α)(1 − α)(a)2 (c)2 ]1/2 }.
Proof. We employ the technique as used by Silverman and Silvia [8]. Since
∂|f3 (reiθ )|2
b(c)
4(1 − b)(c)2
3
(3.6)
= 2(1 − α)r sin θ
+
r cos θ
∂θ
(2 + β − α)(a) (3 + 2β − α)(a)2
b(1 − b)(1 − α)(c)(c)2
2
−
r
(2 + β − α)(3 + 2β − α)(a)(a)2
we can see that
∂|f3 (reiθ )|2
(3.7)
=0
∂θ
for θ1 = 0, θ2 = π, and
1 b[(1 − b)(1 − α)(c)2 r2 − (3 + 2β − α)(a)2 ]
−1
(3.8)
θ3 = cos
r
4(1 − b)(2 + β − α)(a)(c + 1)
since θ3 is a valid root only when −1 ≤ cos θ3 ≤ 1. Hence we have a third root if and only
if r0 ≤ r < 1 and 0 ≤ b ≤ b0 . Thus the results of the theorem follow from comparing the
extremal values |f3 (reiθk )|, k = 1, 2, 3 on the appropriate intervals.
Lemma 3.2. Let the functions fn (z) be defined by (2.17) and n ≥ 4. Then
(3.9)
J. Inequal. Pure and Appl. Math., 5(4) Art. 85, 2004
|fn (reiθ )| ≤ |f4 (−r)|.
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G. M URUGUSUNDARAMOORTHY
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Proof. Since
fn (z) = z −
and
rn
n
b(1 − α)(c) 2
(1 − b)(1 − α)(c)n−1
z −
zn
(2 + β − α)(a)
[n(1 + β) − (α + β)](a)n−1
is a decreasing function of n, we have
b(1 − α)(c) 2 (1 − b)(1 − α)(c)3 4
r −
r
(2 + β − α)(a)
[4 + 3β − α](a)3
= −f4 (−r),
|fn (reiθ | ≤ r +
which shows (3.9).
Theorem 3.3. Let the function f (z) defined by (2.4) belong to the class T Sb (α, β), then for
0 ≤ r < 1,
|f (reiθ )| ≥ r −
(3.10)
b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3
r −
r
(2 + β − α)(a)
[3 + 2β − α](a)2
with equality for f3 (z) at z = r, and
n
o
iθ
|f (re )| ≤ max max |f3 (re )|, −f4 (−r) ,
iθ
(3.11)
θ
where max |f3 (reiθ )| is given by Lemma 3.1.
θ
Proof. The proof of Theorem 3.3 is obtained by comparing the bounds of Lemma 3.1 and
Lemma 3.2.
Remark 3.4. Taking b = 1 in Theorem 3.3 we obtain the following result.
Corollary 3.5. Let the function f (z) defined by (1.7) be in the class T S(α, β). Then for |z| =
r < 1, we have
r−
(3.12)
(1 − α)(c)
(1 − α)(c)
r2 ≤ |f (z)| ≤ r +
r2 .
(2 + β − α)(a)
(2 + β − α)(a)
Lemma 3.6. Let the function f3 (z) be defined by (3.1). Then, for 0 ≤ r < 1, and 0 ≤ b ≤ 1,
|f30 (reiθ )| ≥ 1 −
(3.13)
2b(1 − α)(c)
3(1 − b)(1 − α)(c)2 2
r−
r
(2 + β − α)(a)
(3 + 2β − α)(a)2
with equality for θ = 0. For either 0 ≤ b < b1 and 0 ≤ r ≤ r1 or b1 ≤ b ≤ 1,
|f30 (reiθ )| ≤ 1 +
(3.14)
2b(1 − α)(c)
3(1 − b)(1 − α)(c)2 2
r−
r
(2 + β − α)(a)
(3 + 2β − α)(a)2
with equality for θ = π, where
1
6(1 − α)(c)(c2 )
× {−[(3 + 2β − α)(a)2 (c) + 6(2 + β − α)(a)(c)2 − 3(1 − α)(c)(c)2 ]
(3.15) b1 =
+ {((3 + 2β − α)(a)2 (c) + 6(2 + β − α)(a)(c)2 − 3(1 − α)(c)(c)2 )2
+ 72(2 + β − α)(1 − α)(a)(c)(c22 )}1/2 }
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and
(3.16) r1 =
1
{−3(1 − b)(2 + β − α)(a)(c + 1)
3b(1 − b)(1 − α)(c2 )
+ [8(1 − b)2 (2 + β − α)2 (a)2 (c + 1)2
+ 3b2 (1 − b)(3 + 2β − α)(1 − α)(a)2 (c)2 ]1/2 }.
Proof. The proof of Lemma 3.6 is much akin to the proof of Lemma 3.1.
Theorem 3.7. Let the function f (z) defined by (2.4) belong to the class T Sb (α, β), then for
0 ≤ r < 1,
|f 0 (reiθ )| ≥ 1 −
(3.17)
2b(1 − α)(c)
3(1 − b)(1 − α)(c)2 2
r−
r
(2 + β − α)(a)
[3 + 2β − α](a)2
with equality for f30 (z) at z = r, and
n
o
|f 0 (reiθ )| ≤ max max |f30 (reiθ )|, −f40 (−r) ,
(3.18)
θ
where max |f30 (reiθ )| is given by Lemma 3.6.
θ
Remark 3.8. Putting b = 1 in Theorem 3.7 we obtain the following result.
Corollary 3.9. Let the function f (z) defined by (1.2) be in the class T S(α, β). Then for |z| =
r < 1, we have
1−
(3.19)
2(1 − α)(c)
2(1 − α)(c)
r ≤ |f 0 (z)| ≤ 1 +
r.
(2 + β − α)(a)
(2 + β − α)(a)
4. T HE C LASS T Sbn ,k (α, β)
Instead of fixing just the second coefficient, we can fix finitely many coefficients. Let
T Sbn ,k (α, β) denote the class of functions in T Sb (α, β) of the form
f (z) = z −
(4.1)
k
X
n=2
where 0 ≤
Pk
n=2 bn
∞
X
bn (1 − α)(c)n−1
n
z −
an z n ,
[n(1 + β) − (α + β)](a)n−1
n=k+1
= b ≤ 1. Note that T Sb2 ,2 (α, β) = T Sb (α, β).
Theorem 4.1. The extreme points of the class T Sbn ,k (α, β) are
fk (z) = z −
k
X
n=2
bn (1 − α)(c)n−1
zn
[n(1 + β) − (α + β)](a)n−1
and
fn (z) = z −
k
X
n=2
∞
X
bn (1 − α)(c)n−1
(1 − b)(1 − α)(c)n−1
n
z −
zn
[n(1 + β) − (α + β)](a)n−1
[n(1 + β) − (α + β)](a)n−1
n=k+1
The details of the proof are omitted, since the characterization of the extreme points enables
us to solve the standard extremal problems in the same manner as was done for T Sb (α, β).
J. Inequal. Pure and Appl. Math., 5(4) Art. 85, 2004
http://jipam.vu.edu.au/
10
G. M URUGUSUNDARAMOORTHY AND N. M AGESH
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J. Inequal. Pure and Appl. Math., 5(4) Art. 85, 2004
http://jipam.vu.edu.au/