Paper-26-Acta24.2010. 137KB Jun 04 2011 12:03:13 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 24/2010
pp. 295-306
A SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS ASSOCIATED
WITH A FRACTIONAL CALCULUS OPERATOR
INVOLVING CAPUTO’S FRACTIONAL DIFFERENTIATION
Jamal Salah and Maslina Darus
Abstract. In this paper, we introduce a new class of functions which are analytic
and univalent with negative coefficients defined by using certain fractional operators
descibed in the Caputo sense. Characterization property, the results on modified
Hadamard product and integral transforms are discussed. Further, distortion theorem and radii of starlikeness and convexity are also determined here.
Keywords: Caputo’s differentiation operator; uniformly starlike; Hadamard product
1. Introduction
In recent years, considerable interest in fractional calculus operators has been stimulated due to their applications in the theory of analytic functions. There are many
definitions of fractional integration and differentiation can be found in various books
([11]-[14]). For the purpose of this paper, the Caputo’s definition of fractional differentiation will be used to introduce a new operator.
Denote by A the class of functions of the form
f (z) = z +
∞
X
an z n
(1)
n=2
which are analytic and univalent in the open disk U = {z : |z| < 1}. Also denote by
T the subclass of A consisting of functions of the form
f (z) = z −
∞
X
n=2
an z n , an ≥ 0.
(2)
Perhaps, Silverman [10] was the first to define the class T . For further properties
for the class T , see also [9].
A function f ∈ A is said to be in the class of uniformly convex functions of order α,
denoted by U CV (α) if
zf ” (z)
ℜ 1+ ′
−α
f (z)
”
zf (z)
≥β ′
− 1 .
f (z)
295
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J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
And is said to be in a corresponding subclass of U CV (α) denoted by Sp (α) if
ℜ
(
−1 ≤ α ≤ 1 and z ∈ U .
′
zf (z)
−α
f (z)
)
′
zf (z)
≥β
− 1 ,
f (z)
(4)
The class of uniformly convex and uniformly starlike functions has been studied by
Goodman,
P∞ seen ([2],[3]) and Ma and Minda [5]. If f of the form (1) and g(z) =
z + n=2 bn z are two functions in A. Then the Hadamard product of f and g is
denoted by f ∗ g and is given by
(f ∗ g) (z) = z +
∞
X
an bn z n
(5)
n=2
Now we look at the Caputo’s definition which shall be used throughout the paper.
Definition 1.[15] Caputo’s definition of the fractional-order derivative is defined as
1
D f (t) =
Γ (n − α)
α
Z
t
a
f (n) (τ )
dτ
(t − τ )α+1−n
(6)
where n − 1 < Re (α) ≤ n, n ∈ N , and the parameter α is allowed to be real or even
complex, a is the initial value of the function f .
Definition 2. The generalization operator of Salagean[17] derivative operator and
Libera integral operator [16], was given by Owa [6].
P
Γ(2−λ)Γ(n+1)
n
Ωλ f (z) = Γ (2 − λ) z λ Dzλ f (z) = z + ∞
n=2 Γ(n−λ+1) an z , for any real λ.
Remark 1. We note that
Ω0 f (z) = f (z) = z +
∞
X
an z n ,
n=2
Ω1 f (z) = Ωf (z) = zf ′ (z) = z +
∞
X
nan z n
n=2
and
Ωk f (z) = Ω(Ωk−1 f (z)) = zf ′ (z) = z +
∞
X
n=2
296
nk an z n
(k = 1, 2, 3...)
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
and this is the Salagean derivative operator [17].
Remark 2. We also note that
Ω
−1
2
f (z) =
z
z
Z
f (t)dt = z +
0
∞
X
n=2
2
an z n
n+1
and
Ω
−k
f (z) = Ω
−1
−k+1
(Ω
′
f (z)) = zf (z) = z +
∞
X
n=2
(
2 k
) an z n
n+1
(k = 1, 2, 3...)
and this is the Libera integral operator [16].
Now, using the previous definitions we can introduce the following operator:
Z
Γ (2 + η − λ) λ−η z Ωη f (ξ)
z
Jη,λ f (z) =
dξ
λ+1−η
Γ (η − λ)
0 (z − ξ)
(7)
where η(real number) and (η − 1 < λ ≤ η < 2).
By simple calculations we can write
Jη,λ f (z) = z +
∞
X
(Γ (n + 1))2 Γ (2 + η − λ) Γ (2 − η)
Γ (n + η − λ + 1) Γ (n − η + 1)
n=2
for f (z) ∈ A and has the form (1).
an z n
(8)
′
Further, note that J0,0 f (z) = f (z) and J1,1 f (z) = zf (z).
Now using the operator introduced in (7), we can define the following subclass of
analytic function. For −1 ≤ α < 1, a function f ∈ A is said to be in the class
∗ (α) if and only if
Sη,λ
ℜ
(
′
z (Jη,λ f (z))
−α
Jη,λ f (z)
)
z (J f (z))′
η,λ
≥
− 1 , z ∈ U.
Jη,λ f (z)
∗ (α)
Now let’s write T R (η, λ, α) = Sη,λ
Note that if η = λ = 0, we have
T
T.
∗
Sη,λ
(α) = Sp (α) .
297
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J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
And for η = λ = 1, we have
∗
(α) = U CV (α) .
Sη,λ
The classes Sp (α)and U CV (α) are introduced and studied by various authors including [1],[7],and [8].
2. Characterization Property
Definition 3. A function f is in T R (η, λ, α)if f satisfies the analytic characterization
ℜ
(
′
z (Jη,λ f (z))
−α
Jη,λ f (z)
where 0 ≤ α < 1,η − 1 < λ ≤ η < 2.
)
z (J f (z))′
η,λ
>
− 1
Jη,λ f (z)
(10)
Theorem 1. A function f defined by (2) is in the class T R (η, λ, α) if and only if
∞
X
(Γ (n + 1))2 Γ (2 − η) Γ (2 + η − λ) 2n − 1 − α
·
|an | ≤ 1.
Γ (n + η − λ + 1) Γ (n − η + 1)
1−α
(11)
n=2
Proof. It suffices to show that
(
)
′
z (J f (z))′
z
(J
f
(z))
η,λ
η,λ
− 1 ≤ ℜ
−α
Jη,λ f (z)
Jη,λ f (z)
and we have
(
)
′
′
z
J
f
(z)
z
J
f
(z)
( η,λ )
( η,λ )
Jη,λ f (z) − 1 ≤ ℜ
Jη,λ f (z) − 1 + (1 − α)
that is
(
)
′
′
′
z (Jη,λ f (z))
z (Jη,λ f (z))
z (Jη,λ f (z))
−
1
−
ℜ
−
1
≤
2
−
1
Jη,λ f (z)
≤
Jη,λ f (z)
Jη,λ f (z)
P∞
n=2
P(n−1)φ(n)|an |
1− ∞
n=2 φ(n)|an |
where
φ (n) =
(Γ(n+1))2 Γ(2+η−λ)Γ(2−η)
Γ(n+η−λ+1)Γ(n−η+1) .
The above expression is bounded by (1 − α) and hence the assertion of the result.
298
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
Now we want to show that f ∈ T R (η, λ, α) satisfies (11) if f ∈ T R (η, λ, α), then
(10) yields
P
n−1
1− P∞
n=2 nφ(n)an z
∞
n−1
1− n=2 φ(n)an z
−α≥
P
n−1
1− ∞
nz
n=2
P ∞(n−1)φ(n)an−1
.
1− n=2 φ(n)an z
Letting z → 1 along the real axis leads to the inequality
1 − α.
P∞
n=2 (2n
− 1 − α) φ (n) an ≤
Corollary 1. Let a function f , defined by (2) belongs to the class T R (η, λ, α).
Then
an ≤
1−α
Γ (n + η − λ + 1) Γ (n − η + 1)
·
, for n ≥ 2.
2
2n
−1−α
(Γ (n + 1)) Γ (2 + η − λ) Γ (2 − η)
Next we consider the growth and distortion theorems for the class T R (η, λ, α). We
shall omit the proof as the techniques are similar to various other papers.
Theorem 2. Let the function f , defined by (2) be in the class T R (η, λ, α). Then
|z| − |z|2
(2 + η − λ) (2 − η) (1 − α)
(2 + η − λ) (2 − η) (1 − α)
≤ |Jη,λ f (z)| ≤ |z| + |z|2
(12)
4 (3 − α)
4 (3 − α)
′
(2 + η − λ) (2 − η) (1 − α)
(2 + η − λ) (2 − η) (1 − α)
1 − |z|
≤ (Jη,λ f (z)) ≤ 1 + |z|
(13)
2 (3 − α)
2 (3 − α)
The bounds (12) and (13) are attained for the functions given by
f (z) = z −
(2 + η − λ) (2 − η) (1 − α) 2
z .
4 (3 − α)
(14)
Theorem 3. Let a function f , be defined by (2) and
g (z) = z −
∞
X
bn z n
(15)
n=2
be in the class T R (η, λ, α). then the function h, defined by
h (z) = (1 − β) f (z) + βg (z) = z −
299
∞
X
n=2
cn z n
(16)
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
where cn = (1 − β) an + βbn ,and 0 ≤ β ≤ 1,is also in the class T R (η, λ, α) .
Proof. The result follows easily by using (11) and (16).
Now we define the following functions fj (z) ,(j = 1, 2, 3, ......m) of the form
fj (z) = z −
∞
X
n=2
an,j z n , an,j ≥ 0, z ∈ U.
(17)
Theorem 4. (Closure theorem)Let the functions fj (z) , (j = 1, 2, 3, ......m) defined by (16), be in the class T R (η, λ, αj ) (j =
1, 2, 3, ......m)
respectively. Then the
P
m
1 P∞
n
function h (z) defined by h (z) = z − m n=2
j=1 an,j z
is in the class T R (η, λ, α) where
α = min {αj } with0 ≤ αj < 1.
1≤j≤m
(18)
Proof. Since fj ∈ T R (η, λ, αj ) (j = 1, 2, 3, ......m) by applying Theorem 1, we observe that
P
P∞
P∞
m
1
1 Pm
φ
(n)
(2n
−
1
−
α)
a
n=2
j=1 n,j = m
j=1 ( n=2 φ (n) (2n − 1 − α) an,j ) .
m
In view of Theorem 1 again implies that h (z) ∈ T R (η, λ, α).
3. Results involving convolution
Theorem 5. For functions fj (z) (j = 1, 2) defined by (17) let f1 (z) ∈ T R(η, λ, α)
and f2 (z) ∈ T R(η, λ, β). then f1 ∗ f2 ∈ T R(η, λ, γ), where
γ ≤1−
n ≥ 2 and φ (n) =
2 (n − 1) (1 − α) (1 − β)
,
(2n − 1 − α) (2n − 1 − β) φ (n) − (1 − α) (1 − β)
(Γ(n+1))2 Γ(2+η−λ)Γ(2−η)
Γ(n+η−λ+1)Γ(n−η+1)
Proof. In view of Theorem 1, it suffices to prove that
∞
X
2n − 1 − γ
n=2
1−γ
φ (n) an,1 an,2 ≤ 1.
It follows from Theorem 1 and the Cauchy-Shwarz inequality that
300
(19)
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
∞
X
n=2
√
√
2n − 1 − α · 2n − 1 − β
√
p
φ (n) an,1 an,2 ≤ 1.
(1 − α) (1 − β)
Thus we need to find γ such that
∞
X
2n − 1 − γ
1−γ
n=2
φ (n) an,1 an,2 ≤
or
√
an,1 an,2 ≤
∞
X
n=2
√
∞
X
n=2
√
√
2n − 1 − α · 2n − 1 − β
√
p
φ (n) an,1 an,2 ≤ 1
(1 − α) (1 − β)
√
2n − 1 − α · 2n − 1 − β
1−γ
p
·
2n − 1 − γ
(1 − α) (1 − β)
by virtue of (19), it suffices to find φ (n) such that
p
(1 − α) (1 − β)
√
≤
√
2n − 1 − α 2n − 1 − βφ (n)
which yield
γ ≤1−
√
√
1−γ
2n − 1 − α 2n − 1 − β
p
·
2n
−1−γ
(1 − α) (1 − β)
2 (n − 1) (1 − α) (1 − β)
, n ≥ 2.
(2n − 1 − α) (2n − 1 − β) φ (n) − (1 − α) (1 − β)
Theorem 6. Let the functions fj (z) , (j = 1, 2) defined by (16) be in the class
T R(η, λ, α). Then (f1 ∗ f2 ) (z) ∈ T R(η, λ, δ) where δ ≤ 1 −
2(n−1)(1−α)2
(2n−1−α)2 φ(n)−(1−α)2
Proof. By taking β = α in the above theorem, the result follows.
Theorem 7.
P Let then function f defined by (2) be in the class T R(η, λ, α), and let
g (z) = z − ∞
n=2 bn z , for |bn | ≤ 1. Then (f ∗ g) (z) ∈ T R (η, λ, α) .
Proof.
∞
X
n=2
φ (n) (2n − 1 − α) |an bn | =
≤
∞
X
n=2
∞
X
n=2
φ (n) (2n − 1 − α) an |bn |
φ (n) (2n − 1 − α) an ≤ 1 − α.
301
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
Hence it follows that (f ∗ g) (z) ∈ T R (η, λ, α) .
Corollary P
2. Let the function f defined by (2) be in the class T R(η, λ, α). Also let
n
g (z) = z − ∞
n=2 bn z for 0 ≤ bn ≤ 1. Then (f ∗ g) (z) ∈ T R (η, λ, α) .
Theorem 8. Let the functions fj (z) , (j = 1, 2) defined
be in the class
P by (16)
n
2 + a2
T R(η, λ, α). Then the function h defined by h (z) = z − ∞
a
n,1
n,2 z is in
n=2
the class T R(η, λ, µ) where
µ≤1−
4 (1 − α)2
,n ≥ 2
(2n − 1 − α)2 φ (n) − 2 (1 − α)2
(20)
Proof In view of Theorem 1, it suffices to prove that
∞
X
φ (n)
n=2
2n − 1 − µ 2
an,1 + a2n,2 ≤ 1
1−µ
(21)
From (16) and Theorem 1 we get,
2
∞
∞
X
X
2n − 1 − α 2 2
2n − 1 − α
φ (n)
an,j ≤ 1
an,j ≤
φ (n)
1−α
1−α
n=2
(22)
n=2
On comparing (21) and (22), it can be seen that inequality (20) will be satisfied if
1
2n − 1 − α
2n − 1 − µ 2
2
φ (n)
an,1 + an,2 ≤
a2n,1 + a2n,2
φ (n)
1−µ
2
1−α
that is if
µ≤1−
4 (1 − α)2
(2n − 1 − α)2 φ (n) − 2 (1 − α)2
which completes the proof.
4. Integral Transform of the class T R(η, λ, α)
We define the integral transform
Vµ (f ) (z) =
Z
1
µ (t)
0
302
f (tz)
dt
t
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
R1
where µ (t) is a real valued, non-negative weight function normalized so that 0 µ (t) dt =
1.
δ
1 δ−1
c
, c > −1, δ ≥ 0, which gives the
Special case of µ (t) is µ (t) = (c+1)
µ(δ) t log t
Komatu operator.
For further details on integral transform, see also [4].
Theorem 9. Let f ∈ T R(η, λ, α). Then Vµ (f ) ∈ T R(η, λ, α).
Proof By definition, we have
(c + 1)δ
Vµ (f ) =
µ (δ)
Z
1
(−1)δ−1 tc (log t)δ−1
0
(−1)δ−1 (c + 1)δ
lim
=
µ (δ)
r→0+
"Z
1
tc (log t)δ−1
r
z−
z−
∞
X
an z n tn−1
n=2
∞
X
an z n tn−1
n=2
!
!
dt
dt
#
with simple calculations, we get
Vµ (f ) (z) = z −
∞
X
c+1 δ
n=2
c+n
an z n .
We need to prove that
∞
X
n=2
2n − 1 − α
φ (n) ·
1−α
c+1
c+n
δ
an < 1.
(23)
On the other hand f ∈ T R(η, λ, α) if and only if
∞
X
n=2
hence
c+1
c+n
φ (n) ·
2n − 1 − α
an < 1
1−α
< 1. Therefore (23) holds and the proof is complete.
Theorem 10. (Radius of starlikeness) Let f ∈ T R(η, λ, α).then Vµ (f )is starlike
of order 0 ≤ γ ≺ 1 in |z| ≺ R1 , where
R1 = inf
n
"
c+n
c+1
δ
# 1
n−1
1 − γ (2n − 1 − α)
φ (n)
·
(n − γ) (1 − α)
and
303
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
φ (n) =
(Γ (n + 1))2 Γ (2 + η − λ) Γ (2 − η)
Γ (n + η − λ + 1) Γ (n − η + 1) .
Proof. It suffices to prove that
z (V (f ) (z))′
µ
− 1 < 1 − γ
Vµ (f ) (z)
we have
(24)
δ
P∞
c+1
n−1
z (V (f ) (z))′
n=2 (1 − n) c+n
a
z
n
µ
− 1 =
δ
P
Vµ (f ) (z)
c+1
n−1
1− ∞
a
z
n
n=2 c+n
≤
δ
c+1
an |z|n−1
(n
−
1)
n=2
c+n
P∞ c+1 δ
1 − n=2 c+n an |z|n−1
P∞
the last expression is less that 1 − γ since
c + n δ (1 − γ) (2n − 1 − α)
ϕ (n)
|z|n−1 <
c+1
(n − γ) (1 − α)
so the proof is complete.
Theorem 11. If f ∈ T R(η, λ, α). Then Vµ (f )is convex of order 0 ≤ γ < 1,in
|z| < R2 , where
# 1
"
n−1
c + n δ (1 − γ) (2n − 1 − α)
φ (n)
R2 = inf
n
c+1
n (n − γ) (1 − α)
′
using the fact that f is convex if and only if zf is starlike, the proof can be easily
derived.
AcknowledgementThe work presented here was supported by MOHE: UKM-ST06-FRGS0107-2009
304
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
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Theory and Applications, Gordon and Breach, Yverdon, 1993.
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Jamal Salah and Maslina Darus∗
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi 43600 Selangor D. Ehsan, Malaysia
Emails: [email protected]∗ [email protected]
306
ISSN: 1582-5329
No. 24/2010
pp. 295-306
A SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS ASSOCIATED
WITH A FRACTIONAL CALCULUS OPERATOR
INVOLVING CAPUTO’S FRACTIONAL DIFFERENTIATION
Jamal Salah and Maslina Darus
Abstract. In this paper, we introduce a new class of functions which are analytic
and univalent with negative coefficients defined by using certain fractional operators
descibed in the Caputo sense. Characterization property, the results on modified
Hadamard product and integral transforms are discussed. Further, distortion theorem and radii of starlikeness and convexity are also determined here.
Keywords: Caputo’s differentiation operator; uniformly starlike; Hadamard product
1. Introduction
In recent years, considerable interest in fractional calculus operators has been stimulated due to their applications in the theory of analytic functions. There are many
definitions of fractional integration and differentiation can be found in various books
([11]-[14]). For the purpose of this paper, the Caputo’s definition of fractional differentiation will be used to introduce a new operator.
Denote by A the class of functions of the form
f (z) = z +
∞
X
an z n
(1)
n=2
which are analytic and univalent in the open disk U = {z : |z| < 1}. Also denote by
T the subclass of A consisting of functions of the form
f (z) = z −
∞
X
n=2
an z n , an ≥ 0.
(2)
Perhaps, Silverman [10] was the first to define the class T . For further properties
for the class T , see also [9].
A function f ∈ A is said to be in the class of uniformly convex functions of order α,
denoted by U CV (α) if
zf ” (z)
ℜ 1+ ′
−α
f (z)
”
zf (z)
≥β ′
− 1 .
f (z)
295
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J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
And is said to be in a corresponding subclass of U CV (α) denoted by Sp (α) if
ℜ
(
−1 ≤ α ≤ 1 and z ∈ U .
′
zf (z)
−α
f (z)
)
′
zf (z)
≥β
− 1 ,
f (z)
(4)
The class of uniformly convex and uniformly starlike functions has been studied by
Goodman,
P∞ seen ([2],[3]) and Ma and Minda [5]. If f of the form (1) and g(z) =
z + n=2 bn z are two functions in A. Then the Hadamard product of f and g is
denoted by f ∗ g and is given by
(f ∗ g) (z) = z +
∞
X
an bn z n
(5)
n=2
Now we look at the Caputo’s definition which shall be used throughout the paper.
Definition 1.[15] Caputo’s definition of the fractional-order derivative is defined as
1
D f (t) =
Γ (n − α)
α
Z
t
a
f (n) (τ )
dτ
(t − τ )α+1−n
(6)
where n − 1 < Re (α) ≤ n, n ∈ N , and the parameter α is allowed to be real or even
complex, a is the initial value of the function f .
Definition 2. The generalization operator of Salagean[17] derivative operator and
Libera integral operator [16], was given by Owa [6].
P
Γ(2−λ)Γ(n+1)
n
Ωλ f (z) = Γ (2 − λ) z λ Dzλ f (z) = z + ∞
n=2 Γ(n−λ+1) an z , for any real λ.
Remark 1. We note that
Ω0 f (z) = f (z) = z +
∞
X
an z n ,
n=2
Ω1 f (z) = Ωf (z) = zf ′ (z) = z +
∞
X
nan z n
n=2
and
Ωk f (z) = Ω(Ωk−1 f (z)) = zf ′ (z) = z +
∞
X
n=2
296
nk an z n
(k = 1, 2, 3...)
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
and this is the Salagean derivative operator [17].
Remark 2. We also note that
Ω
−1
2
f (z) =
z
z
Z
f (t)dt = z +
0
∞
X
n=2
2
an z n
n+1
and
Ω
−k
f (z) = Ω
−1
−k+1
(Ω
′
f (z)) = zf (z) = z +
∞
X
n=2
(
2 k
) an z n
n+1
(k = 1, 2, 3...)
and this is the Libera integral operator [16].
Now, using the previous definitions we can introduce the following operator:
Z
Γ (2 + η − λ) λ−η z Ωη f (ξ)
z
Jη,λ f (z) =
dξ
λ+1−η
Γ (η − λ)
0 (z − ξ)
(7)
where η(real number) and (η − 1 < λ ≤ η < 2).
By simple calculations we can write
Jη,λ f (z) = z +
∞
X
(Γ (n + 1))2 Γ (2 + η − λ) Γ (2 − η)
Γ (n + η − λ + 1) Γ (n − η + 1)
n=2
for f (z) ∈ A and has the form (1).
an z n
(8)
′
Further, note that J0,0 f (z) = f (z) and J1,1 f (z) = zf (z).
Now using the operator introduced in (7), we can define the following subclass of
analytic function. For −1 ≤ α < 1, a function f ∈ A is said to be in the class
∗ (α) if and only if
Sη,λ
ℜ
(
′
z (Jη,λ f (z))
−α
Jη,λ f (z)
)
z (J f (z))′
η,λ
≥
− 1 , z ∈ U.
Jη,λ f (z)
∗ (α)
Now let’s write T R (η, λ, α) = Sη,λ
Note that if η = λ = 0, we have
T
T.
∗
Sη,λ
(α) = Sp (α) .
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And for η = λ = 1, we have
∗
(α) = U CV (α) .
Sη,λ
The classes Sp (α)and U CV (α) are introduced and studied by various authors including [1],[7],and [8].
2. Characterization Property
Definition 3. A function f is in T R (η, λ, α)if f satisfies the analytic characterization
ℜ
(
′
z (Jη,λ f (z))
−α
Jη,λ f (z)
where 0 ≤ α < 1,η − 1 < λ ≤ η < 2.
)
z (J f (z))′
η,λ
>
− 1
Jη,λ f (z)
(10)
Theorem 1. A function f defined by (2) is in the class T R (η, λ, α) if and only if
∞
X
(Γ (n + 1))2 Γ (2 − η) Γ (2 + η − λ) 2n − 1 − α
·
|an | ≤ 1.
Γ (n + η − λ + 1) Γ (n − η + 1)
1−α
(11)
n=2
Proof. It suffices to show that
(
)
′
z (J f (z))′
z
(J
f
(z))
η,λ
η,λ
− 1 ≤ ℜ
−α
Jη,λ f (z)
Jη,λ f (z)
and we have
(
)
′
′
z
J
f
(z)
z
J
f
(z)
( η,λ )
( η,λ )
Jη,λ f (z) − 1 ≤ ℜ
Jη,λ f (z) − 1 + (1 − α)
that is
(
)
′
′
′
z (Jη,λ f (z))
z (Jη,λ f (z))
z (Jη,λ f (z))
−
1
−
ℜ
−
1
≤
2
−
1
Jη,λ f (z)
≤
Jη,λ f (z)
Jη,λ f (z)
P∞
n=2
P(n−1)φ(n)|an |
1− ∞
n=2 φ(n)|an |
where
φ (n) =
(Γ(n+1))2 Γ(2+η−λ)Γ(2−η)
Γ(n+η−λ+1)Γ(n−η+1) .
The above expression is bounded by (1 − α) and hence the assertion of the result.
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J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
Now we want to show that f ∈ T R (η, λ, α) satisfies (11) if f ∈ T R (η, λ, α), then
(10) yields
P
n−1
1− P∞
n=2 nφ(n)an z
∞
n−1
1− n=2 φ(n)an z
−α≥
P
n−1
1− ∞
nz
n=2
P ∞(n−1)φ(n)an−1
.
1− n=2 φ(n)an z
Letting z → 1 along the real axis leads to the inequality
1 − α.
P∞
n=2 (2n
− 1 − α) φ (n) an ≤
Corollary 1. Let a function f , defined by (2) belongs to the class T R (η, λ, α).
Then
an ≤
1−α
Γ (n + η − λ + 1) Γ (n − η + 1)
·
, for n ≥ 2.
2
2n
−1−α
(Γ (n + 1)) Γ (2 + η − λ) Γ (2 − η)
Next we consider the growth and distortion theorems for the class T R (η, λ, α). We
shall omit the proof as the techniques are similar to various other papers.
Theorem 2. Let the function f , defined by (2) be in the class T R (η, λ, α). Then
|z| − |z|2
(2 + η − λ) (2 − η) (1 − α)
(2 + η − λ) (2 − η) (1 − α)
≤ |Jη,λ f (z)| ≤ |z| + |z|2
(12)
4 (3 − α)
4 (3 − α)
′
(2 + η − λ) (2 − η) (1 − α)
(2 + η − λ) (2 − η) (1 − α)
1 − |z|
≤ (Jη,λ f (z)) ≤ 1 + |z|
(13)
2 (3 − α)
2 (3 − α)
The bounds (12) and (13) are attained for the functions given by
f (z) = z −
(2 + η − λ) (2 − η) (1 − α) 2
z .
4 (3 − α)
(14)
Theorem 3. Let a function f , be defined by (2) and
g (z) = z −
∞
X
bn z n
(15)
n=2
be in the class T R (η, λ, α). then the function h, defined by
h (z) = (1 − β) f (z) + βg (z) = z −
299
∞
X
n=2
cn z n
(16)
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
where cn = (1 − β) an + βbn ,and 0 ≤ β ≤ 1,is also in the class T R (η, λ, α) .
Proof. The result follows easily by using (11) and (16).
Now we define the following functions fj (z) ,(j = 1, 2, 3, ......m) of the form
fj (z) = z −
∞
X
n=2
an,j z n , an,j ≥ 0, z ∈ U.
(17)
Theorem 4. (Closure theorem)Let the functions fj (z) , (j = 1, 2, 3, ......m) defined by (16), be in the class T R (η, λ, αj ) (j =
1, 2, 3, ......m)
respectively. Then the
P
m
1 P∞
n
function h (z) defined by h (z) = z − m n=2
j=1 an,j z
is in the class T R (η, λ, α) where
α = min {αj } with0 ≤ αj < 1.
1≤j≤m
(18)
Proof. Since fj ∈ T R (η, λ, αj ) (j = 1, 2, 3, ......m) by applying Theorem 1, we observe that
P
P∞
P∞
m
1
1 Pm
φ
(n)
(2n
−
1
−
α)
a
n=2
j=1 n,j = m
j=1 ( n=2 φ (n) (2n − 1 − α) an,j ) .
m
In view of Theorem 1 again implies that h (z) ∈ T R (η, λ, α).
3. Results involving convolution
Theorem 5. For functions fj (z) (j = 1, 2) defined by (17) let f1 (z) ∈ T R(η, λ, α)
and f2 (z) ∈ T R(η, λ, β). then f1 ∗ f2 ∈ T R(η, λ, γ), where
γ ≤1−
n ≥ 2 and φ (n) =
2 (n − 1) (1 − α) (1 − β)
,
(2n − 1 − α) (2n − 1 − β) φ (n) − (1 − α) (1 − β)
(Γ(n+1))2 Γ(2+η−λ)Γ(2−η)
Γ(n+η−λ+1)Γ(n−η+1)
Proof. In view of Theorem 1, it suffices to prove that
∞
X
2n − 1 − γ
n=2
1−γ
φ (n) an,1 an,2 ≤ 1.
It follows from Theorem 1 and the Cauchy-Shwarz inequality that
300
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J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
∞
X
n=2
√
√
2n − 1 − α · 2n − 1 − β
√
p
φ (n) an,1 an,2 ≤ 1.
(1 − α) (1 − β)
Thus we need to find γ such that
∞
X
2n − 1 − γ
1−γ
n=2
φ (n) an,1 an,2 ≤
or
√
an,1 an,2 ≤
∞
X
n=2
√
∞
X
n=2
√
√
2n − 1 − α · 2n − 1 − β
√
p
φ (n) an,1 an,2 ≤ 1
(1 − α) (1 − β)
√
2n − 1 − α · 2n − 1 − β
1−γ
p
·
2n − 1 − γ
(1 − α) (1 − β)
by virtue of (19), it suffices to find φ (n) such that
p
(1 − α) (1 − β)
√
≤
√
2n − 1 − α 2n − 1 − βφ (n)
which yield
γ ≤1−
√
√
1−γ
2n − 1 − α 2n − 1 − β
p
·
2n
−1−γ
(1 − α) (1 − β)
2 (n − 1) (1 − α) (1 − β)
, n ≥ 2.
(2n − 1 − α) (2n − 1 − β) φ (n) − (1 − α) (1 − β)
Theorem 6. Let the functions fj (z) , (j = 1, 2) defined by (16) be in the class
T R(η, λ, α). Then (f1 ∗ f2 ) (z) ∈ T R(η, λ, δ) where δ ≤ 1 −
2(n−1)(1−α)2
(2n−1−α)2 φ(n)−(1−α)2
Proof. By taking β = α in the above theorem, the result follows.
Theorem 7.
P Let then function f defined by (2) be in the class T R(η, λ, α), and let
g (z) = z − ∞
n=2 bn z , for |bn | ≤ 1. Then (f ∗ g) (z) ∈ T R (η, λ, α) .
Proof.
∞
X
n=2
φ (n) (2n − 1 − α) |an bn | =
≤
∞
X
n=2
∞
X
n=2
φ (n) (2n − 1 − α) an |bn |
φ (n) (2n − 1 − α) an ≤ 1 − α.
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J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
Hence it follows that (f ∗ g) (z) ∈ T R (η, λ, α) .
Corollary P
2. Let the function f defined by (2) be in the class T R(η, λ, α). Also let
n
g (z) = z − ∞
n=2 bn z for 0 ≤ bn ≤ 1. Then (f ∗ g) (z) ∈ T R (η, λ, α) .
Theorem 8. Let the functions fj (z) , (j = 1, 2) defined
be in the class
P by (16)
n
2 + a2
T R(η, λ, α). Then the function h defined by h (z) = z − ∞
a
n,1
n,2 z is in
n=2
the class T R(η, λ, µ) where
µ≤1−
4 (1 − α)2
,n ≥ 2
(2n − 1 − α)2 φ (n) − 2 (1 − α)2
(20)
Proof In view of Theorem 1, it suffices to prove that
∞
X
φ (n)
n=2
2n − 1 − µ 2
an,1 + a2n,2 ≤ 1
1−µ
(21)
From (16) and Theorem 1 we get,
2
∞
∞
X
X
2n − 1 − α 2 2
2n − 1 − α
φ (n)
an,j ≤ 1
an,j ≤
φ (n)
1−α
1−α
n=2
(22)
n=2
On comparing (21) and (22), it can be seen that inequality (20) will be satisfied if
1
2n − 1 − α
2n − 1 − µ 2
2
φ (n)
an,1 + an,2 ≤
a2n,1 + a2n,2
φ (n)
1−µ
2
1−α
that is if
µ≤1−
4 (1 − α)2
(2n − 1 − α)2 φ (n) − 2 (1 − α)2
which completes the proof.
4. Integral Transform of the class T R(η, λ, α)
We define the integral transform
Vµ (f ) (z) =
Z
1
µ (t)
0
302
f (tz)
dt
t
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
R1
where µ (t) is a real valued, non-negative weight function normalized so that 0 µ (t) dt =
1.
δ
1 δ−1
c
, c > −1, δ ≥ 0, which gives the
Special case of µ (t) is µ (t) = (c+1)
µ(δ) t log t
Komatu operator.
For further details on integral transform, see also [4].
Theorem 9. Let f ∈ T R(η, λ, α). Then Vµ (f ) ∈ T R(η, λ, α).
Proof By definition, we have
(c + 1)δ
Vµ (f ) =
µ (δ)
Z
1
(−1)δ−1 tc (log t)δ−1
0
(−1)δ−1 (c + 1)δ
lim
=
µ (δ)
r→0+
"Z
1
tc (log t)δ−1
r
z−
z−
∞
X
an z n tn−1
n=2
∞
X
an z n tn−1
n=2
!
!
dt
dt
#
with simple calculations, we get
Vµ (f ) (z) = z −
∞
X
c+1 δ
n=2
c+n
an z n .
We need to prove that
∞
X
n=2
2n − 1 − α
φ (n) ·
1−α
c+1
c+n
δ
an < 1.
(23)
On the other hand f ∈ T R(η, λ, α) if and only if
∞
X
n=2
hence
c+1
c+n
φ (n) ·
2n − 1 − α
an < 1
1−α
< 1. Therefore (23) holds and the proof is complete.
Theorem 10. (Radius of starlikeness) Let f ∈ T R(η, λ, α).then Vµ (f )is starlike
of order 0 ≤ γ ≺ 1 in |z| ≺ R1 , where
R1 = inf
n
"
c+n
c+1
δ
# 1
n−1
1 − γ (2n − 1 − α)
φ (n)
·
(n − γ) (1 − α)
and
303
J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
φ (n) =
(Γ (n + 1))2 Γ (2 + η − λ) Γ (2 − η)
Γ (n + η − λ + 1) Γ (n − η + 1) .
Proof. It suffices to prove that
z (V (f ) (z))′
µ
− 1 < 1 − γ
Vµ (f ) (z)
we have
(24)
δ
P∞
c+1
n−1
z (V (f ) (z))′
n=2 (1 − n) c+n
a
z
n
µ
− 1 =
δ
P
Vµ (f ) (z)
c+1
n−1
1− ∞
a
z
n
n=2 c+n
≤
δ
c+1
an |z|n−1
(n
−
1)
n=2
c+n
P∞ c+1 δ
1 − n=2 c+n an |z|n−1
P∞
the last expression is less that 1 − γ since
c + n δ (1 − γ) (2n − 1 − α)
ϕ (n)
|z|n−1 <
c+1
(n − γ) (1 − α)
so the proof is complete.
Theorem 11. If f ∈ T R(η, λ, α). Then Vµ (f )is convex of order 0 ≤ γ < 1,in
|z| < R2 , where
# 1
"
n−1
c + n δ (1 − γ) (2n − 1 − α)
φ (n)
R2 = inf
n
c+1
n (n − γ) (1 − α)
′
using the fact that f is convex if and only if zf is starlike, the proof can be easily
derived.
AcknowledgementThe work presented here was supported by MOHE: UKM-ST06-FRGS0107-2009
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J. Salah, M. Darus - A subclass of uniformly convex functions associated with a...
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Jamal Salah and Maslina Darus∗
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi 43600 Selangor D. Ehsan, Malaysia
Emails: [email protected]∗ [email protected]
306