DOI: 10.1515/auom-2015-0001
An. Şt. Univ. Ovidius Constanţa
Vol. 23(1),2015, 9–24
Properties on a subclass of univalent functions
defined by using a multiplier transformation
and Ruscheweyh derivative
Alina Alb Lupaş
Abstract
In this paper we have introduced and studied the subclass RI(d, α, β)
γ
of univalent functions defined by the linear operator RIn,λ,l
f (z) defined
n
by using the Ruscheweyh derivative R f (z) and multiplier transformaγ
γ
tion I (n, λ, l) f (z), as RIn,λ,l
: A → A, RIn,λ,l
f (z) = (1 − γ)Rn f (z) +
γI (n, λ, l) f (z), z ∈ U, where An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 +
. . . , z ∈ U } is the class of normalized analytic functions with A1 = A.
The main object is to investigate several properties such as coefficient
estimates, distortion theorems, closure theorems, neighborhoods and
the radii of starlikeness, convexity and close-to-convexity of functions
belonging to the class RI(d, α, β).
1
Introduction
Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and
H(U ) the space of holomorphic functions in U .
Let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U } with A1 = A.
Key Words: univalent function, Starlike functions, Convex functions, Distortion
theorem.
2010 Mathematics Subject Classification: Primary 30C45, Secondary 30A20, 34A40.
Received: 3 May, 2014.
Revised: 24 June, 2014.
Accepted: 29 June, 2014.
9
PROPERTIES OA SUBCLASS OF UNIVALENT FUNCTIONS DEFINED
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Definition 1. (Ruscheweyh [20]) For f ∈ A, n ∈ N, the operator Rn is
defined by Rn : A → A,
R0 f (z)
=
f (z)
R1 f (z)
=
zf 0 (z) , ...
(n + 1) Rn+1 f (z)
=
0
z (Rn f (z)) + nRn f (z) ,
P∞
Remark 1. If f ∈ A, f (z) = z + j=2 aj z j , then
n
R f (z) = z +
∞
X
(n + j − 1)!
j=2
n! (j − 1)!
z ∈ U.
aj z j , z ∈ U.
Definition 2. For f ∈ A, n ∈ N, λ, l ≥ 0, the operator I (n, λ, l) f (z) is
defined by the following infinite series
I (n, λ, l) f (z) = z +
n
∞ X
λ (j − 1) + l + 1
j=2
l+1
aj z j .
Remark 2. It follows from the above definition that
I (0, λ, l) f (z) = f (z),
0
(l + 1) I (n + 1, λ, l) f (z) = (l + 1 − λ) I (n, λ, l) f (z) + λz (I (n, λ, l) f (z)) ,
z ∈ U.
Remark 3. For l = 0, λ ≥ 0, the operator Dλn = I (n, λ, 0) was introduced
and studied by Al-Oboudi [16], which is reduced to the Sălăgean differential
operator [21] for λ = 1.
γ
Definition 3. [7] Let γ, λ, l ≥ 0, n ∈ N. Denote by RIn,λ,l
the operator given
γ
γ
n
by RIn,λ,l : A → A, RIn,λ,l f (z) = (1 − γ)R f (z) + γI (n, λ, l) f (z), z ∈ U.
P∞
Remark 4. If f ∈ A, f (z) = z + j=2 aj z j , then
n
o
P∞ n γ
+ (1 − γ) (n+j−1)!
aj z j , z ∈ U.
RIn,λ,l
f (z) = z + j=2 γ 1+λ(j−1)+l
l+1
n!(j−1)!
This operator was studied also in [13], [14].
0
Remark 5. For α = 0, RIm,λ,l
f (z) = Rm f (z), where z ∈ U and for α = 1,
1
RIm,λ,l
f (z) = I (m, λ, l) f (z), where z ∈ U , which was studied in [3], [4], [10],
α
m
[9]. For l = 0, we obtain RIm,λ,0
f (z) = RDλ,α
f (z) which was studied in [5],
α
f (z) =
[6], [11], [12], [17], [18] and for l = 0 and λ = 1, we obtain RIm,1,0
Lm
f
(z)
which
was
studied
in
[1],
[2],
[8],
[15].
α
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We follow the works of A.R. Juma and H. Ziraz [19].
Definition 4. Let the function f ∈ A. Then f (z) is said to be in the class
RI(d, α, β) if it satisfies the following criterion:
γ
γ
0
2
00
1 z(RIn,λ,l f (z)) + αz RIn,λ,l f (z))
− 1)| < β,
| (
γ
γ
d (1 − α)RIn,λ,l f (z) + αz(RIn,λ,l f (z))0
(1)
where d ∈ C − {0}, 0 ≤ α ≤ 1, 0 < β ≤ 1, z ∈ U .
In this paper we shall first deduce a necessary and sufficient condition for
a function f (z) to be in the class RI(d, α, β). Then obtain the distortion
and growth theorems, closure theorems, neighborhood and radii of univalent
starlikeness, convexity and close-to-convexity of order δ, 0 ≤ δ < 1, for these
functions.
2
Coefficient Inequality
Theorem 1. Let the function f ∈ A. Then f (z) is said to be in the class
RI(d, α, β) if and only if
∞
X
(1 + α(j − 1))(j − 1 + β|d|)·
j=2
n
1 + λ (j − 1) + l
(n + j − 1)!
γ
+ (1 − γ)
aj ≤ β|d|,
l+1
n! (j − 1)!
(2)
where d ∈ C − {0}, 0 ≤ α ≤ 1, 0 < β ≤ 1, z ∈ U .
Proof. Let f (z) ∈ RI(d, α, β). Assume that inequality (2) holds true. Then
we find that
γ
γ
z(RIn,λ,l
f (z))0 + αz 2 (RIn,λ,l
f (z))00
|
− 1|
γ
γ
(1 − α)RIn,λ,l
f (z) + αz(RIn,λ,l
f (z))0
n
o
n P∞
1+λ(j−1)+l
(n+j−1)!
+
(1
−
γ)
aj z j
(1
+
α(j
−
1))(j
−
1)
γ
j=2
l+1
n!(j−1)!
n n
o
|
=|
P∞
+ (1 − γ) (n+j−1)!
aj z j
z + j=2 (1 + α(j − 1)) γ 1+λ(j−1)+l
l+1
n!(j−1)!
n n
o
P∞
1+λ(j−1)+l
+ (1 − γ) (n+j−1)!
aj |z|j−1
j=2 (1 + α(j − 1))(j − 1) γ
l+1
n!(j−1)!
n
o
≤
n
P∞
1 − j=2 (1 + α(j − 1)) γ 1+λ(j−1)+l
+ (1 − γ) (n+j−1)!
aj |z|j−1
l+1
n!(j−1)!
< β|d|.
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Choosing values of z on real axis and letting z → 1− , we have
n
∞
X
1 + λ (j − 1) + l
(n + j − 1)!
(1 + α(j − 1))(j − 1 + β|d|) γ
+ (1 − γ)
aj
l+1
n! (j − 1)!
j=2
≤ β|d|.
Conversely, assume that f (z) ∈ RI(d, α, β), then we get the following inequality
γ
γ
z(RIn,λ,l
f (z))0 + αz 2 (RIn,λ,l
f (z))00
− 1|} > −β|d|
Re{
γ
γ
(1 − α)RIn,λ,l f (z) + αz(RIn,λ,l f (z))0
∞
P
z+
j=2
Re{
n 1+λ(j−1)+l n
j(1 + α(j − 1)) γ
+ (1 − γ)
l+1
∞
P
z+
n 1+λ(j−1)+l n
(1 + α(j − 1)) γ
+ (1 − γ)
l+1
j=2
β|d|z +
∞
P
(n+j−1)!
n!(j−1)!
(n+j−1)!
n!(j−1)!
o
o
aj z j
− 1 + β|d|} > 0
aj z j
n 1+λ(j−1)+l n
(1 + α(j − 1))(j − 1 + β|d|) γ
+ (1 − γ)
l+1
j=2
Re
n ∞
P
1+λ(j−1)+l n
z+
(1 + α(j − 1)) γ
+ (1 − γ)
l+1
j=2
(n+j−1)!
n!(j−1)!
(n+j−1)!
n!(j−1)!
o
o
aj z j
> 0.
aj z j
iθ
Since Re(−e ) ≥ −|eiθ | = −1, the above inequality reduces to
β|d|r −
∞
P
n 1+λ(j−1)+l n
(1 + α(j − 1))(j − 1 + β|d|) γ
+ (1 − γ)
l+1
j=2
r−
∞
P
n 1+λ(j−1)+l n
(1 + α(j − 1)) γ
+ (1 − γ)
l+1
j=2
(n+j−1)!
n!(j−1)!
(n+j−1)!
n!(j−1)!
o
o
aj r j
> 0.
aj r j
Letting r → 1− and by the mean value theorem we have desired inequality
(2).
This completes the proof of Theorem 1
Corollary 1. Let the function f ∈ A be in the class RI(d, α, β). Then
aj ≤
3
β|d|
n n
(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ)
l+1
(n+j−1)!
n!(j−1)!
o,
j ≥ 2.
Distortion Theorems
Theorem 2. Let the function f ∈ A be in the class RI(d, α, β). Then for
|z| = r < 1, we have
r−
β|d|
h n
i r2 ≤ |f (z)|
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
PROPERTIES OA SUBCLASS OF UNIVALENT FUNCTIONS DEFINED
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≤r+
13
β|d|
h n
i r2 .
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
The result is sharp for the function f (z) given by
f (z) = z +
β|d|
h n
i z2.
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
Proof. Given that f (z) ∈ RI(d, α, β), from the equation (2) and since
n
1+λ+l
+ (1 − γ) (n + 1)
(1 + α)(1 + β|d|) γ
l+1
is non decreasing and positive for j ≥ 2,then we have
n
X
∞
1+λ+l
aj ≤
+ (1 − γ) (n + 1)
(1 + α)(1 + β|d|) γ
l+1
j=2
n
∞
X
1 + λ (j − 1) + l
(n + j − 1)!
(1 + α(j − 1))(j − 1 + β|d|) γ
+ (1 − γ)
aj
l+1
n! (j − 1)!
j=2
≤ β|d|,
which is equivalent to,
∞
X
aj ≤
j=2
β|d|
h n
i.
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
Using (3), we obtain
f (z) = z +
∞
X
aj z j
j=2
|f (z)| ≤ |z| +
∞
X
aj |z|j ≤ r +
j=2
≤r+
∞
X
j=2
aj rj ≤ r + r2
∞
X
aj
j=2
β|d|
h n
i r2 .
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
Similarly,
|f (z)| ≥ r2 −
β|d|
h n
i r2 .
1+λ+l
+ (1 − γ) (n + 1)
(1 + α)(1 + β|d|) γ l+1
This completes the proof of Theorem 2.
(3)
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Theorem 3. Let the function f ∈ A be in the class RI(d, α, β). Then for
|z| = r < 1, we have
−
2β|d|
h n
i r ≤ |f 0 (z)|
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
≤
2β|d|
h n
i r.
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
The result is sharp for the function f (z) given by
f (z) = z +
β|d|
n
i z2.
h 1+λ+l
+ (1 − γ) (n + 1)
(1 + α)(1 + β|d|) γ l+1
Proof. From (3)
f 0 (z) = 1 +
∞
X
jaj z j−1
j=2
|f 0 (z)| ≤ 1 −
∞
X
jaj |z|j−1 ≤ 1 +
j=2
≤1+
∞
X
jaj rj−1
j=2
2β|d|
h n
i r.
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
Similarly,
|f 0 (z)| ≥ 1 −
2β|d|
h i r.
n
(1 + α)(1 + β|d|) γ 1+λ+l
+ (1 − γ) (n + 1)
l+1
This completes the proof of Theorem 3.
4
Closure Theorems
Theorem 4. Let the functions fk , k = 1, 2, ..., m, defined by
fk (z) = z +
∞
X
aj,k z j ,
aj,k ≥ 0,
j=2
be in the class RI(d, α, β). Then the function h(z) defined by
h(z) =
m
X
k=1
µk fk (z),
µk ≥ 0,
(4)
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15
is also in the class RI(d, α, β), where
m
X
µk = 1.
k=1
Proof. We can write
h(z) =
m
X
µm z +
m X
∞
X
µk aj,k z j = z +
k=1 j=2
k=1
∞ X
m
X
µk aj,k z j .
j=2 k=1
Furthermore, since the functions fk (z), k = 1, 2, ..., m, are in the class
RI(d, α, β), then from Theorem 1 we have
∞
X
(1 + α(j − 1))(j − 1 + β|d|)·
j=2
n
(n + j − 1)!
1 + λ (j − 1) + l
+ (1 − γ)
aj,k ≤ β|d|.
γ
l+1
n! (j − 1)!
Thus it is enough to prove that
∞
X
(1 + α(j − 1))(j − 1 + β|d|)·
j=2
!
n
X
m
1 + λ (j − 1) + l
(n + j − 1)!
γ
+ (1 − γ)
µk aj,k =
l+1
n! (j − 1)!
k=1
m
X
k=1
µk
∞
X
(1 + α(j − 1))(j − 1 + β|d|)·
j=2
n
1 + λ (j − 1) + l
(n + j − 1)!
γ
+ (1 − γ)
aj,k
l+1
n! (j − 1)!
≤
m
X
µk β|d| = β|d|.
k=1
Hence the proof is complete.
Corollary 2. Let the functions fk , k = 1, 2, defined by (4) be in the class
RI(d, α, β). Then the function h(z) defined by
h(z) = (1 − ζ)f1 (z) + ζf2 (z),
is also in the class RI(d, α, β).
0 ≤ ζ ≤ 1,
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Theorem 5. Let
f1 (z) = z,
and
fj (z) = z+
β|d|
n n
o zj ,
1+λ(j−1)+l
(n+j−1)!
(1 + α(j − 1))(j − 1 + β|d|) γ
+ (1 − γ) n!(j−1)!
l+1
j ≥ 2.
Then the function f (z) is in the class RI(d, α, β) if and only if it can be
expressed in the form:
f (z) = µ1 f1 (z) +
∞
X
µj fj (z),
j=2
where µ1 ≥ 0, µj ≥ 0, j ≥ 2 and µ1 +
P∞
j=2
µj = 1.
Proof. Assume that f (z) can be expressed in the form
f (z) = µ1 f1 (z) +
∞
X
µj fj (z) =
j=2
∞
X
z+
j=2
β|d|
n n
o µj z j .
1+λ(j−1)+l
(n+j−1)!
(1 + α(j − 1))(j − 1 + β|d|) γ
+ (1 − γ) n!(j−1)!
l+1
Thus
n n
o
∞ (1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ) (n+j−1)!
X
l+1
n!(j−1)!
β|d|
j=2
·
β|d|
n n
o µj
(n+j−1)!
(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+
(1
−
γ)
l+1
n!(j−1)!
=
∞
X
µj = 1 − µ1 ≤ 1.
j=2
Hence f (z) ∈ RI(d, α, β).
Conversely, assume that f (z) ∈ RI(d, α, β).
Setting
n
o
n (n+j−1)!
+
(1
−
γ)
(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
l+1
n!(j−1)!
µj =
aj ,
β|d|
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since
µ1 = 1 −
∞
X
17
µj .
j=2
Thus
f (z) = µ1 f1 (z) +
∞
X
µj fj (z).
j=2
Hence the proof is complete.
Corollary 3. The extreme points of the class RI(d, α, β) are the functions
f1 (z) = z,
and
fj (z) = z+
β|d|
n n
o zj ,
(n+j−1)!
(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+
(1
−
γ)
l+1
n!(j−1)!
j ≥ 2.
5
Inclusion and Neighborhood Results
We define the δ- neighborhood of a function f (z) ∈ A by
Nδ (f ) = {g ∈ A : g(z) = z +
∞
X
bj z j and
j=2
∞
X
j|aj − bj | ≤ δ}.
(5)
j=2
In particular, for e(z) = z
Nδ (e) = {g ∈ A : g(z) = z +
∞
X
j=2
bj z j and
∞
X
j|bj | ≤ δ}.
(6)
j=2
Furthermore, a function f ∈ A is said to be in the class RIξ (d, α, β) if there
exists a function h(z) ∈ RI(d, α, β) such that
f (z)
(7)
h(z) − 1 < 1 − ξ, z ∈ U, 0 ≤ ξ < 1.
Theorem 6. If
n
1 + λ (j − 1) + l
(n + j − 1)!
+ (1 − γ)
γ
l+1
n! (j − 1)!
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n
1+λ+l
≥ γ
+ (1 − γ) (n + 1) , j ≥ 2
l+1
and
δ=
2β|d|
h n
i,
(1 + α)(1 + β|d|) γ 1+λ+l
+
(1
−
γ)
(n
+
1)
l+1
then
RI(d, α, β) ⊂ Nδ (e).
Proof. Let f ∈ RI(d, α, β). Then in view of assertion (2) of Theorem 1 and
thecondition n
o
h n
i
n
(n+j−1)!
1+λ+l
+
(1
−
γ)
≥
γ
+
(1
−
γ)
(n
+
1)
for
γ 1+λ(j−1)+l
l+1
n!(j−1)!
l+1
j ≥ 2, we get
n
X
∞
1+λ+l
(1 + α)(1 + β|d|) γ
aj ≤
+ (1 − γ) (n + 1)
l+1
j=2
∞
X
n
1 + λ (j − 1) + l
(n + j − 1)!
(1+α(j−1))(j−1+β|d|) γ
+ (1 − γ)
aj
l+1
n! (j − 1)!
j=2
≤ β|d|,
which implise
∞
X
aj ≤
j=2
β|d|
h i.
n
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
Applying assertion (2) of Theorem 1 in conjunction with (8), we obtain
n
X
∞
1+λ+l
aj ≤ β|d|,
(1 + α)(1 + β|d|) γ
+ (1 − γ) (n + 1)
l+1
j=2
n
X
∞
1+λ+l
2(1 + α)(1 + β|d|) γ
+ (1 − γ) (n + 1)
aj ≤ 2β|d|
l+1
j=2
∞
X
j=2
jaj ≤
2β|d|
h n
i = δ,
1+λ+l
(1 + α)(1 + β|d|) γ l+1
+ (1 − γ) (n + 1)
by virtue of (5), we have f ∈ Nδ (e).
This completes the proof of the Theorem 6.
(8)
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Theorem 7. If h ∈ RI(d, α, β) and
h n
i
1+λ+l
(1
+
α)(1
+
β|d|)
γ
+
(1
−
γ)
(n
+
1)
l+1
δ
h i
ξ =1−
,
2 (1 + α)(1 + β|d|) γ 1+λ+l n + (1 − γ) (n + 1) − β|d|
(9)
l+1
then
Nδ (h) ⊂ RIξ (d, α, β).
Proof. Suppose that f ∈ Nδ (h), we then find from (5) that
∞
X
j|aj − bj | ≤ δ,
j=2
which readily implies the following coefficient inequality
∞
X
|aj − bj | ≤
j=2
δ
.
2
(10)
Next, since h ∈ RI(d, α, β) in the view of (8), we have
∞
X
j=2
bj ≤
β|d|
n
i.
h 1+λ+l
+ (1 − γ) (n + 1)
(1 + α)(1 + β|d|) γ l+1
Using 10) and (11), we get
P∞
f (z)
j=2 |aj − bj |
h(z) − 1 ≤ 1 − P∞ bj ≤
j=2
(11)
δ
2 1−
β|d|
n
(1+α)(1+β|d|)[γ ( 1+λ+l
l+1 ) +(1−γ)(n+1)]
n
i
h 1+λ+l
+
(1
−
γ)
(n
+
1)
(1
+
α)(1
+
β|d|)
γ
l+1
δ
h i
= 1 − ξ,
≤
2 (1 + α)(1 + β|d|) γ 1+λ+l n + (1 − γ) (n + 1) − β|d|
l+1
provided that ξ is given by (9), thus by condition (7), f ∈ RIξ (d, α, β), where
ξ is given by (9).
6
Radii of Starlikeness,
Convexity and Close-to-Convexity
Theorem 8. Let the function f ∈ A be in the class RI(d, α, β). Then f is
univalent starlike of order δ, 0 ≤ δ < 1, in |z| < r1 , where
r1 =
PROPERTIES OA SUBCLASS OF UNIVALENT FUNCTIONS DEFINED
BY USING A MULTIPLIER TRANSFORMATION AND RUSCHEWEYH
DERIVATIVE
inf
j
n n

 (1 − δ)(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ)
l+1
20
(n+j−1)!
n!(j−1)!
o 1
 j−1
β|d|(1 − δ)


The result is sharp for the function f (z) given by
fj (z) = z +
β|d|
n
n + (1 − γ)
(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
l+1
(n+j−1)!
n!(j−1)!
o zj ,
j ≥ 2.
Proof. It suffices to show that
0
zf (z)
≤ 1 − δ,
−
1
f (z)
|z| < r1 .
Since
P∞
0
P∞
j−1
zf (z)
j=2 (j − 1)aj z j−1 j=2 (j − 1)aj |z|
=
P
P
≤
|
.
−
1
∞
f (z)
1 + ∞ aj z k−1 1 − j=2 aj |z|j−1
j=2
To prove the theorem, we must show that
P∞
j−1
j=2 (j − 1)aj |z|
P∞
≤ 1 − δ.
1 − j=2 aj |z|j−1
It is equivalent to
∞
X
(j − δ)aj |z|j−1 ≤ 1 − δ,
j=2
using Theorem 1, we obtain
|z| ≤
n
n 
 (1 − δ)(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ)
l+1
(n+j−1)!
n!(j−1)!
o 1
 j−1
β|d|(1 − δ)

.

Hence the proof is complete.
Theorem 9. Let the function f ∈ A be in the class RI(d, α, β). Then f is
univalent convex of order δ, 0 ≤ δ ≤ 1, in |z| < r2 , where
r2 = inf
j
n n

 (1 − δ)(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ)
l+1

(n+j−1)!
n!(j−1)!
2(j − δ)β|d|
o 1
 k−p
.

The result is sharp for the function f (z) given by
fj (z) = z +
β|d|
n n
(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ)
l+1
j ≥ 2.
(n+j−1)!
n!(j−1)!
o zj ,
(12)
.
PROPERTIES OA SUBCLASS OF UNIVALENT FUNCTIONS DEFINED
BY USING A MULTIPLIER TRANSFORMATION AND RUSCHEWEYH
DERIVATIVE
Proof. It suffices to show that
00 zf (z) f 0 (z)) ≤ 1 − δ,
21
|z| < r2 .
Since
P∞
00 P∞
j−1
zf (z) j=2 j(j − 1)aj z j−1 j=2 j(j − 1)aj |z|
=
P
P
≤
.
∞
f 0 (z) 1 + ∞ jaj z j−1 1 − j=2 jaj |z|j−1
j=2
To prove the theorem, we must show that
P∞
j−1
j=2 j(j − 1)aj |z|
P∞
≤ 1 − δ,
1 − j=2 jaj |z|j−1
∞
X
j(j − δ)aj |z|j−1 ≤ 1 − δ,
j=2
using Theorem 1, we obtain
j−1
|z|
≤
n 1+λ(j−1)+l n
(1 − δ)(1 + α(j − 1))(j − 1 + β|d|) γ
+ (1 − γ)
l+1
(n+j−1)!
n!(j−1)!
o
2(j − δ)β|d|
,
or
|z| ≤
n n

 (1 − δ)(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ)
l+1
(n+j−1)!
n!(j−1)!
o 1
 j−1
2(j − δ)β|d|

.

Hence the proof is complete.
Theorem 10. Let the function f ∈ A be in the class RI(d, α, β). Then f is
univalent close-to-convex of order δ, 0 ≤ δ < 1, in |z| < r3 , where
r3 = inf
j
n n

 (1 − δ)(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ)
l+1
jβ|d|

Proof. It suffices to show that
Then
o 1
 j−1

The result is sharp for the function f (z) given by (12).
|f 0 (z) − 1| ≤ 1 − δ,
(n+j−1)!
n!(j−1)!
|z| < r3 .
X
X
∞
∞
0
j−1 |f (z) − 1| = jaj z ≤
jaj |z|j−1 .
j=2
j=2
.
PROPERTIES OA SUBCLASS OF UNIVALENT FUNCTIONS DEFINED
BY USING A MULTIPLIER TRANSFORMATION AND RUSCHEWEYH
DERIVATIVE
Thus |f 0 (z) − 1| ≤ 1 − δ if
inequality holds true if
j−1
|z|
≤
P∞
jaj
j−1
j=2 1−δ |z|
22
≤ 1. Using Theorem 1, the above
n 1+λ(j−1)+l n
+ (1 − γ)
(1 − δ)(1 + α(j − 1))(j − 1 + β|d|) γ
l+1
(n+j−1)!
n!(j−1)!
o
jβ|d|
or
|z| ≤
n
n 
 (1 − δ)(1 + α(j − 1))(j − 1 + β|d|) γ 1+λ(j−1)+l
+ (1 − γ)
l+1

jβ|d|
(n+j−1)!
n!(j−1)!
o 1
 j−1
.

Hence the proof is complete.
References
[1] A. Alb Lupaş, On special differential subordinations using Sălăgean and
Ruscheweyh operators, Mathematical Inequalities and Applications, Volume 12, Issue 4, 2009, 781-790.
[2] A. Alb Lupaş, On a certain subclass of analytic functions defined by
Salagean and Ruscheweyh operators, Journal of Mathematics and Applications, No. 31, 2009, 67-76.
[3] A. Alb Lupaş, A special comprehensive class of analytic functions defined
by multiplier transformation, Journal of Computational Analysis and Applications, Vol. 12, No. 2, 2010, 387-395.
[4] A. Alb Lupaş, A new comprehensive class of analytic functions defined
by multiplier transformation, Mathematical and Computer Modelling 54
(2011) 2355–2362.
[5] A. Alb Lupaş, On special differential subordinations using a generalized
Sălăgean operator and Ruscheweyh derivative, Journal of Computational
Analysis and Applications, Vol. 13, No.1, 2011, 98-107.
[6] A. Alb Lupaş, On a certain subclass of analytic functions defined by a generalized Sălăgean operator and Ruscheweyh derivative, Carpathian Journal of Mathematics, 28 (2012), No. 2, 183-190.
[7] A. Alb Lupaş, On special differential subordinations using multiplier
transformation and Ruscheweyh derivative, Romai Journal, Vol. 6, Nr.
2, 2010, p. 1-14.
PROPERTIES OA SUBCLASS OF UNIVALENT FUNCTIONS DEFINED
BY USING A MULTIPLIER TRANSFORMATION AND RUSCHEWEYH
DERIVATIVE
23
[8] A. Alb Lupaş, D. Breaz, On special differential superordinations using
Sălăgean and Ruscheweyh operators, Geometric Function Theory and Applications’ 2010 (Proc. of International Symposium, Sofia, 27-31 August
2010), 98-103.
[9] A. Alb Lupaş, Certain special differential superordinations using multiplier transformation, International Journal of Open Problems in Complex
Analysis, Vol. 3, No. 2, July, 2011, 50-60.
[10] A. Alb Lupaş, On special differential superordinations using multiplier
transformation, Journal of Computational Analysis and Applications,
Vol. 13, No.1, 2011, 121-126.
[11] A. Alb Lupaş, On special differential superordinations using a
generalized Sălăgean operator and Ruscheweyh derivative, Computers and Mathematics with Applications 61, 2011, 1048-1058,
doi:10.1016/j.camwa.2010.12.055.
[12] A. Alb Lupaş, Certain special differential superordinations using a generalized Sălăgean operator and Ruscheweyh derivative, Analele Universitatii
Oradea, Fasc. Matematica, Tom XVIII, 2011, 167-178.
[13] A. Alb Lupaş, On special differential superordinations using multiplier
transformation and Ruscheweyh derivative, International Journal of Research and Reviews in Applied Sciences 9 (2), November 2011, 211-222.
[14] A. Alb Lupaş, Certain special differential superordinations using multiplier transformation and Ruscheweyh derivative, Journal of Computational Analysis and Applications, Vol. 13, No.1, 2011, 108-115.
[15] A. Alb Lupaş, Some differential subordinations using Ruscheweyh derivative and Sălăgean operator, Advances in Difference Equations.2013,
2013:150., DOI: 10.1186/1687-1847-2013-150.
[16] F.M. Al-Oboudi, On univalent functions defined by a generalized Sălăgean
operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436.
[17] L. Andrei, Differential subordinations using Ruscheweyh derivative and
generalized Sălăgean operator, Advances in Difference Equation, 2013,
2013:252, DOI: 10.1186/1687-1847-2013-252.
[18] L. Andrei, V. Ionescu, Some differential superordinations using
Ruscheweyh derivative and generalized Sălăgean operator, Journal of
Computational Analysis and Applications, Vol. 17, No. 3, 2014, 437-444
PROPERTIES OA SUBCLASS OF UNIVALENT FUNCTIONS DEFINED
BY USING A MULTIPLIER TRANSFORMATION AND RUSCHEWEYH
DERIVATIVE
24
[19] A.R. Juma, H. Zirar, Properties on a subclass of p-valent functions defined
by new operator Vpλ , Analele Univ. Oradea, Fasc. Math, Tom XXI (2014),
Issue No. 1, 73–82.
[20] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math.
Soc., 49(1975), 109-115.
[21] G. St. Sălăgean, Subclasses of univalent functions, Lecture Notes in
Math., Springer Verlag, Berlin, 1013 (1983), 362-372.
Alina ALB LUPAŞ,
Department of Mathematics and Computer Science,
University of Oradea,
1 Universitatii Street, 410087, Oradea, Romania.
Email: dalb@uoradea.ro