SOME PROPERTIES OF ANALYTIC FUNCTIONS
DEFINED BY A LINEAR OPERATOR
Some Properties of
Analytic Functions
B.A. FRASIN
Department of Mathematics
Al al-Bayt University
P.O. Box: 130095 Mafraq, Jordan
EMail: bafrasin@yahoo.com
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
Title Page
Contents
Received:
29 August, 2006
Accepted:
03 April, 2007
Communicated by:
A. Sofo
2000 AMS Sub. Class.:
30C45.
Key words:
Analytic functions, Hadamard product, Linear operator.
Abstract:
The object of the present paper is to derive some properties of analytic functions
defined by the Carlson - Shaffer linear operator L(a, c)f (z) .
Acknowledgements:
The author would like to thank the referee for his valuable suggestions.
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Contents
1
Introduction and Definitions
3
2
Main Results
6
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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1.
Introduction and Definitions
Let A denote the class of functions of the form:
∞
X
(1.1)
f (z) = z +
an z n ,
n=2
which are analytic in the open unit disc U = {z : |z| < 1} . For two functions f (z)
and g(z) given by
(1.2)
f (z) = z +
∞
X
an z n
and
g(z) = z +
n=2
∞
X
bn z n ,
(f ∗ g)(z) := z +
∞
X
Contents
n
an b n z .
Define the function φ(a, c; z) by
φ(a, c; z) :=
∞
X
(a)n
n=0
(c)n
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z
n+1
,
−
(a ∈ R; c ∈ R\Z−
0 ; Z0 := {0, −1, −2, . . .}, z ∈ U),
where (λ)n is the Pochhammer symbol given in terms of Gamma functions,
Γ(λ + n)
Γ(λ)
1,
n = 0,
=
λ(λ + 1)(λ + 2) . . . (λ + n − 1), n ∈ N : {1, 2, . . .}.
(λ)n :=
vol. 8, iss. 2, art. 53, 2007
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n=2
(1.4)
B.A. Frasin
n=2
their Hadamard product (or convolution) is defined by
(1.3)
Some Properties of
Analytic Functions
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Corresponding to the function φ(a, c; z), Carlson and Shaffer [1] introduced a
linear operator L(a, c) : A → A by
L(a, c)f (z) := φ(a, c; z) ∗ f (z),
(1.5)
or, equivalently, by
(1.6)
L(a, c)f (z) := z +
∞
X
(a)n
n=1
(c)n
an+1 z n+1
(z ∈ U).
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
It follows from (1.6) that
(1.7)
Some Properties of
Analytic Functions
z(L(a, c)f (z))0 = aL(a + 1, c)f (z) − (a − 1)L(a, c)f (z),
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and
Contents
L(1, 1)f (z) = f (z),
0
L(2, 1)f (z) = zf (z),
1
L(3, 1)f (z) = zf 0 (z) + z 2 f 00 (z).
2
Many properties of analytic functions defined by the Carlson-Shaffer linear operator were studied by (among others) Owa and Srivastava [10], Ding [5], Kim and
Lee [6], Ravichandran et al. ([8, 9]), Shanmugam et al. [7] and Frasin ([2, 3]).
In this paper we shall derive some properties of analytic functions defined by the
linear operator L(a, c)f (z).
In order to prove our main results, we recall the following lemma:
Lemma 1.1 ([4]). Let Φ(u, v) be a complex valued function,
Φ : D → C,
(D ⊂ C2 ; C is the complex plane),
and let u = u1 + iu2 and v = v1 + iv2 . Suppose that the function Φ(u, v) satisfies:
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(i) Φ(u, v) is continuous in D;
(ii) (1, 0) ∈ D and Re(Φ(1, 0)) > 0;
(iii) Re(Φ(iu2 , v1 )) ≤ 0 for all (iu2 , v1 ) ∈ D and such that v1 ≤ −(1 + u22 )/2.
Let p(z) = 1 + p1 z + p2 z 2 + · · · be regular in U such that (p(z), zp0 (z)) ∈ D for
all z ∈ U. If Re(Φ(p(z), zp0 (z))) > 0 (z ∈ U), then Re(p(z)) > 0 (z ∈ U).
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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2.
Main Results
Theorem 2.1. Let α ∈ C, β ∈ C, (Re(β) ≥ 0), (α − β) ∈ R, and suppose that
L(a, c)f (z)
(a + 1)L(a + 2, c)f (z)
(2.1)
Re
α−β
+ aα > γ
L(a + 1, c)f (z)
L(a + 1, c)f (z)
for some γ(γ < (α − β)(a + 1)) and 2(α − β) + Re(β) 6= 0, then
2γ − 2a(α − β) + Re(β)
L(a, c)f (z)
(2.2)
Re
>
(z ∈ U).
L(a + 1, c)f (z)
2(α − β) + Re(β)
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
Proof. Define the function p(z) by
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L(a, c)f (z)
:= δ + (1 − δ)p(z),
L(a + 1, c)f (z)
(2.3)
Contents
where
δ=
(2.4)
2γ − 2a(α − β) + Re(β)
< 1.
2(α − β) + Re(β)
(2.5)
0
0
z(L(a, c)f (z))
z(L(a + 1, c)f (z))
(1 − δ)zp (z)
−
=
L(a, c)f (z)
L(a + 1, c)f (z)
δ + (1 − δ)p(z)
by making use of the familiar identity (1.7) in (2.5), we get
(2.6)
a
(1 − δ)zp0 (z)
(a + 1)L(a + 2, c)f (z)
=1+
−
L(a + 1, c)f (z)
δ + (1 − δ)p(z) δ + (1 − δ)p(z)
or, equivalently,
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Then p(z) = 1 + b1 z + b2 z + · · · is regular in U. It follows from (2.3) that
0
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L(a, c)f (z)
(a + 1)L(a + 2, c)f (z)
(2.7)
α−β
+ aα
L(a + 1, c)f (z)
L(a + 1, c)f (z)
= (δ + a)(α − β) + (1 − δ)(α − β)p(z) + β(1 − δ)zp0 (z).
Therefore, we have
L(a, c)f (z)
(a + 1)L(a + 2, c)f (z)
α−β
+ aα − γ
(2.8) Re
L(a + 1, c)f (z)
L(a + 1, c)f (z)
= Re {(δ + a)(α − β) − γ + (1 − δ)(α − β)p(z) + β(1 − δ)zp0 (z)} > 0.
If we define the function Φ(u, v) by
(2.9)
Φ(u, v) = (δ + a)(α − β) − γ + (1 − δ)(α − β)u + β(1 − δ)v
with u = u1 + iu2 and v = v1 + iv2 , then
(i) Φ(u, v) is continuous in D = C2 ;
(ii) (1, 0) ∈ D and Re(Φ(1, 0)) = (α − β)(a + 1) − γ > 0;
(iii) For all (iu2 , v1 ) ∈ D and such that v1 ≤ −(1 + u22 )/2,
Re(Φ(iu2 , v1 )) = Re{(δ + a)(α − β) − γ + β(1 − δ)v1 }
(1 − δ)(1 + u22 ) Re(β)
≤ (δ + a)(α − β) − γ −
2
(1 − δ)u22 Re(β)
=−
≤ 0.
2
Therefore, the function Φ(u, v) satisfies the conditions in Lemma 1.1. Thus we have
Re(p(z)) > 0 (z ∈ U), that is
2γ − 2a(α − β) + Re(β)
L(a, c)f (z)
>
.
Re
L(a + 1, c)f (z)
2(α − β) + Re(β)
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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−
Letting β = −α in Theorem 2.1, we have
Corollary 2.2. Let α ∈ C, (Re(α) < 0), and suppose that
L(a, c)f (z)
− (a + 1)L(a + 2, c)f (z)
(2.10)
Re
α+α
+ aα > γ
L(a + 1, c)f (z)
L(a + 1, c)f (z)
for some γ (γ < 2(a + 1) Re(α)), then
L(a, c)f (z)
2γ − (4a + 1) Re(α)
(2.11)
Re
>
L(a + 1, c)f (z)
3 Re(α)
Some Properties of
Analytic Functions
B.A. Frasin
(z ∈ U).
vol. 8, iss. 2, art. 53, 2007
Letting a = c = 1 in Theorem 2.1, we have
Corollary 2.3. Let α, β ∈ C, (Re(β) ≥ 0), (α − β) ∈ R, and suppose that
f (z)
zf 00 (z)
(2.12)
Re
α − 2β 1 +
+α >γ
zf 0 (z)
2f 0 (z)
for some γ(γ < 2(α − β)) and 2(α − β) + Re(β) 6= 0, then
f (z)
2γ − 2(α − β) + Re(β)
(2.13)
Re
>
(z ∈ U).
0
zf (z)
2(α − β) + Re(β)
Theorem 2.4. Let α ∈ C, β ∈ C, (Re(β) ≥ 0), (α − β) ∈ R, and suppose that
(a + 1)L(a + 2, c)f (z)
L(a, c)f (z)
α−β
+ aα < γ
(2.14)
Re
L(a + 1, c)f (z)
L(a + 1, c)f (z)
for some γ(γ > (α − β)(a + 1)) and 2(α − β) + Re(β) 6= 0, then
L(a, c)f (z)
2γ − 2a(α − β) + Re(β)
(2.15)
Re
<
L(a + 1, c)f (z)
2(α − β) + Re(β)
(z ∈ U).
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Proof. Define the function p(z) by
(2.16)
L(a, c)f (z)
:= δ + (1 − δ)p(z),
L(a + 1, c)f (z)
where
(2.17)
δ=
2γ − 2a(α − β) + Re(β)
> 1.
2(α − β) + Re(β)
Some Properties of
Analytic Functions
B.A. Frasin
Then p(z) = 1 + b1 z + b2 z + · · · is regular in U. It follows from (2.16) that
L(a, c)f (z)
(a + 1)L(a + 2, c)f (z)
(2.18) Re γ −
α−β
− aα
L(a + 1, c)f (z)
L(a + 1, c)f (z)
= Re {γ − (δ + a)(α − β) − (1 − δ)(α − β)p(z) − β(1 − δ)zp0 (z)} > 0.
If we define the function Φ(u, v) by
(2.19)
Φ(u, v) = γ − (δ + a)(α − β) − (1 − δ)(α − β)u − β(1 − δ)v
with u = u1 + iu2 and v = v1 + iv2 , then
(i) Φ(u, v) is continuous in D = C2 ;
(ii) (1, 0) ∈ D and Re(Φ(1, 0)) = γ − (α − β)(a + 1) > 0;
(iii) For all (iu2 , v1 ) ∈ D and such that v1 ≤ −(1 + u22 )/2,
Re(Φ(iu2 , v1 )) = Re{γ − (δ + a)(α − β) − β(1 − δ)v1 }
(1 − δ)(1 + u22 ) Re(β)
≤ γ − (δ + a)(α − β) +
2
(1 − δ)u22 Re(β)
≤ 0,
=
2
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applying Lemma 1.1, we have Re(p(z)) > 0 (z ∈ U), that is
L(a, c)f (z)
2γ − 2a(α − β) + Re(β)
Re
<
.
L(a + 1, c)f (z)
2(α − β) + Re(β)
Theorem 2.5. Let a > −1, µ > 0, α ∈ C, β ∈ C, (Re(β) ≥ 0), (α + β) ∈ R, and
suppose that
(
µ )
L(a + 1, c)f (z)
L(a + 2, c)f (z)
(2.20)
Re
α+β
>γ
z
L(a + 1, c)f (z)
for some γ(γ < α + β) and 2µ(α + β)(a + 1) + Re(β) 6= 0, then
(
µ )
L(a + 1, c)f (z)
2µγ(a + 1) + Re(β)
(2.21) Re
>
z
2µ(α + β)(a + 1) + Re(β)
vol. 8, iss. 2, art. 53, 2007
Contents
(z ∈ U).
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2µγ(a + 1) + Re(β)
δ=
> 1.
2µ(α + β)(a + 1) + Re(β)
Then p(z) = 1 + b1 z + b2 z + · · · is regular in U. Also, by a simple computation and
by making use of the familiar identity (1.7) we find from (2.22) that
(2.23)
B.A. Frasin
Title Page
Proof. Define the function p(z) by
µ
L(a + 1, c)f (z)
:= δ + (1 − δ)p(z),
(2.22)
z
where
Some Properties of
Analytic Functions
βL(a + 2, c)f (z)
β(1 − δ)zp0 (z)
=
+ β,
L(a + 1, c)f (z)
µ(a + 1)(δ + (1 − δ)p(z))
Close
and by using (2.22) and (2.23), we get
µ L(a + 1, c)f (z)
βL(a + 2, c)f (z)
(2.24)
+α
z
L(a + 1, c)f (z)
β(1 − δ)zp0 (z)
=
+ (α + β)(δ + (1 − δ)p(z)).
µ(a + 1)
Therefore, we have
(
)
µ L(a + 1, c)f (z)
βL(a + 2, c)f (z)
(2.25) Re
+α −γ
z
L(a + 1, c)f (z)
β(1 − δ) 0
zp (z) > 0.
= Re (α + β)δ − γ + (1 − δ)(α + β)p(z) +
µ(a + 1)
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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If we define the function Φ(u, v) by
(2.26)
Φ(u, v) = (α + β)δ − γ + (1 − δ)(α + β)u +
β(1 − δ)
v
µ(a + 1)
with u = u1 + iu2 and v = v1 + iv2 , then
(i) Φ(u, v) is continuous in D = C2 ;
(ii) (1, 0) ∈ D and Re(Φ(1, 0)) = (α + β) − γ > 0;
(iii) For all (iu2 , v1 ) ∈ D and such that v1 ≤ −(1 + u22 )/2,
β(1 − δ)
Re(Φ(iu2 , v1 )) = Re{(α + β)δ − γ +
v1 }
µ(a + 1)
(1 − δ)(1 + u22 ) Re(β)
≤ (α + β)δ − γ −
2µ(a + 1)
2
(1 − δ)u2 Re(β)
=−
≤ 0.
2µ(a + 1)
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Therefore, the function Φ(u, v) satisfies the conditions in Lemma 1.1. Thus we
have Re(p(z)) > 0 (z ∈ U), that is,
(
µ )
L(a + 1, c)f (z)
2µγ(a + 1) + Re(β)
Re
>
(z ∈ U).
z
2µ(α + β)(a + 1) + Re(β)
Some Properties of
Analytic Functions
−
Letting α = β in Theorem 2.5, we have
B.A. Frasin
Corollary 2.6. Let a > −1, µ > 0, β ∈ C, (Re(β) > 0), and suppose that
µ L(a + 1, c)f (z)
L(a + 2, c)f (z)
(2.27)
Re
α+β
>γ
z
L(a + 1, c)f (z)
for some γ(γ < 2 Re(β)), then
(
µ )
L(a + 1, c)f (z)
2µγ(a + 1) + Re(β)
(2.28)
Re
>
z
(4µ(a + 1) + 1) Re(β)
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(z ∈ U).
Letting a = c = 1 in Theorem 2.5, we have,
Corollary 2.7. Let µ > 0, α ∈ C, β ∈ C, (Re(β) ≥ 0), (α + β) ∈ R, and suppose
that
zf 00 (z)
µ
0
(2.29)
Re (f (z)) α + β 1 +
>γ
2f 0 (z)
for some γ(γ < α + β) and 4µ(α + β) + Re(β) 6= 0, then
n
o
µ
4µγ + Re(β)
0
(2.30)
Re (f (z)) >
(z ∈ U).
4µ(α + β) + Re(β)
vol. 8, iss. 2, art. 53, 2007
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Employing the same manner as in the proofs of Theorems 2.4 and 2.5, we have:
Theorem 2.8. Let a > −1, µ > 0, α ∈ C, β ∈ C, (Re(β) ≥ 0), (α + β) ∈ R, and
suppose that
(
µ )
L(a + 1, c)f (z)
L(a + 2, c)f (z)
(2.31)
Re
α+β
<γ
z
L(a + 1, c)f (z)
for some γ(γ > α + β) and 2µ(α + β)(a + 1) + Re(β) 6= 0, then
(
µ )
2µγ(a + 1) + Re(β)
L(a + 1, c)f (z)
(2.32) Re
<
z
2µ(α + β)(a + 1) + Re(β)
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
(z ∈ U).
Theorem 2.9. Let λ > 0, α ∈ C, β ∈ C, (Re(β) ≥ 0), (α + β) ∈ R, and suppose
that
(
λ
L(a + 1, c)f (z)
(2.33) Re
L(a, c)f (z)
L(a + 2, c)f (z)
L(a + 1, c)f (z)
× λβ(a + 1)
− aλβ
+ λα
>γ
L(a + 1, c)f (z)
L(a, c)f (z)
for some γ(γ < λ(α + β)) and 2λ(α + β) + Re(β) 6= 0, then
λ
L(a + 1, c)f (z)
2γ + Re(β)
>
(2.34)
Re
L(a, c)f (z)
2λ(α + β) + Re(β)
Proof. Define the function p(z) by
λ
L(a + 1, c)f (z)
(2.35)
:= δ + (1 − δ)p(z),
L(a, c)f (z)
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(z ∈ U).
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where
δ=
(2.36)
2γ + Re(β)
< 1.
2λ(α + β) + Re(β)
Then p(z) = 1 + b1 z + b2 z + · · · is regular in U . Also, by a simple computation
and by making use of the familiar identity (1.7), we find from (2.35) that
(2.37)
λ
L(a + 1, c)f (z)
L(a, c)f (z)
L(a + 2, c)f (z)
L(a + 1, c)f (z)
× λβ(a + 1)
− aλβ
+ λα
L(a + 1, c)f (z)
L(a, c)f (z)
= λ(α + β)(δ + (1 − δ)p(z)) + β(1 − δ)zp0 (z)
Therefore, we have
(
λ
L(a + 1, c)f (z)
(2.38) Re
L(a, c)f (z)
L(a + 2, c)f (z)
L(a + 1, c)f (z)
× λβ(a + 1)
− aλβ
+ λα − γ
L(a + 1, c)f (z)
L(a, c)f (z)
= Re {λ(α + β)(δ + (1 − δ)p(z)) + β(1 − δ)zp0 (z) − γ} > 0.
If we define the function Φ(u, v) by
(2.39)
Φ(u, v) = λδ(α + β) − γ + λ(1 − δ)(α + β)u + β(1 − δ)v
with u = u1 + iu2 and v = v1 + iv2 , then
(i) Φ(u, v) is continuous in D = C2 ;
(ii) (1, 0) ∈ D and Re(Φ(1, 0)) = λ(α + β) − γ > 0;
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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(iii) For all (iu2 , v1 ) ∈ D and such that v1 ≤ −(1 + u22 )/2,
Re(Φ(iu2 , v1 )) = Re{λδ(α + β) − γ + β(1 − δ)v1 }
(1 − δ)(1 + u22 )
≤ λδ(α + β) − γ −
Re(β)
2
(1 − δ)u22 Re(β)
=−
≤ 0.
2
Therefore, the function Φ(u, v) satisfies the conditions in Lemma 1.1. Thus we have
λ
2γ + Re(β)
L(a + 1, c)f (z)
>
Re
(z ∈ U).
L(a, c)f (z)
2λ(α + β) + Re(β)
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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−
Letting α = β in Theorem 2.9, we have:
Corollary 2.10. Let λ > 0, α ∈ C, β ∈ C, (Re(β) > 0), (α + β) ∈ R, and suppose
that
(
λ
L(a + 1, c)f (z)
(2.40) Re
L(a, c)f (z)
−
L(a + 2, c)f (z)
L(a + 1, c)f (z)
× λβ(a + 1)
− aλβ
+ λβ
>γ
L(a + 1, c)f (z)
L(a, c)f (z)
for some γ(γ < λ(α + β)), then
λ
2γ + Re(β)
L(a + 1, c)f (z)
(2.41)
Re
>
L(a, c)f (z)
(4λ + 1) Re(β)
(z ∈ U).
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Letting a = c = λ = 1 in Theorem 2.9, we have:
Corollary 2.11. Let α ∈ C, β ∈ C, (Re(β) ≥ 0), (α + β) ∈ R, and suppose that:
0 00
zf (z)
f (z) f 0 (z)
(2.42)
Re
(2β + α) + βz
−
>γ
f (z)
f 0 (z)
f (z)
for some γ(γ < α + β) and 2(α + β) + Re(β) 6= 0, then f (z) is starlike of order δ,
2γ+Re(β)
where δ = 2(α+β)+Re(β)
.
Employing the same manner as in the proofs of Theorems 2.4 and 2.9, we have:
Theorem 2.12. Let λ > 0, α ∈ C, β ∈ C, (Re(β) ≥ 0), (α + β) ∈ R, and suppose
that:
(
λ
L(a + 1, c)f (z)
(2.43) Re
L(a, c)f (z)
L(a + 2, c)f (z)
L(a + 1, c)f (z)
× λβ(a + 1)
− aλβ
+ λα
<γ
L(a + 1, c)f (z)
L(a, c)f (z)
for some γ(γ > λ(α + β)) and 2λ(α + β) + Re(β) 6= 0, then
λ
L(a + 1, c)f (z)
2γ + Re(β)
(2.44)
Re
<
L(a, c)f (z)
2λ(α + β) + Re(β)
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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(z ∈ U).
Theorem 2.13. Let a > −1, α ∈ C, β ∈ C, (Re(β) ≥ 0), (α−β) ∈ R, and suppose
that
z
L(a + 2, c)f (z)
(2.45)
Re
(a + 1)α − β
>γ
L(a + 1, c)f (z)
L(a + 1, c)f (z)
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for some γ(γ < (α − β)(a + 1)), then
z
2γ + Re(β)
>
(2.46)
Re
L(a + 1, c)f (z)
2(a + 1)(α − β) + Re(β)
(z ∈ U).
Proof. Define the function p(z) by
(2.47)
z
:= δ + (1 − δ)p(z),
L(a + 1, c)f (z)
Some Properties of
Analytic Functions
B.A. Frasin
where
(2.48)
vol. 8, iss. 2, art. 53, 2007
δ=
2γ + Re(β)
> 1.
2(a + 1)(α − β) + Re(β)
Then p(z) = 1 + b1 z + b2 z + · · · is regular in U. Also, by a simple computation and
by making use of the familiar identity (1.7), we find from (2.47) that
z
L(a + 2, c)f (z)
(a + 1)α − β
(2.49)
L(a + 1, c)f (z)
L(a + 1, c)f (z)
= (a + 1)(α − β)(δ + (1 − δ)p(z)) + β(1 − δ)zp0 (z).
Therefore, we have
L(a + 2, c)f (z)
z
(a + 1)α − β
−γ
(2.50) Re
L(a + 1, c)f (z)
L(a + 1, c)f (z)
= Re{(a + 1)(α − β)(δ + (1 − δ)p(z)) + β(1 − δ)zp0 (z) − γ} > 0.
If we define the function Φ(u, v) by
(2.51)
Φ(u, v) = δ(a + 1)(α − β) − γ + (a + 1)(α − β)(1 − δ)u + β(1 − δ)v
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with u = u1 + iu2 and v = v1 + iv2 , then
(i) Φ(u, v) is continuous in D = C2 ;
(ii) (1, 0) ∈ D and Re(Φ(1, 0)) = (α − β)(a + 1) − γ > 0;
(iii) For all (iu2 , v1 ) ∈ D and such that v1 ≤ −(1 + u22 )/2,
Re(Φ(iu2 , v1 )) = Re{δ(a + 1)(α − β) − γ + β(1 − δ)v1 }
(1 − δ)(1 + u22 )
≤ δ(a + 1)(α − β) − γ −
Re(β)
2
(1 − δ)u22 Re(β)
=−
≤ 0.
2
Therefore, the function Φ(u, v) satisfies the conditions in Lemma 1.1. Thus we have
z
2γ + Re(β)
Re
>
(z ∈ U).
L(a + 1, c)f (z)
2(a + 1)(α − β) + Re(β)
−
Letting β = −α in Theorem 2.13, we have:
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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Corollary 2.14. Let a > −1, a 6= 0, α ∈ C, (Re(α) < 0), and suppose that
z
− L(a + 2, c)f (z)
(2.52)
Re
(a + 1)α + α
>γ
L(a + 1, c)f (z)
L(a + 1, c)f (z)
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for some γ(γ < 2(a + 1) Re(α)), then
z
2γ − Re(α)
(2.53)
Re
>
L(a + 1, c)f (z)
(4a + 3) Re(α)
Letting a = c = 1 in Theorem 2.13, we have:
(z ∈ U).
Corollary 2.15. Let a > −1, α ∈ C, β ∈ C, (Re(β) ≥ 0), (α − β) ∈ R, and
suppose that
1
zf 00 (z)
(2.54)
Re
2α − β −
β
>γ
f 0 (z)
2f 0 (z)
for some γ(γ < 2(α − β)) and 4(α − β) + Re(β) 6= 0, then
1
2γ + Re(β)
(2.55)
Re
>
(z ∈ U).
f 0 (z)
4(α − β) + Re(β)
Some Properties of
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B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
Employing the same manner as in the proofs of Theorem 2.4 and 2.13, we have:
Theorem 2.16. Let a > −1, α ∈ C, β ∈ C, (Re(β) ≥ 0), (α−β) ∈ R, and suppose
that
z
L(a + 2, c)f (z)
(2.56)
Re
(a + 1)α − β
<γ
L(a + 1, c)f (z)
L(a + 1, c)f (z)
for some γ(γ > (α − β)(a + 1)) and 2(a + 1)(α − β) + Re(β) 6= 0, then
z
2γ + Re(β)
(2.57)
Re
<
(z ∈ U).
L(a + 1, c)f (z)
2(a + 1)(α − β) + Re(β)
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References
[1] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15(4) (1984), 737–745.
[2] B.A. FRASIN, Subordinations results for a class of analytic functions defined
by linear operator, J. Inequal. Pure Appl. Math., 7(4) (2006), Art. 134 [ONLINE: http://jipam.vu.edu.au/article.php?sid=754].
[3] B.A. FRASIN, Subclasses of analytic functions defined by Carlson-Shaffer linear operator, to appear in Tamsui Oxford J. Math.
[4] S. S. MILLER AND P.T. MOCANU, Second order differential inequalities in
the complex plane, J. Math. Anal. Appl., 65 (1978), 289–305.
[5] S. DING, Some properties of a class of analytic functions, J. Math. Anal. Appl.,
195 (1995), 71–81.
[6] Y.C. KIM AND K.S. LEE, Some applications of fractional integral operators
and Ruscheweyh derivatives, J. Math. Anal. Appl., 197 (1996), 505–517.
[7] T.N. SHANMUGAM, V. RAVICHANDRAN AND S. SIVASUBRAMANIAN,
Differential sandwich theorems for some subclasses of analytic functions, Aust.
J. Math. Anal. Appl., 3(1) (2006), 1–11.
[8] V. RAVICHANDRAN, H. SELVERMAN, S.S KUMAR AND K.G. SUBRAMANIAN, On differential subordinations for a class of analytic functions defined by a linear operator, International Journal of Mathematics and Mathematical Sciences, 42 (2004), 2219–2230.
[9] V. RAVICHANDRAN, N. SEENIVASAGAN AND H.M. SRIVASTAVA, Some
inequalities associated with a linear operator defined for a class of multivalent
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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functions, J. Inequal. Pure Appl. Math., 4(4) (2003), Art.70 [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=311].
[10] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39 (1987), 1057–1077.
Some Properties of
Analytic Functions
B.A. Frasin
vol. 8, iss. 2, art. 53, 2007
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