MATEMATIQKI VESNIK
originalni nauqni rad
research paper
62, 1 (2010), 51–61
March 2010
CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS
DEFINED BY A FAMILY OF LINEAR OPERATORS
A. O. Mostafa
Abstract. In this paper, we obtain some applications of first order differential subordination
and superordination results involving Dziok-Srivastava operator and other linear operators for
certain normalized analytic functions in the open unit disc.
1. Introduction
Let H(U ) be the class of analytic functions in the unit disk U = {z ∈ C : |z| <
1} and let H[a, k] be the subclass of H(U ) consisting of functions of the form
f (z) = a + ak z k + ak+1 z k+1 + · · ·
(a ∈ C).
(1.1)
Also, let A be the subclass of H(U ) consisting of functions of the form
f (z) = z +
∞
P
ak z k .
(1.2)
k=2
If f , g ∈ H(U ), we say that f is subordinate to g, written f (z) ≺ g(z) if there
exists a Schwarz function w, which (by definition) is analytic in U with w(0) = 0
and |w(z)| < 1for all z ∈ U, such that f (z) = g(w(z)), z ∈ U . Furthermore, if the
function g is univalent in U , then we have the following equivalence, (cf., e.g. ,[3],
[10]; see also [11]):
f (z) ≺ g(z) ⇔ f (0) = g(0) and f (U ) ⊂ g(U ).
Let p, h ∈ H(U ) and let ϕ(r, s, t; z) : C 3 × U → C. If p and ϕ(p(z), zp0 (z),
z 2 p00 (z); z) are univalent and if p satisfies the second order superordination
h(z) ≺ ϕ(p(z), zp0 (z), z 2 p00 (z); z),
(1.3)
then p is a solution of the differential superordination (1.3). Note that if f is
subordinate to g, then g is superordinate to f . An analytic function q is called a
2010 AMS Subject Classification: 30C45.
Keywords and phrases: Analytic functions; differential subordination; superordination;
sandwich theorems; Dziok-Srivastava operator.
51
52
A. O. Mostafa
subordinant if q(z) ≺ p(z) for all p satisfying (1.3). A univalent subordinant qe that
satisfies q ≺ qe for all subordinants of (1.3) is called the best subordinant. Recently
Miller and Mocanu [12] obtained conditions on the functions h, q and ϕ for which
the following implication holds:
h(z) ≺ ϕ(p(z), zp0 (z), z 2 p00 (z); z) ⇒ q(z) ≺ p(z).
(1.4)
Using the results of Miller and Mocanu [12], Bulboacă considered certain classes of first order differential superordinations as well as superordination-preserving
integral operators [4]. Ali et al. [1], have used the results of Bulboacă to obtain
sufficient conditions for normalized analytic functions to satisfy
q1 (z) ≺
zf 0 (z)
≺ q2 (z),
f (z)
here q1 and q2 are given univalent functions in U . Also, Tuneski [17] obtained a suff 00 (z)f (z)
ficient condition for starlikeness of f in terms of the quantity
. Recently,
(f 0 (z))2
Shanmugam et al. [15] obtained sufficient conditions for the normalized analytic
function f to satisfy
f (z)
q1 (z) ≺
≺ q2 (z)
zf 0 (z)
and
z 2 f 0 (z)
q1 (z) ≺
≺ q2 (z).
{f (z)}2
They also obtained results for functions defined by using Carlson-Shaffer operator.
For complex numbers α1 , α2 , . . . , αq and β1 , β2 , . . . , βs (βj ∈
/ Z0− =
{0, −1, −2, . . . }, j = 1, 2, . . . , s), we define the generalized hypergeometric function q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z) by (see, for example, [16]) the following infinite
series
∞
P
(α1 )k . . . (αq )k k
z
(β
1 )k . . . (βs )k (1)k
k=0
(q ≤ s + 1; s, q ∈ N0 = N ∪ {0}; z ∈ U ),
q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z)
where
½
(d)k =
=
1,
d(d + 1) . . . (d + k − 1),
(1.5)
(k = 0; d ∈ C\{0}),
(k ∈ N ; d ∈ C).
Dziok and Srivastava [8] considered a linear operator Hq,s (α1 , . . . , αq ;
β1 , . . . , βs ) : A → A, defined by the following Hadamard product
Hq,s (α1 , . . . , αq ; β1 , . . . , βs )f (z) = [z q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z)] ∗ f (z),
(q ≤ s + 1; s, q ∈ N0 ; z ∈ U ).
(1.6)
We observe that for a function f of the form (1.2), we have
Hq,s (α1 , . . . , αq ; β1 , . . . , βs )f (z) = z +
∞
P
(α1 )k−1 . . . (αq )k−1
ak z k .
k=2 (β1 )k−1 . . . (βs )k−1 (1)k−1
(1.7)
Subclasses of analytic functions defined by a family of linear operators
53
If, for convenience, we write
Hq,s (α1 ) = Hq,s (α1 , . . . , αq ; β1 , . . . , βs ),
(1.8)
then one can easily verify from the definition (1.7) that
z(Hq,s (α1 )f (z))0 = α1 Hq,s (α1 + 1)f (z) − (α1 − 1)Hq,s (α1 )f (z) (f ∈ A).
(1.9)
It should be remarked that the linear operator Hq,s (α1 )f (z) is a generalization
of many other linear operators considered earlier. In particular, for f ∈ A, we have:
(i) Hq,s (a, 1; c)f (z) = L(a, c)f (z) (a > 0, c > 0), where L(a, c) is the CarlsonShaffer operator (see [5]);
(ii) Hq,s (λ + 1, c; a)f (z) = I λ (a, c)f (z) (a, c ∈ R\Z0− ; λ > −1), where I λ (a, c)
is the Cho-Kwon-Srivastava operator (see [6]);
(iii) Hq,s (η, 1; λ + 1)f (z) = Iλ,η f (z) (λ > −1; η > 0), where Iλ,η is the ChoiSaigo-Srivastava operator (see [7]);
R z η−1
(iv) Hq,s (η + 1, 1; η + 2)f (z) = Fη (f )(z) = η+1
t
f (t)dt (η > −1) where
zη
0
Fη is the Libera operator (see [9]);
(v) Hq,s (δ+1, 1; 1)f (z) = Dδ f (z) (δ > −1), where Dδ f (z) is the δ-Ruscheweyh
derivative of f (z) (see [13]).
In this paper, we obtain sufficient conditions for the normalized analytic function f defined by using Dziok-Srivastava operator to satisfy
µ
¶µ
z
q1 (z) ≺
≺ q2 (z)
Hq,s (α1 )f (z)
and q1 and q2 are given univalent functions in U .
2. Definitions and preliminaries
In order to prove our results, we shall make use of the following known results.
Definition 1. [12] Denote by Q the set of all functions f that are analytic
and injective on U \E(f ), where
E(f ) = { ξ ∈ ∂U : lim f (z) = ∞ },
z→ξ
and are such that f 0 (ξ) 6= 0 for ξ ∈ ∂U \ E(f ).
Lemma 1. [11] Let q be univalent in the unit disk U and θ and ϕ be analytic
in a domain D containing q(U ) with ϕ(w) 6= 0 when w ∈ q(U ). Set
ψ(z) = zq 0 (z)ϕ(q(z)) and h(z) = θ(q(z)) + ψ(z).
Suppose that:
(i) ψ(z) is starlike univalent in U ,
(2.1)
54
A. O. Mostafa
n
(ii) Re
zh0 (z)
ψ(z)
o
> 0 for z ∈ U .
If p is analytic with p(0) = q(0), p(U ) ⊆ D and
θ(p(z)) + zp0 (z)ϕ(p(z)) ≺ θ(q(z)) + zq 0 (z)ϕ(q(z)),
(2.2)
then p(z) ≺ q(z) and q is the best dominant.
Lemma 2. [2] Let q be convex univalent in U and ϑ and φ be analytic in a
domain D containing q(U ). Suppose that:
(i) Re{ϑ0 (q(z))/φ(q(z))} > 0 for z ∈ U ,
(ii) Q(z) = zq 0 (z)φ(q(z)) is starlike univalent in U .
If p(z) ∈ H[q(0), 1]∩Q, with p(U ) ⊆ D, and ϑ(p(z))+zp0 (z)φ(p(z)) is univalent
in U and
ϑ(q(z)) + zq 0 (z)φ(q(z)) ≺ ϑ(p(z)) + zp0 (z)φ(p(z)),
(2.3)
then q(z) ≺ p(z) and q is the best subordinant.
3. Applications to Dziok-Srivastava operator and sandwich theorems
Unless otherwise mentioned, we shall assume in the reminder of this paper
that, γ, ξ, δ ∈ C and β, µ ∈ C ∗ = C \ {0}.
Theorem 1. Let q be analytic univalent in U with q(z) 6= 0. Suppose that
is starlike univalent in U . Let γ, ξ, δ ∈ C; β, µ ∈ C ∗ satisfy:
½
¾
ξ
2δ
zq 0 (z) zq 00 (z)
Re 1 + q(z) + (q(z))2 −
+ 0
> 0,
(3.1)
β
β
q(z)
q (z)
zq 0 (z)
q(z)
and
Ψ(α1 , γ, ξ, δ, β, µ, f ) =
γ + ξ(
z
z
Hq,s (α1 + 1)f (z)
)µ + δ(
)2µ + βµα1 [1 −
].
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
(3.2)
If q satisfies the following subordination:
Ψ(α1 , γ, ξ, δ, β, µ, f ) ≺ γ + ξq(z) + δ(q(z))2 + β
then
(
z
)µ ≺ q(z)
Hq,s (α1 )f (z)
zq 0 (z)
,
q(z)
(µ ∈ C ∗ )
(3.3)
(3.4)
and q is the best dominant.
Proof. Define a function p by
p(z) = (
z
)µ (z ∈ U ; µ ∈ C ∗ ).
Hq,s (α1 )f (z)
(3.5)
55
Subclasses of analytic functions defined by a family of linear operators
Then the function p is analytic in U and p(0) = 1. Therefore, differentiating
(3.5) logarithmically with respect to z and using the identity (1.9) in the resulting
equation, we have
γ + ξ(
z
z
Hq,s (α1 + 1)f (z)
)µ + δ(
)2µ + βµα1 [1 −
]
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
zp0 (z)
= γ + ξp(z) + δ(p(z))2 + β
. (3.6)
p(z)
Using (3.6) and (3.3), we have
γ + ξp(z) + δ(p(z))2 + β
zp0 (z)
zq 0 (z)
≺ γ + ξq(z) + δ(q(z))2 + β
.
p(z)
q(z)
(3.7)
Setting
β
,
w
it can be easily observed that θ is analytic in C, ϕ is analytic in C ∗ and ϕ(w) 6= 0
(w ∈ C ∗ ). Hence, the result now follows by using Lemma 1.
Putting q(z) = (1 + Az)/(1 + Bz) (−1 ≤ B < A ≤ 1) in Theorem 1, we have
the following corollary.
θ(w) = γ + ξw + δw2 and ϕ(w) =
Corollary 1. Let −1 ≤ B < A ≤ 1 and
½
µ
¶
µ
¶2
¾
ξ 1 + Az
2δ 1 + Az
(A + B + 3AB)z
Re 1 +
+
−
>0
β 1 + Bz
β 1 + Bz
(1 + Az)(1 + Bz)
holds. If f (z) ∈ A, and
γ + ξ(
z
z
Hq,s (α1 + 1)f (z)
)µ + δ(
)2µ + βµα1 [1 −
]
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
1 + Az 2
(A − B)z
1 + Az
+ δ(
) +β
,
≺γ+ξ
1 + Bz
1 + Bz
(1 + Az)(1 + Bz)
then
and
µ
z
Hq,s (α1 )f (z)
¶µ
≺
1 + Az
1 + Bz
(µ ∈ C ∗ )
1+Az
1+Bz
is the best dominant.
³
´ν
1+z
Putting q(z) = 1−z
(0 < ν ≤ 1) in Theorem 1, we obtain the following
corollary.
Corollary 2. Assume that (3.1) holds. If f ∈ A, and
γ + ξ(
z
z
Hq,s (α1 + 1)f (z)
)µ + δ(
)2µ + βµα1 [1 −
]
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
µ
¶ν
¶2ν
µ
1+z
1+z
2νz
≺γ+ξ
,
+δ
+β
1−z
1−z
(1 − z)2
56
A. O. Mostafa
then
µ
³
and
1+z
1−z
´ν
z
Hq,s (α1 )f (z)
¶µ
µ
≺
1+z
1−z
¶ν
(µ ∈ C ∗ ; 0 < ν ≤ 1)
is the best dominant.
Taking α1 = a > 0, β1 = c > 0, αj = 1 (j = 2, . . . , s + 1) and βj = 1
(j = 2, . . . , s), in Theorem 1, we have the following corollary which improves the
result obtained by Shanmugam et al. [14, Theorem 3.1].
Corollary 3. Let q be analytic univalent in U with q(z) 6= 0 and condition
0
(z)
is starlike univalent in U and
(3.1) holds. Suppose also that zqq(z)
z
z
L(a + 1, c)f (z)
)µ + δ(
)2µ + βµa[1 −
].
L(a, c)f (z)
L(a, c)f (z)
L(a, c)f (z)
(3.8)
If q satisfies the following subordination:
ζ(γ, ξ, δ, β, µ) = γ + ξ(
ζ(γ, ξ, δ, β, µ) ≺ γ + ξq(z) + δ(q(z))2 + β
then
(
z
)µ ≺ q(z)
L(a, c)f (z)
zq 0 (z)
,
q(z)
(µ ∈ C ∗ )
and q is the best dominant.
Taking α1 = λ + 1, α2 = c, β1 = a (a, c ∈ R \ Z0− ; λ > −1), αj = 1
(j = 3, . . . , s + 1) and βj = 1(j = 2, . . . , s), in Theorem 1, we have
Corollary 4. Let q be analytic univalent in U with q(z) 6= 0. Suppose that
is starlike univalent in U . Further, assume that (3.1) holds. If f ∈ A, and
zq 0 (z)
q(z)
γ + ξ(
then
z
I λ (a, c)f (z)
)µ + δ(
µ
z
I λ (a, c)f (z)
z
I λ (a, c)f (z)
I λ+1 (a, c)f (z)
]
I λ (a, c)f (z)
zq 0 (z)
≺ γ + ξq(z) + δ(q(z))2 + β
,
q(z)
)2µ + βµ(λ + 1)[1 −
¶µ
≺ q(z)
(µ ∈ C ∗ )
and q is the best dominant.
Taking α1 = η, β1 = λ + 1 (λ > −1; η > 0), αj = 1 (j = 2, . . . , s + 1) and
βj = 1(j = 2, . . . , s) in Theorem 1, we have
57
Subclasses of analytic functions defined by a family of linear operators
Corollary 5. Let q be analytic univalent in U with q(z) 6= 0. Suppose that
is starlike univalent in U . Further, assume that (3.1) holds. If f ∈ A, and
zq 0 (z)
q(z)
γ + ξ(
z
z
Iλ,η+1 f (z)
)µ + δ(
)2µ + βµη[1 −
]
Iλ,η f (z)
Iλ,η f (z)
Iλ,η f (z)
≺ γ + ξq(z) + δ(q(z))2 + β
then
µ
z
Iλ,η f (z)
zq 0 (z)
,
q(z)
¶µ
≺ q(z)
(µ ∈ C ∗ )
and q is the best dominant.
Taking α1 = η + 1, β1 = η + 2 (η > −1), αj = 1 (j = 2, . . . , s + 1) and
βj = 1(j = 2, . . . , s) in Theorem 1, we have
Corollary 6. Let q be analytic univalent in U with q(z) 6= 0. Suppose that
is starlike univalent in U . Further, assume that (3.1) holds. If f ∈ A, and
zq 0 (z)
q(z)
γ + ξ(
z
z
f (z)
)µ + δ(
)2µ + βµ(1 + η)[1 −
]
Fη f (z)
Fη f (z)
Fη f (z)
≺ γ + ξq(z) + δ(q(z))2 + β
then
(
z
)µ ≺ q(z)
Fη f (z)
zq 0 (z)
,
q(z)
(µ ∈ C ∗ )
and q is the best dominant.
Now, by appealing to Lemma 2 it can be easily prove the following theorem.
0
(z)
be starlike
Theorem 2. Let q be convex univalent in U , q(z) 6= 0 and zqq(z)
univalent in U . Assume that
½
¾
2δ
ξ
Re
(q(z))2 + q(z) > 0 (z ∈ U ).
(3.9)
β
β
µ
¶µ
z
If f ∈ A, 0 6=
∈ H[q(0), 1] ∩ Q, Ψ(α1 , γ, ξ, δ, β, µ, f ) is univalent
Hq,s (α1 )f (z)
in U , and
γ + ξq(z) + δ(q(z))2 + β
zq 0 (z)
≺ Ψ(α1 , γ, ξ, δ, β, µ, f ),
q(z)
where Ψ(α1 , γ, ξ, δ, β, µ, f ) is given by (3.2), then
µ
¶µ
z
q(z) ≺
(µ ∈ C ∗ )
Hq,s (α1 )f (z)
and q is the best subordinant.
(3.10)
58
A. O. Mostafa
Proof. Taking
ϑ(w) = γ + ξw + δw2 and ϕ(w) =
β
,
w
it is easily observed that ϑ is analytic in C, ϕ is analytic in C ∗ and ϕ(w) 6= 0
(w ∈ C ∗ ). Since q is convex (univalent) function it follows that
½ 0
¾
½
¾
ϑ (q(z))
2δ
ξ
Re
= Re
(q(z))2 + q(z) q 0 (z) > 0 (z ∈ U ).
ϕ(q(z))
β
β
Thus the assertion (3.10) of Theorem 2 follows by an application of Lemma 2.
Taking α1 = a > 0, β1 = c > 0, αj = 1 (j = 2, . . . , s + 1) and βj = 1
(j = 2, . . . , s) in Theorem 2, we have the following corollary which improves the
result of Shanmugam et. al. [14, Theorem 3.6].
zq 0 (z)
be starq(z)
z
like univalent in U . Assume that (3.9) holds. If f ∈ A, 0 =
6 (
)µ ∈
L(a, c)f (z)
H[q(0), 1] ∩ Q, ζ(γ, ξ, δ, β, µ) is univalent in U and
Corollary 7. Let q be convex univalent in U , q(z) 6= 0 and
γ + ξq(z) + δ(q(z))2 + β
zq 0 (z)
≺ ζ(γ, ξ, δ, β, µ),
q(z)
where ζ(γ, ξ, δ, β, µ) is given by (3.8), then
µ
¶µ
z
q(z) ≺
(µ ∈ C ∗ )
L(a, c)f (z)
and q is the best subordinant.
Taking α1 = λ + 1, α2 = c, β1 = a (a, c ∈ R \ Z0− ; λ > −1), αj = 1
(j = 3, . . . , s + 1) and βj = 1 (j = 2, . . . , s), in Theorem 2, we have
zq 0 (z)
be starq(z)
z
like univalent in U . Assume that (3.9) holds. If f ∈ A, 0 =
6 ( λ
)µ ∈
I (a, c)f (z)
H[q(0), 1] ∩ Q,
Corollary 8. Let q be convex univalent in U , q(z) 6= 0 and
γ + ξ(
z
z
I λ+1 (a, c)f (z)
µ
2µ
)
+
δ(
)
+
βµ(λ
+
1)[1
−
]
I λ (a, c)f (z)
I λ (a, c)f (z)
I λ (a, c)f (z)
is univalent in U , and
γ + ξq(z) + δ(q(z))2 + β
≺ γ + ξ(
z
I λ (a, c)f (z)
zq 0 (z)
q(z)
)µ + δ(
z
I λ (a, c)f (z)
)2µ + βµ(λ + 1)[1 −
I λ+1 (a, c)f (z)
]
I λ (a, c)f (z)
59
Subclasses of analytic functions defined by a family of linear operators
then
µ
q(z) ≺
z
I λ (a, c)f (z)
¶µ
(µ ∈ C ∗ )
and q is the best subordinant.
Letting α1 = η, β1 = λ + 1 (λ > −1; η > 0), αj = 1 (j = 2, . . . , s + 1) and
βj = 1 (j = 2, . . . , s), in Theorem 2, we have
zq 0 (z)
Corollary 9. Let q be convex univalent in U , q(z) 6= 0 and
be starlike
q(z)
z
)µ ∈ H[q(0), 1]∩Q,
univalent in U . Assume that (3.9) holds. If f ∈ A, 0 6= (
Iλ,η f (z)
γ + ξ(
z
z
Iλ,η+1 f (z)
)µ + δ(
)2µ + βµη[1 −
]
Iλ,η f (z)
Iλ,η f (z)
Iλ,η f (z)
is univalent in U , and
γ + ξq(z) + δ(q(z))2 + β
zq 0 (z)
q(z)
≺ γ + ξ(
z
z
Iλ,η+1 f (z)
)µ + δ(
)2µ + βµη[1 −
]
Iλ,η f (z)
Iλ,η f (z)
Iλ,η f (z)
then
µ
q(z) ≺
z
Iλ,η f (z)
¶µ
(µ ∈ C ∗ )
and q is the best subordinant.
Taking α1 = η + 1, β1 = η + 2 (η > −1), αj = 1 (j = 2, . . . , s + 1) and βj = 1
(j = 2, . . . , s), in Theorem 2, we have
0
(z)
Corollary 10. Let q be convex univalent in U , q(z) 6= 0 and zqq(z)
be
¶µ
µ
z
∈
starlike univalent in U . Assume that (3.9) holds. If f ∈ A, 0 6=
Fµ f (z)
H[q(0), 1] ∩ Q,
γ + ξ(
z
z
f (z)
)µ + δ(
)2µ + βµ(1 + η)[1 −
]
Fη f (z)
Fη f (z)
Fη f (z)
is univalent in U , and
γ+ξq(z)+δ(q(z))2 +β
zq 0 (z)
z
z
f (z)
≺ γ+ξ(
)µ +δ(
)2µ +βµ(1+η)[1−
]
q(z)
Fη f (z)
Fη f (z)
Fη f (z)
then
µ
q(z) ≺
and q is the best dominant.
z
Fη f (z)
¶µ
(µ ∈ C ∗ )
60
A. O. Mostafa
Combining Theorem 1 and Theorem 2, we get the following sandwich theorem.
Theorem 3. Let q1 be convex univalent in U and q2 be univalent in U, q1 (z) 6=
0 and q2 (z) 6=µ0 in U . Suppose
¶µthat q2 and q1 satisfy (3.1) and (3.9), respectively.
z
If f ∈ A, 0 6=
∈ H[q(0), 1] ∩ Q and
Hq,s (α1 )f (z)
γ + ξ(
z
Hq,s (α1 + 1)f (z)
z
)µ + δ(
)2µ + βµα1 [1 −
]
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
is univalent in U . Then
γ + ξq1 (z) + δ(q1 (z))2 + β
≺ γ + ξ(
zq10 (z)
q1 (z)
z
z
Hq,s (α1 + 1)f (z)
)µ + δ(
)2µ + βµα1 [1 −
]
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
Hq,s (α1 )f (z)
zq 0 (z)
≺ γ + ξq2 (z) + δ(q2 (z))2 + β 2
q2 (z)
implies
µ
¶µ
z
≺ q2 (z) (µ ∈ C ∗ )
Hq,s (α1 )f (z)
and q1 and q2 are, respectively, the best subordinant and the best dominant.
q1 (z) ≺
Taking α1 = a > 0, β1 = c > 0, αj = 1 (j = 2, . . . , s + 1) and βj = 1
(j = 2, . . . , s) in Theorem 3, we have the following corollary which improves the
result of Shanmugam et al. [14, Theorem 3.7].
Corollary 11. Let q1 be convex univalent in U and q2 be univalent in
U, q1 (z) 6= 0 and q2 (z) =
6 0µin U . Suppose
¶µ that q2 and q1 satisfy (3.1) and (3.9),
z
respectively. If f ∈ A, 0 6=
∈ H[q(0), 1] ∩ Q and
L(a, c)f (z)
γ + ξ(
z
z
L(a + 1, c)f (z)
)µ + δ(
)2µ + βµa[1 −
]
L(a, c)f (z)
L(a, c)f (z)
L(a, c)f (z)
is univalent in U . Then
γ + ξq1 (z) + δ(q1 (z))2 + β
≺ γ + ξ(
zq10 (z)
q1 (z)
z
z
L(a + 1, c)f (z)
)µ + δ(
)2µ + βµa[1 −
]
L(a, c)f (z)
L(a, c)f (z)
L(a, c)f (z)
zq 0 (z)
≺ γ + ξq2 (z) + δ(q2 (z))2 + β 2
q2 (z)
implies
µ
¶µ
z
≺ q2 (z) (µ ∈ C ∗ )
L(a, c)f (z)
and q1 and q2 are, respectively, the best subordinant and the best dominant.
q1 (z) ≺
Subclasses of analytic functions defined by a family of linear operators
61
Remarks. Combining: (i) Corollary 4 and Corollary 8; (ii) Corollary 5 and
Corollary 9; (iii) Corollary 6 and Corollary 10, we obtain similar sandwich theorems
for the corresponding operators.
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(received 21.12.2008, in revised form 01.02.2009)
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
E-mail: adelaeg254@yahoo.com