An. Şt. Univ. Ovidius Constanţa
Vol. 21(1), 2013, 5–18
Differential sandwich theorems of p−valent
analytic functions involving a linear operator
M. K. Aouf and T. M. Seoudy
Abstract
In this paper we derive some subordination and superordination results for certain p−valent analytic functions in the open unit disc, which
are acted upon by a class of a linear operator. Some of our results improve and generalize previously known results.
1
Introduction
Let H(U ) denotes the class of analytic functions in the open unit disc U =
{z ∈ C : |z| < 1} and let H[a, p] denotes the subclass of the functions f ∈ H(U )
of the form:
f (z) = a + ap z p + ap+1 z p+1 + ...(a ∈ C; p ∈ N = {1, 2, ..}).
Also, let A(p) be the subclass of the functions f ∈ H(U ) of the form:
f (z) = z p +
∞
∑
ak z k
(p ∈ N),
(1.1)
k=p+1
and set A ≡ A(1). For functions f (z) ∈ A(p), given by (1.1), and g(z) given
by
∞
∑
g(z) = z p +
bk z k
(p ∈ N),
(1.2)
k=p+1
Key Words: Analytic function, Hadamard product, differential subordination, superordination, linear operator.
2010 Mathematics Subject Classification: 30C45.
Received: April, 2011.
Revised: April, 2011.
Accepted: February, 2012.
5
6
M. K. Aouf and T. M. Seoudy
the Hadamard product (or convolution) of f (z) and g(z) is defined by
(f ∗ g)(z) = z p +
∞
∑
ak bk z k = (g ∗ f )(z).
(1.3)
k=p+1
For f, g ∈ H(U ), we say that the function f is subordinate to g, if there
exists a Schwarz function w, i.e, w ∈ H(U ) with w(0) = 0 and |w(z)| < 1,
z ∈ U, such that f (z) = g(w(z)) for all z ∈ U. This subordination is usually
denoted by f (z) ≺ g(z). It is well-known that, if the function g is univalent
in U , then f (z) ≺ g(z) is equivalent to f (0) = g(0) and f (U ) ⊂ g(U ) (see [5]
and [9]).
Supposing that h and k are two analytic functions in U , let
ϕ(r, s, t; z) : C3 × U → C.
′
′′
If h and φ(h(z), zh (z), z 2 h (z); z) are univalent functions in U and if h satisfies
the second-order superordination
′
′′
k(z) ≺ φ(h(z), zh (z), z 2 h (z); z),
(1.4)
then h is called to be a solution of the differential superordination (1.4). A
function q ∈ H(U ) is called a subordinant of (1.4), if q(z) ≺ h(z) for all the
functions h satisfying (1.4). A univalent subordinant qe that satisfies q(z) ≺
qe(z) for all of the subordinants q of (1.4), is said to be the best subordinant.
Recently, Miller and Mocanu [10] obtained sufficient conditions on the
functions k, q and φ for which the following implication holds:
′
′′
k(z) ≺ φ(h(z), zh (z), z 2 h (z); z) ⇒ q(z) ≺ h(z).
Using these results, Bulboaca [3] considered certain classes of first-order
differential superordinations, as well as superordination-preserving integral operators [4]. Ali et al. [1], using the results from [3], obtained sufficient conditions for certain normalized analytic functions to satisfy
′
q1 (z) ≺
zf (z)
≺ q2 (z),
f (z)
where q1 and q2 are given univalent normalized functions in U .
For complex parameters
(
)
α1 , ..., αq and β1 , ..., βs βj ∈
/ Z−
0 = {0, −1, −2, ...} ; j = 1, 2, ..., s ,
we now define the generalized hypergeometric function q Fs (α1 ,..., αq ; β1 , ..., βs ; z)
by (see, for example, [15, p.19])
q Fs (α1 , ..., αq ; β1 , ..., βs ; z)
=
∞
∑
(α1 )k ...(αq )k z k
.
(β1 )k ...(βs )k k!
k=0
(1.5)
7
Differential Sandwich Theorems of p−Valent Analytic Functions
(q ≤ s + 1; q, s ∈ N0 = N ∪ {0}; z ∈ U ),
where (θ)ν is the Pochhammer symbol defined, in terms of the Gamma function
Γ, by
Γ(θ + ν)
(θ)ν =
=
Γ(θ)
{
1
(ν = 0; θ ∈ C∗ = C\{0}),
θ(θ + 1)....(θ + ν − 1)
(ν ∈ N; θ ∈ C).
(1.6)
Let
h(α1 , ..., αq ; β1 , ..., βs ; z) =
=
z p q Fs (α1 , ..., αq ; β1 , ..., βs ; z)
∞
∑
zp +
Γp,q,s (α1 ) z k ,
k=p+1
where
Γp,q,s (α1 ) =
(α1 )k−p ...(αq )k−p
,
(β1 )k−p ...(βs )k−p (1)k−p
(1.7)
and using the Hadamard product, El-Ashwah and Aouf [7] defined the following operator
m,ℓ
Ip,λ
(α1 , ..., αq ; β1 , ..., βs ) : A(p) → A(p)
by
0,ℓ
Ip,λ
(α1 , ..., αq ; β1 , ..., βs )f (z) =
f (z) ∗ h(α1 , ..., αq ; β1 , ..., βs ; z);
1,ℓ
Ip,λ
(α1 , ..., αq ; β1 , ..., βs )f (z) =
(1 − λ)(f (z) ∗ h(α1 , ..., αq ; β1 , ..., βs ; z))
+
′
λ
(z ℓ f (z) ∗ h(α1 , ..., αq ; β1 , ..., βs ; z)) ;
ℓ−1
(p + ℓ)z
and
m,ℓ
1,ℓ
m−1,ℓ
Ip,λ
(α1 , ..., αq ; β1 , ..., βs )f (z) = Ip,q,s,λ
(Ip,q,s,λ
(α1 , ..., αq ; β1 , ..., βs )f (z)).
(1.8)
If f ∈ A(p), then from (1.1) and (1.8), we can easily see that
m,ℓ
Ip,λ
(α1 , ..., αq ; β1 , ..., βs )f (z) = z p +
]m
∞ [
∑
p + ℓ + λ(k − p)
Γp,q,s (α1 ) ak z k ,
p+ℓ
k=p+1
(1.9)
(p ∈ N; m ∈ N0 = N ∪ {0}; ℓ ≥ 0; λ ≥ 0; z ∈ U )
It can be easily verified from the definition (1.9) that:
′
m,ℓ
m,ℓ
m,ℓ
z(Ip,q,s,λ
(α1 )f (z)) = α1 Ip,q,s,λ
(α1 + 1)f (z) − (α1 − p)Ip,q,s,λ
(α1 )f (z), (1.10)
8
M. K. Aouf and T. M. Seoudy
where
m,ℓ
m,ℓ
(α1 )f (z) = Ip,λ
(α1 , ..., αq ; β1 , ..., βs )f (z).
Ip,q,s,λ
m,ℓ
It should be remarked that the linear operator Ip,q,s,λ
(α1 ) is a generalization of many other linear operators considered earlier. In particular, we
have
0,ℓ
Ip,q,s,λ
(α1 )f (z) = Hp,q,s (α1 )f (z),
where the linear operator Hp,q,s (α1 ) was investigated by Dziok and Srivastava
[8], and also we have
0,ℓ
Ip,2,1,λ
(a, 1; c)f (z) = Lp (a, c)f (z)(a ∈ R; c ∈ R\Z−
0 ),
where the linear operator Lp (a, c) was studied by Saitoh [13] which yields the
operator L(a, c)f (z) introduced by Carlson and Shaffer [6] for p = 1.
2
Preliminaries
In order to prove our subordination and superordination results, we make use
of the following known definition and results.
Definition [10]. Denote by Q the set of all functions f (z) that are analytic
and injective on U \E(f ), where
{
}
E(f ) = ζ : ζ ∈ ∂ and lim f (z) = ∞
(2.1)
z→ζ
′
and are such that f (ζ) ̸= 0 for ζ ∈ ∂U \E(f ).
Lemma 1 [9]. Let the function q(z) be univalent in the unit disc U and let
θ and φ be analytic in a domain D containing q(U ) with φ(w) ̸= 0 when
′
w ∈ q(U ). Set Q(z) = zq (z)φ(q(z)) and h(z) = θ(q(z)) + Q(z). Suppose that
(i) Q(z)
( is′ starlike
) univalent in U ,
zh (z)
(ii) ℜ
> 0 for z ∈ U.
Q(z)
If p is analytic with p(0) = q(0), p(U ) ⊆ D and
′
′
θ(p(z)) + zp (z)φ(p(z)) ≺ θ(q(z)) + zq (z)φ(q(z)),
(2.2)
then p(z) ≺ q(z) and q(z) is the best dominant.
Lemma 2 [5]. Let q(z) be convex univalent in the unit disc U and let θ and
φ be analytic
D containing q(U ). Suppose that
}
{ ′ in a domain
θ (q(z))
> 0 for z ∈ U ;
(i) ℜ
φ(q(z))
9
Differential Sandwich Theorems of p−Valent Analytic Functions
′
(ii) zq (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)) + zp (z)φ(p(z)) is univalent in U , and
′
′
θ(q(z)) + zq (z)φ(q(z)) ≺ θ(p(z)) + zp (z)φ(p(z)),
(2.4)
then q(z) ≺ p(z) and q(z) is the best subordinant
The following lemma gives us a necessary and sufficient condition for the
univalence of a special function which will be used in some particular case.
−2ab
Lemma 3 [12].The function q(z) = (1 − z)
(a, b ∈ C∗ ) is univalent in the
unit disc U if and only if |2ab − 1 | ≤ 1 or |2ab + 1 | ≤ 1.
3
Main Results
Unless otherwise mentioned, we assume throughout this paper that p ∈ N, m ∈
N0 , ℓ ≥ 0; λ ≥ 0 and the power understood as principal values.
Theorem 1. Let q(z) be univalent in U such that q(0) = 1, q(z) ̸= 0 and
′
zq (z)
q(z)
is starlike in U. Let f ∈ A(p) and suppose that f and q satisfy the next
conditions:
]µ [
]η
[ m,ℓ
Ip,q,s,λ (α1 )f (z)
zp
̸= 0 (µ ∈ C∗ ; η ∈ C; z ∈ U ),
m,ℓ
zp
Ip,q,s,λ
(α1 + 1)f (z)
(3.1)
and
{
}
′
′′
ζ
2δ
zq (z) zq (z)
2
ℜ 1 + q (z) +
[q (z)] −
+ ′
> 0 (ζ, δ ∈ C; γ ∈ C∗ ; z ∈ U ).
γ
γ
q(z)
q (z)
(3.2)
If
′
zq (z)
2
Ψ (z) ≺ χ + ζq (z) + δ [q (z)] + γ
,
(3.3)
q(z)
where
[
Ψ (z) =
χ+ζ
m,ℓ
Ip,q,s,λ
(α1 )f (z)
]µ [
zp
]η
m,ℓ
Ip,q,s,λ
(α1 + 1)f (z)
]2µ [
]2η
[ m,ℓ
Ip,q,s,λ (α1 )f (z)
zp
+δ
m,ℓ
zp
Ip,q,s,λ
(α1 + 1)f (z)
[ m,ℓ
]
Ip,q,s,λ (α1 + 1)f (z)
+γµα1
−1
m,ℓ
Ip,q,s,λ
(α1 )f (z)
zp
10
M. K. Aouf and T. M. Seoudy
[
+γη (α1 + 1) 1 −
[
then
m,ℓ
Ip,q,s,λ
(α1 )f (z)
m,ℓ
Ip,q,s,λ
(α1 + 2)f (z)
,
m,ℓ
Ip,q,s,λ
(α1 + 1)f (z)
]µ [
(3.4)
]η
zp
≺ q (z) ,
m,ℓ
Ip,q,s,λ
(α1 + 1)f (z)
zp
]
and q is the best dominant of (3.3).
Proof. Let
[
h(z) =
m,ℓ
Ip,q,s,λ
(α1 )f (z)
]µ [
]η
zp
(z ∈ U ).
m,ℓ
Ip,q,s,λ
(α1 + 1)f (z)
zp
(3.5)
According to (3.1) the function h(z) is analytic in U , and differentiating (3.5)
logarithmically with respect to z, we obtain
[
]
[
′
′ ]
′
m,ℓ
m,ℓ
z(Ip,q,s,λ
(α1 )f (z))
z(Ip,q,s,λ
(α1 + 1)f (z))
zh (z)
=µ
−p +η p−
.
m,ℓ
m,ℓ
h(z)
Ip,q,s,λ
Ip,q,s,λ
(α1 )f (z)
(α1 + 1)f (z)
By using the identity (1.10), we obtain
[ m,ℓ
]
[
]
′
m,ℓ
Ip,q,s,λ (α1 + 1)f (z)
Ip,q,s,λ
(α1 + 2)f (z)
zh (z)
= µα1
− 1 +η (α1 + 1) 1 − m,ℓ
.
m,ℓ
h(z)
Ip,q,s,λ
(α1 )f (z)
Ip,q,s,λ (α1 + 1)f (z)
In order to prove our result we will use Lemma 1. In this lemma consider
θ(w) = χ + ζw + δw2
and φ(w) =
γ
,
w
then θ is analytic in C and φ(w) ̸= 0 is analytic in C∗ .Also, if we let
′
zq (z)
Q(z) = zq (z)φ(q(z)) = γ
,
q(z)
′
and
′
2
g(z) = θ(q(z)) + Q(z) = χ + ζq (z) + δ [q (z)] + γ
zq (z)
.
q(z)
We see that Q(z) is starlike function in U . From (3.2), we also have
{ ′
}
{
}
′
′′
zg (z)
ζ
2δ
zq (z) zq (z)
2
ℜ
= ℜ 1 + q (z) +
[q (z)] −
+ ′
> 0 (z ∈ U ),
Q(z)
γ
γ
q(z)
q (z)
Differential Sandwich Theorems of p−Valent Analytic Functions
11
and then, by using Lemma 1 we deduce that the subordination (3.3) implies
h(z) ≺ q(z), and the function q is the best dominant of (3.3).
Putting q = 2, s = p = 1, m = 0, α1 = a (a ∈ C) , α2 = 1 and β1 = c
(c ∈ C\Z−
0 ) in Theorem 1, we obtain the following result which improves the
corresponding work of Shammugam et al. [14,Theorem 3.1].
Corollary 1. Let q(z) be univalent in U such that q(0) = 1, q(z) ̸= 0 and
′
zq (z)
q(z)
is starlike in U. Let f ∈ A such that
[
L (a, c) f (z)
z
]µ [
z
L (a + 1, c) f (z)
]η
(µ ∈ C∗ ; z ∈ U ),
̸= 0
(3.6)
and suppose that q satisfies (3.2).If
′
2
Λ (z) ≺ χ + ζq (z) + δ [q (z)] + γ
zq (z)
,
q(z)
(3.7)
where
[
]µ [
]η
L (a, c) f (z)
z
z
L (a + 1, c) f (z)
[
]2µ [
]2η
L (a, c) f (z)
z
+δ
z
L (a + 1, c) f (z)
[
]
L (a + 1, c) f (z)
+γµa
−1
L (a, c) f (z)
[
]
L (a + 2, c) f (z)
+γη (a + 1) 1 −
,
L (a + 1, c) f (z)
Λ (z) =
then
[
χ+ζ
L (a, c) f (z)
z
]µ [
z
L (a + 1, c) f (z)
(3.8)
]η
≺ q (z) ,
and q is the best dominant of (3.7).
1+Az
Putting q (z) = 1+Bz
(−1 ≤ B < A ≤ 1) in Corollary 1, we obtain the
following result which improves the corresponding work of Shammugam et al.
[14, Corollary 3.2].
Corollary 2. Assume that
{
[
]
[
]2 }
ζ 1 + Az
2δ 1 + Az
1 − ABz 2
+
+
>0
ℜ
(1 + Az) (1 + Bz) γ 1 + Bz
γ 1 + Bz
(ζ, δ ∈ C; γ ∈ C∗ ; z ∈ U )
12
M. K. Aouf and T. M. Seoudy
holds. Let f ∈ A such that (3.6) holds. If
[
]2
1 + Az
1 + Az
γ (A − B) z
Λ (z) ≺ χ + ζ
+δ
+
,
1 + Bz
1 + Bz
(1 + Az) (1 + Bz)
(3.9)
where Λ (z) is given by (3.8), then
[
]µ [
]η
L (a, c) f (z)
1 + Az
z
≺
,
z
L (a + 1, c) f (z)
1 + Bz
and
1+Az
1+Bz
is the best dominant of (3.9).
(
)ϑ
1+z
Putting q (z) = 1−z
(0 < ϑ ≤ 1) in Corollary 1, we obtain the following
result which improves the corresponding work of Shammugam et al. [14,
Corollary 3.3].
Corollary 3. Assume that
{
[
]ϑ
[
]2ϑ }
1 − 3z 2
ζ 1+z
2δ 1 + z
ℜ
+
+
> 0 (ζ, δ ∈ C; γ ∈ C∗ ; z ∈ U )
1 − z2
γ 1−z
γ 1−z
holds. Let f ∈ A such that (3.6) holds. If
(
Λ (z) ≺ χ + ζ
1+z
1−z
)ϑ
(
+δ
1+z
1−z
)2ϑ
+
2γϑz
(0 < ϑ ≤ 1) ,
(1 − z 2 )
(3.10)
where Λ (z) is given by (3.8), then
[
(
and
1+z
1−z
L (a, c) f (z)
z
]µ [
z
L (a + 1, c) f (z)
]η
(
≺
1+z
1−z
)ϑ
,
)ϑ
is the best dominant of (3.10).
Putting q (z) = eµAz (|µA| < π) in Corollary 1, we obtain the following
result which improves the corresponding work of Shammugam et al. [14,
Corollary 3.4].
Corollary 4. Assume that
{
}
ζ
2δ
ℜ 1 + eµAz q (z) + e2µAz > 0 (ζ, δ ∈ C; γ ∈ C∗ ; z ∈ U )
γ
γ
holds and let f ∈ A such that (3.6) holds. If
Λ (z) ≺ χ + ζeµAz + δe2µAz + γAµz
(|µA| < π) ,
(3.11)
Differential Sandwich Theorems of p−Valent Analytic Functions
13
where Λ (z) is given by (3.8), then
[
]µ [
]η
L (a, c) f (z)
z
≺ eµAz ,
z
L (a + 1, c) f (z)
and eµAz is the best dominant of (3.11).
Putting q = s + 1, αi = 1(i = 1, .., s + 1), βj = 1(j = 1, .., s), m = ζ =
1
δ = 0, χ = p = 1, γ = ab
(a, b ∈ C∗ ), µ = a, η = 0 and q(z) = (1 − z)−2ab in
Theorem 1, then combining this to gather with Lemma 3 we obtain the next
result due to Obradovic et al. [11, Theorem 1].
Corollary 5 [11]. Let a, b ∈ C∗ such that |2ab − 1| ≤ 1 or |2ab + 1| ≤ 1. Let
f ∈ A and suppose that f (z)
z ̸= 0 for all z ∈ U. If
1+
1
b
(
then
(
)
1+z
zf ′ (z)
−1 ≺
,
f (z)
1−z
f (z)
z
)a
≺ (1 − z)−2ab
(3.12)
and (1 − z)−2ab is the best dominant of (3.12).
Remark 1. For a = 1, Corollary 5 reduces to the recent result of Srivastava
and Lashin [16].
Putting q = s + 1, αi = 1(i = 1, .., s + 1), βj = 1(j = 1, .., s), m = ζ = δ =
µ(A−B)
in Theorem 2, and using
0, χ = p = γ = 1, η = 0 and q(z) = (1 + Bz) B
Lemma 2 we obtain the next result.
Corollary 6. Let −1 ≤ A < B ≤ 1 with B ̸= 0, and suppose that µ(A−B)
−
1
≤
B
+ 1 ≤ 1 . Let f ∈ A such that f (z)
1 or µ(A−B)
B
z ̸= 0 for all z ∈ U , and let
∗
µ ∈ C . If
( ′
)
zf (z)
1 + [B + µ(A − B)]z
1+µ
−1 ≺
,
f (z)
1 + Bz
(
then
µ(A−B)
f (z)
z
)µ
≺ (1 + Bz)
µ(A−B)
B
,
(3.13)
and (1 + Bz) B
is the best dominant of (3.13).
Putting q = s + 1, αi = 1(i = 1, .., s + 1), βj = 1(j = 1, .., s), m = ζ =
iτ
δ = 0, χ = p = 1, γ = abecos τ (a, b ∈ C∗ ; |τ | < π2 ), µ = a, η = 0 and q(z) =
−iτ
(1 − z)−2ab cos τ e in Theorem 1, we obtain the following result due to Aouf
et al. [2, Theorem 1].
14
M. K. Aouf and T. M. Seoudy
Corollary 7 [2]. Let a, b ∈ C∗ , |τ | < π2 and suppose that 2ab cos τ e−iτ −1 ≤
1 or 2ab cos τ e−iτ +1 ≤ 1. Let f ∈ A and suppose that f (z)
̸= 0 for all
z
z ∈ U. If
( ′
)
eiτ
1+z
zf (z)
1+
−1 ≺
,
b cos τ
f (z)
1−z
then
(
and (1 − z)−2ab cos τ e
−iτ
f (z)
z
)a
≺ (1 − z)−2ab cos τ e
−iτ
(3.14)
is the best dominant of (3.14).
′
(z)
Theorem 2. Let q be convex in U such that q (0) = 1 and zqq(z)
is starlike
in U. Further assume that
{
}
′
q (z) q (z)
ℜ (ζ + 2δq (z))
> 0 (ζ, δ ∈ C; γ ∈ C∗ ) .
(3.15)
γ
Let f ∈ A(p) such that
[
0 ̸=
m,ℓ
Ip,q,s,λ
(α1 )f (z)
]µ [
]η
zp
m,ℓ
Ip,q,s,λ
(α1 + 1)f (z)
zp
∈ H[q(0), 1] ∩ Q.
(3.16)
If Ψ (z) given by (3.4) is univalent in U and satisfies the following superordination condition
′
2
χ + ζq (z) + δ [q (z)] + γ
then
[
q(z) ≺
m,ℓ
Ip,q,s,λ
(α1 )f (z)
zp
zq (z)
≺ Ψ (z) ,
q(z)
]µ [
zp
m,ℓ
Ip,q,s,λ
(α1 + 1)f (z)
(3.17)
]η
,
and q is the best subordinant of (3.17).
Putting q = 2, s = p = 1, m = 0, α1 = a (a ∈ C) , α2 = 1 and β1 = c
(c ∈ C\Z−
0 ) in Theorem 2, we obtain the following result which improves the
corresponding work of Shammugam et al. [14, Theorem 3.11].
′
Corollary 8. Let q be convex in U such that q (0) = 1 and
in U. Further assume that (3.15) holds. Let f ∈ A such that
[
L (a, c) f (z)
0 ̸=
z
]µ [
z
L (a + 1, c) f (z)
zq (z)
q(z)
is starlike
]η
∈ H[q(0), 1] ∩ Q.
(3.18)
15
Differential Sandwich Theorems of p−Valent Analytic Functions
If Λ (z) given by (3.8) is univalent in U and satisfies the following superordination condition
′
zq (z)
χ + ζq (z) + δ [q (z)] + γ
≺ Λ (z) ,
q(z)
2
[
then
L (a, c) f (z)
q(z) ≺
z
]µ [
z
L (a + 1, c) f (z)
(3.19)
]η
,
and q is the best subordinant of (3.19).
Combining Theorems 1 and 2, we obtain the following two sandwich results:
Theorem 3. Let qi be two convex functions in U such that qi (0) = 1 and
′
zqi (z)
qi (z)
(i = 1, 2) is starlike in U. Suppose that q1 (z) satisfies (3.15) and q2 (z)
satisfies (3.2) . Let f ∈ A(p) and suppose that
[ m,ℓ
]µ [
]η
Ip,q,s,λ (α1 )f (z)
zp
∈ H[q(0), 1] ∩ Q.
m,ℓ
zp
Ip,q,s,λ
(α1 + 1)f (z)
If Ψ (z) given by (3.4) is univalent in U , and
′
2
χ + ζq1 (z) + δ [q1 (z)] + γ
then
[
q1 (z) ≺
′
zq1 (z)
zq (z)
2
≺ Ψ (z) ≺ χ + ζq2 (z) + δ [q2 (z)] + γ 2 ,
q1 (z)
q2 (z)
(3.20)
m,ℓ
Ip,q,s,λ
(α1 )f (z)
]µ [
zp
m,ℓ
Ip,q,s,λ
(α1 + 1)f (z)
zp
]η
≺ q2 (z) ,
and q1 and q2 are, respectively, the best subordinant and the best dominant of
(3.20).
Putting q = 2, s = p = 1, m = 0, α1 = a (a ∈ C) , α2 = 1 and β1 = c
(c ∈ C\Z−
0 ) in Theorem 3, we obtain the following result which improves the
corresponding work of Shammugam et al. [14, Theorem 3.12].
Corollary 9. Let qi be two convex functions in U such that qi (0) = 1 and
′
zqi (z)
qi (z)
(i = 1, 2) is starlike in U. Suppose that q1 (z) satisfies (3.15) and q2 (z)
]µ
[
(z)
satisfies (3.2) . Let f ∈ A and suppose that L(a+1,c)f
∈ H[q(0), 1] ∩ Q.
z
If Λ (z) given by (3.8) is univalent in U , and
′
2
χ + ζq1 (z) + δ [q1 (z)] + γ
′
zq1 (z)
zq (z)
2
≺ Λ (z) ≺ χ + ζq2 (z) + δ [q2 (z)] + γ 2 ,
q1 (z)
q2 (z)
(3.21)
16
then
M. K. Aouf and T. M. Seoudy
[
L (a + 1, c) f (z)
q1 (z) ≺
z
]µ
≺ q2 (z) ,
and q1 and q2 are, respectively, the best subordinant and the best dominant of
(3.21).
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Differential Sandwich Theorems of p−Valent Analytic Functions
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M. K. Aouf,
Department of Mathematics,
Faculty of Science,
Mansoura University,
Mansoura 35516, Egypt.
Email: mkaouf127@yahoo.com
T. M. Seoudy,
Department of Mathematics,
Faculty of Science,
Fayoum University,
Fayoum 63514, Egypt.
Email: tms00@fayoum.edu.eg
18
M. K. Aouf and T. M. Seoudy