A N N A L E S
U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A
VOL. LXIII, 2009 SECTIO A 29–38
MOHAMED K. AOUF and RABHA M. EL-ASHWAH
Inclusion properties of certain subclass of analytic functions defined
by multiplier transformations
Abstract. Let A denote the class of analytic functions with normalization f (0) = f0(0) − 1 = 0 in the open unit disk U = {z : |z| < 1}. Set
fλ,`m(z) = z +
∞
X
k=2
` + 1 + λ(k − 1)
` + 1
m
zk (z ∈ U ; m ∈ N0; λ ≥ 0; ` ≥ 0), and define fλ`,µm in terms of the Hadamard product
fλ,`m(z) ∗ fλ,`,µm (z) = z
(1 − z)µ (z ∈ U ; µ > 0).
In this paper, we introduce several new subclasses of analytic functions defined by means of the operator Iλ,`,µm f (z) = fλ,`,µm (z) ∗ f (z) (f ∈ A; m ∈ N0; λ ≥ 0; ` ≥ 0; µ > 0).
Inclusion properties of these classes and the classes involving the general- ized Libera integral operator are also considered.
1. Introduction. Let A denote the class of functions of the form:
(1.1) f (z) = z +
∞
X
k=2
akzk
which are analytic in the open unit disk U = {z : |z| < 1}. If f and g are analytic in U , we say that f is subordinate to g, written f (z) ≺ g(z), if
2000 Mathematics Subject Classification. 30C45.
Key words and phrases. Subordination, analytic, multiplier transformation, Libera integral operator.
there exists an analytic function w in U with w(0) = 0 and |w(z)| < 1 for z ∈ U such that f (z) = g(w(z)). For 0 ≤ η < 1, we denote by S∗(η), K(η) and C the subclasses of A consisting of all analytic functions which are, respectively, starlike of order η, convex of order η and close-to-convex in U (see, e.g. Srivastava and Owa [18]).
For m ∈ N0 = N ∪ {0}, where N = {1, 2, . . . }, λ ≥ 0 and ` ≥ 0, C˘ata¸s [3] defined the multiplier transformations Im(λ, `) on A by the following infinite series
(1.2) Im(λ, `)f (z) = z +
∞
X
k=2
` + 1 + λ(k − 1)
` + 1
m
akzk. It follows from (1.2) that
(1.3) I0(λ, `) = f (z),
(1.4) (` + 1)I2(λ, `)f (z) = (` + 1 − λ)I1(λ, `)f (z) + λz(I1(λ, `)f (z))0, λ > 0, and
(1.5) Im1(λ, `)(Im2(λ, `)f (z)) = Im2(λ, `)(Im1(λ, `)f (z)) for all integers m1 and m2.
We note that:
(i) Im(1, `) = I`m (see Cho and Srivastava [4] and Cho and Kim [5]);
(ii) Im(λ, 0) = Dmλ (m ∈ N0; λ ≥ 0) (see Al-Oboudi [1]);
(iii) Im(1, 0) = Dm (m ∈ N0) (see S˘al˘agean [17]);
(iv) Im(1, 1) = Im (see Uralegaddi and Somanatha [19]).
Let S be the class of all functions ϕ which are analytic and univalent in U and for which ϕ(U ) is convex and ϕ(0) = 1 and Re{ϕ(z)} > 0 (z ∈ U ).
Making use of the principle of subordination between analytic functions, we introduce the subclasses S∗(η; ϕ), K(η; ϕ) and C(η, δ; ϕ, ψ) of the class A for 0 ≤ η, δ < 1 and ϕ, ψ ∈ S (cf., e.g., [6], [8] and [12]), which are defined as follows:
S∗(η; ϕ) = (
f ∈ A : 1 1 − η
zf0(z) f (z) − η
!
≺ ϕ(z), z ∈ U )
,
K(η; ϕ) = (
f ∈ A : 1
1 − η 1 +zf00(z) f0(z) − η
!
≺ ϕ(z), z ∈ U )
, and
C(η, δ; ϕ, ψ) = (
f ∈ A : ∃g ∈ S∗(η, ϕ) s.t.
1 1 − δ
zf0(z) g(z) − δ
!
≺ ψ(z), z ∈ U )
.
We note that, for special choices for the functions ϕ and ψ in the above definitions we obtain the well-known subclasses of A. For examples, we have
(i) S∗
η;1 + z 1 − z
= S∗(η) (0 ≤ η < 1), (ii) K
η;1 + z 1 − z
= K(η) (0 ≤ η < 1) and
(iii) C
0, 0;1 + z 1 − z;1 + z
1 − z
= C.
Setting
fλ,`m(z) = z +
∞
X
k=2
` + 1 + λ(k − 1)
` + 1
m
zk (m ∈ N0, λ ≥ 0, ` ≥ 0), we define a new function fλ,`,µm (z) in terms of the Hadamard product (or convolution) by:
(1.6) fλ,`m(z) ∗ fλ,`,µm (z) = z
(1 − z)µ (µ > 0; z ∈ U ).
Then, motivated essentially by the Choi–Saigo–Srivastava operator [6] (see also [10], [11], [14], and [15]), we now introduce the operators fλ,`,µm : A → A, which are defined here by
(1.7) Iλ,`,µm f (z) = fλ,`,µm ∗ f (z)
(f ∈ A; m ∈ N0; λ ≥ 0; ` ≥ 0; µ > 0). For a function f (z) ∈ A, given by (1.1), it is easily seen from (1.7) that
(1.8) Iλ,`,µm f (z) = z +
∞
X
k=2
` + 1
` + 1 + λ(k − 1)
m
(µ)k−1 (1)k−1akzk (m ∈ N0; λ ≥ 0; ` ≥ 0; z ∈ U ).
We note that:
(i) I1,0,21 f (z) = f (z) and I1,0,20 f (z) = zf0(z), and
(ii) I1,`,µs f (z) = I`,µs f (z) (s ∈ R; see Cho and Kim [5]).
In view of (1.8), we obtain the following relations:
(1.9) λz(Iλ,`,µm+1f (z))0 = (` + 1)Iλ,`,µm f (z) − [λ − (` + 1)] Iλ,`,µm+1f (z) (f ∈ A; m ∈ N0; λ > 0; ` ≥ 0; µ > 0) and
(1.10) z(Iλ,`,µm f (z))0 = µIλ,`,µ+1m f (z) − (µ − 1)Iλ,`,µm f (z)
(f ∈ A; m ∈ N0; λ ≥ 0; ` ≥ 0; µ > 0). Next, by using the operator Iλ,`,µm , we introduce the following classes of analytic functions for ϕ, ψ ∈ S, m ∈ N0, λ ≥ 0, ` ≥ 0, µ > 0 and 0 ≤ η, δ < 1:
Sλ,`,µm (η; ϕ) =f ∈ A : Iλ,`,µm f (z) ∈ S∗(η; ϕ) , (1.11)
Kλ,`,µm (η; ϕ) =f ∈ A : Iλ,`,µm f (z) ∈ K(η; ϕ) (1.12)
and
Cλ,`,µm (η, δ; ϕ, ψ) =f ∈ A : Iλ,`,µm f (z) ∈ C(η, δ; ϕ, ψ) . (1.13)
We also have
(1.14) f (z) ∈ Kλ,`,µm (η; ϕ) ⇔ zf0(z) ∈ Sλ,`,µm (η; ϕ).
In particular, we set Sλ,`,µm
η; 1 + Az 1 + Bz
α
= Sλ,`,µm (η; A, B, α) (0 < α ≤ 1; −1 ≤ B < A ≤ 1) and
Kλ,`,µm
η; 1 + Az 1 + Bz
α
= Kλ,`,µm (η; A, B, α) (0 < α ≤ 1; −1 ≤ B < A ≤ 1).
In this paper, we investigate several inclusion properties of the classes Sλ,`,µm (η; ϕ), Kλ,`,µm (η; ϕ) and Cλ,`,µm (η, δ; ϕ, ψ) associated with the operator Iλ,`,µm . Some applications involving these and other classes of integral oper- ators are also considered.
2. Inclusion properties involving the operator Iλ,`,µm . The following lemmas will be required in our investigation.
Lemma 1 ([7]). Let ϕ be convex, univalent in U with ϕ(0) = 1 and Re {βϕ(z) + ν} > 0 (β, ν ∈ C). If p is analytic in U with p(0) = 1, then
p(z) + zp0(z)
βp(z) + ν ≺ ϕ(z) (z ∈ U ) implies that
p(z) ≺ ϕ(z) (z ∈ U ).
Lemma 2 ([13]). Let ϕ be convex, univalent in U and w be analytic in U with Re{w(z)} ≥ 0. If p(z) is analytic in U and p(0) = ϕ(0), then
p(z) + w(z)zp0(z) ≺ ϕ(z) (z ∈ U ) implies that
p(z) ≺ ϕ(z) (z ∈ U ).
At first, with the help of Lemma 1, we prove the following theorem.
Theorem 1. Let m ∈ N0, λ > 0, ` ≥ 0, ` + 1 > λ and µ ≥ 1. Then Sλ,`,µ+1m (η; ϕ) ⊂ Sλ,`,µm (η; ϕ) ⊂ Sλ,`,µm+1(η; ϕ)
(0 ≤ η < 1; φ ∈ S).
Proof. First of all, we will show that
Sλ,`,µ+1m (η; ϕ) ⊂ Sλ,`,µm (η; ϕ).
Let f ∈ Sλ,`,µ+1m (η; ϕ) and put
(2.1) p(z) = 1
1 − η
z(Iλ,`,µm f (z))0 Iλ,`,µm f (z) − η
! ,
where p(z) is analytic in U with p(0) = 1. Using (1.10) and (2.1), we obtain (2.2) µIλ,`,µ+1m f (z)
Iλ,`,µm f (z) = (1 − η)p(z) + η + (µ − 1).
Differentiating (2.2) logarithmically with respect to z, we obtain (2.3) 1
1 − η
z(Iλ,`,µ+1m f (z))0 Iλ,`,µ+1m f (z) − η
!
= p(z) + zp0(z)
(1 − η)p(z) + η + (µ − 1) (z ∈ U ). Applying Lemma 1 to (2.3), it follows that p ≺ ϕ, that is f ∈ Sλ,`,µm (η; ϕ).
To prove the second part, let f ∈ Sλ,`,µm (η; ϕ) and put h(z) = 1
1 − η
z(Iλ,`,µm+1f (z))0 Iλ,`,µm+1f (z) − η
! ,
where h is analytic in U with h(0) = 1. Then, by using the arguments similar to those detailed above with (1.9), it follows that h ≺ ϕ. This
completes the proof of Theorem 1.
Theorem 2. Let m ∈ N0, λ > 0, ` ≥ 0, ` + 1 > λ and µ ≥ 1. Then Kλ,`,µ+1m (η; ϕ) ⊂ Kλ,`,µm (η; ϕ) ⊂ Kλ,`,µm+1(η; ϕ)
(0 ≤ η < 1; φ ∈ S).
Proof. Applying (1.11) and Theorem 1, we observe that
f ∈ Kλ,`,µ+1m (η; ϕ) ⇔ Iλ,`,µ+1m f (z) ∈ K(η; ϕ) ⇔ z(Iλ,`,µ+1m f (z))0 ∈ S∗(η; ϕ)
⇔ Iλ,`,µ+1m (zf0(z)) ∈ S∗(η; ϕ)
⇔ zf0(z) ∈ Sλ,`,µ+1m (η; ϕ)
⇒ zf0(z) ∈ Sλ,`,µm (η; ϕ)
⇔ Iλ,`,µm (zf0(z)) ∈ Sm(η; ϕ)
⇔ z(Iλ,`,µm (zf (z))0 ∈ Sm(η; ϕ)
⇔ Iλ,`,µm f (z) ∈ K(η; ϕ)
⇔ f (z) ∈ Kλ,`,µm (η; ϕ) and
f (z) ∈ Kλ,`,µm (η; ϕ) ⇔ zf0(z) ∈ S∗(η; ϕ)
⇒ zf0(z) ∈ Sλ,`,µm+1(η; ϕ)
⇔ z(Iλ,`,µm+1f (z))0 ∈ S∗(η; ϕ)
⇔ Iλ,`,µm+1f (z) ∈ K(η; ϕ)
⇔ f (z) ∈ Kλ,`,µm+1(η; ϕ),
which evidently proves Theorem 2.
Taking
ϕ(z) = 1 + Az 1 + Bz
α
(−1 ≤ B < A ≤ 1; 0 < α ≤ 1; z ∈ U ) in Theorem 1 and Theorem 2, we obtain the following corollary.
Corollary 1. Let m ∈ N0, λ > 0, ` ≥ 0, ` + 1 > λ and µ ≥ 1. Then Sλ,`,µ+1m (η; A, B; α) ⊂ Sλ,`,µm (η; A, B; α) ⊂ Sλ,`,µm+1(η; A, B; α) (0 ≤ µ < 1; −1 ≤ B < A ≤ 1; 0 < α ≤ 1), and
Kλ,`,µ+1m (η; A, B; α) ⊂ Kλ,`,µm (η; A, B; α) ⊂ Kλ,`,µm+1(η; A, B; α) (0 ≤ µ < 1; −1 ≤ B < A ≤ 1; 0 < α ≤ 1).
By using Lemma 2, we obtain the following inclusion relation of the class Cλ,`,µm (η, δ; φ, ψ).
Theorem 3. Let m ∈ N0, λ > 0, ` ≥ 0, ` + 1 > λ and µ ≥ 1. Then Cλ,`,µ+1m (η, δ; ϕ, ψ) ⊂ Cλ,`,µm (η, δ; ϕ, ψ) ⊂ Cλ,`,µm+1(η, δ; ϕ, ψ) (0 ≤ η; δ < 1; ϕ, ψ ∈ S).
Proof. We begin by proving that
Cλ,`,µ+1m (η, δ; ϕ, ψ) ⊂ Cλ,`,µm (η, δ; ϕ, ψ).
Let f ∈ Cλ,`,µ+1m (η, δ; ϕ, ψ). Then, in view of the definition of the class Cλ,`,µ+1m (η, δ; ϕ, ψ), there exists a function g ∈ Sλ,`,µ+1m (η; ϕ) such that
1 1 − δ
z(Iλ,`,µ+1m f (z))0 Iλ,`,µ+1m g(z) − δ
!
≺ ψ(z) (z ∈ U ).
Now let
p(z) = 1 1 − δ
z(Iλ,`,µm f (z))0 Iλ,`,µm g(z) − δ
! ,
where p is analytic in U with p(0) = 1. Using the identity (1.10), we obtain (2.4) [(1 − δ)p(z) + δ] Iλ,`,µm g(z) + (µ − 1)Iλ,`,µm f (z) = µIλ,`,µ+1m f (z).
Differentiating (2.4) with respect to z and multiplying by z, we have (2.5) (1 − δ)zp0(z)Iλ,`,µm g(z) + [(1 − δ)p(z) + δ] z(Iλ,`,µm g(z))0
= µz(Iλ,`,µ+1m f (z))0 − (µ − 1)z(Iλ,`,µm f (z))0.
Since g ∈ Sλ,`,µ+1m (η; ϕ), by Theorem 1, we know that g ∈ Smλ,`,µ(η; ϕ). Let q(z) = 1
1 − η
z(Iλ,`,µm g(z))0 Iλ,`,µm g(z) − η
! . Then, using the identity (1.10) once again, we obtain (2.6) µIλ,`,µ+1m g(z)
Iλ,`,µm g(z) = (1 − η)q(z) + η + (µ − 1).
From (2.5) and (2.6), we have 1
1 − δ
z(Iλ,`,µ+1m f (z))0 Iλ,`,µ+1m g(z) − δ
!
= p(z) + zp0(z)
(1 − η)q(z) + η + (µ − 1). Since 0 ≤ η < 1, µ ≥ 1 and q ≺ ϕ in U ,
Re {(1 − η)q(z) + η + µ − 1} > 0
(z ∈ U ). Hence applying Lemma 2, we can show that p ≺ ψ, so that f ∈ Cλ,`,µm (η; δ; ϕ, ψ).
For the second part, by using the arguments similar to those detailed above with (1.9), we obtain
Cλ,`,µm (η, δ; ϕ, ψ) ⊂ Cλ,`,µm+1(η, δ; ϕ, ψ).
This completes the proof of Theorem 3.
3. Inclusion properties involving the integral operator Fc. In this section, we consider the generalized Libera integral operator Fc(see [16], [2]
and [9]) defined by
(3.1) Fc(f ) = Fc(f )(z) = c + 1 zc
z
Z
0
tc−1f (t)dt (c > −1; f ∈ A). We first prove the following theorem.
Theorem 4. Let c, λ ≥ 0, m ∈ N0, ` ≥ 0 and µ > 0. If f ∈ Sλ,`,µm (η; ϕ) (0 ≤ η < 1; ϕ ∈ S), then Fc(f ) ∈ Sλ,`,µm (η; ϕ) (0 ≤ η < 1; ϕ ∈ S).
Proof. Let f ∈ Sλ,`,µm (η; ϕ) and put
(3.2) p(z) = 1
1 − η
z(Iλ,`,µm Fc(f )(z))0 Iλ,`,µm Fc(f )(z) − η
! , where p is analytic in U with p(0) = 1. From (3.1), we have (3.3) z(Iλ,`,µm Fc(f )(z))0 = (c + 1)Iλ,`,µm f (z) − cIλ,`,µm Fc(f )(z).
Then, by using (3.2) and (3.3), we have (3.4) (c + 1) Iλ,`,µm f (z)
Iλ,`,µm Fc(f )(z) = (1 − η)p(z) + η + c.
Differentiating (3.4) logarithmically with respect to z and multiplying by z, we have
p(z) + zp0(z)
(1 − η)p(z) + η + c = 1 1 − η
z(Iλ,`,µm f (z))0 Iλ,`,µm f (z) − η
!
(z ∈ U ).
Hence, by virtue of Lemma 1, we conclude that p ≺ ϕ (z ∈ U ), which
implies that Fc(f ) ∈ Sλ,`,µm (η; ϕ).
Next, we derive an inclusion property involving Fc, which is given by the following theorem.
Theorem 5. Let c, ` ≥ 0, m ∈ N0, λ ≥ 0 and µ > 0. If f ∈ Kλ,`,µm (η; ϕ) (0 ≤ η < 1; ϕ ∈ S), then Fc(f ) ∈ Kλ,`,µm (η; ϕ) (0 ≤ η < 1; ϕ ∈ S).
Proof. By applying Theorem 4, it follows that
f (z) ∈ Kλ,`,µm (η; ϕ) ⇔ zf0(z) ∈ Sλ,`,µm (η; ϕ)
⇒ Fc(zf0(z)) ∈ Sλ,`,µm (η; ϕ)
⇔ z(Fc(f )(z))0 ∈ Sλ,`,µm (η; ϕ)
⇔ Fc(f )(z) ∈ Kλ,`,µm (η; ϕ),
which proves Theorem 5.
From Theorem 4 and Theorem 5, we have the following corollary.
Corollary 2. Let c, ` ≥ 0, m ∈ N0, λ > 0 and µ > 0. If f ∈ Sλ,`,µm (η; A, B; α) (or Kλ,`,µm (η; A, B; α)) (0 ≤ η < 1; −1 ≤ B < A ≤ 1; 0 < α ≤ 1), then Fc(f ) belongs to the class Sλ,`,µm (η; A, B; α) (or Kλ,`,µm (η; A, B; α)) (0 ≤ η < 1;
−1 ≤ B < A ≤ 1; 0 < α ≤ 1).
Finally, we prove the following theorem.
Theorem 6. Let c, ` ≥ 0, m ∈ N0, λ > 0 and µ > 0. If f ∈ Cλ,`,µm (η; δ, ϕ; ψ) (0 ≤ η; δ < 1; ϕ, ψ ∈ S), then Fc(f ) ∈ Cλ,`,µm (η; δ, ϕ; ψ) (0 ≤ η; δ < 1;
ϕ, ψ ∈ S).
Proof. Let f ∈ Cλ,`,µm (η; δ, ϕ; ψ). Then, in view of the definition of the class Cλ,`,µm (η; δ, ϕ; ψ), there exists a function g ∈ Sλ,`,µm (η; ϕ) such that
1 1 − δ
z(Iλ,`,µm f (z))0 Iλ,`,µm g(z) − δ
!
≺ ψ(z) (z ∈ U ).
Thus, we put
p(z) = 1 1 − δ
z(Iλ,`,µm Fc(f )(z))0 Iλ,`,µm Fc(g)(z) − δ
! ,
where p is analytic in U with p(0) = 1. Since g ∈ Sλ,`,µm (η; ϕ), we see from Theorem 4 that Fc(g) ∈ Sλ,`,µm (η; ϕ). Using (3.3), we have
[(1 − δ)p(z) + δ] Iλ,`,µm Fc(g)(z) + cIλ,`,µm Fc(f )(z) = (c + 1)Iλ,`,µm f (z).
Then, by a simple calculations, we get (c + 1)z(Iλ,`,µm f (z))0
Iλ,`,µm Fc(g)(z) = [(1 − δ)p(z) + δ] [(1 − η)q(z) + η + c] + (1 − δ)zp0(z), where
q(z) = 1 1 − η
z(Iλ,`,µm Fc(g)(z))0 Iλ,`,µm Fc(g)(z) − η
! . Hence, we have
1 1 − δ
z(Iλ,`,µm f (z))0 Iλ,`,µm g(z) − δ
!
= p(z) + zp0(z)
(1 − η)q(z) + η + c.
The remaining part of the proof of Theorem 6 is similar to that of Theorem 3
and so we omit it.
Acknowledgments. The authors thank the referees for their valuable sug- gestions to improve the paper.
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M. K. Aouf R. M. El-Ashwah
Math. Dept., Fac. of Sci. Math. Dept., Fac. of Sci.
Mansoura University Mansoura University
Mansoura 35516 Mansoura 35516
Egypt Egypt
e-mail: mkaouf127@yahoo.com e-mail: r elashwah@yahoo.com Received November 1, 2008
