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Some properties of certain multivalent analytic functions involving the Cho–Kwon–Srivastava operator
Mathematical and Computer Modelling 49 (2009) 1969–1984
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Some properties of certain multivalent analytic functions involving the Cho–Kwon–Srivastava operator Zhi-Gang Wang a,b,∗ , R. Aghalary c , M. Darus d , R.W. Ibrahim d a
School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha 410004, Hunan, PR China
b
School of Mathematics and Econometrics, Hunan University, Changsha 410082, Hunan, PR China
c
Department of Mathematics, University of Urmia, Urmia, Iran
d
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor Darul Ehsan, Malaysia
article
a b s t r a c t
info
Article history: Received 8 July 2008 Received in revised form 22 November 2008 Accepted 25 November 2008
By making use of the principle of subordination between analytic functions and the Cho–Kwon–Srivastava operator, we introduce a certain subclass of multivalent analytic functions. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties and sufficient conditions for multivalent starlikeness are proved. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Analytic functions Multivalent functions Differential subordination Superordination Jack’s lemma Hadamard product (or Convolution) Cho–Kwon–Srivastava operator
1. Introduction Let Ap (n) denote the class of functions of the form: f (z ) = z p +
∞ X
ap+k z p+k
(p, n ∈ N := {1, 2, 3, . . .}),
(1.1)
k=n
which are analytic in the open unit disk
U := {z : z ∈ C and
|z | < 1}.
For simplicity, we write
Ap (1) =: Ap and A1 (1) =: A. A function f ∈ Ap (n) is said to be in the class Sp∗,n (%) of p-valent starlike functions of order % in U, if it satisfies the following inequality:
R
zf 0 (z ) f (z )
> % (0 5 % < p ; z ∈ U).
∗ Corresponding author at: School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha 410004, Hunan, PR China. E-mail addresses: [email protected], [email protected] (Z.-G. Wang), [email protected] (R. Aghalary), [email protected] (M. Darus), [email protected] (R.W. Ibrahim). 0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.11.003
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Let H [a, n] be the class of analytic functions of the form: F (z ) = a + an z n + an+1 z n+1 + · · ·
(z ∈ U).
Let f , g ∈ Ap (n), where f is given by (1.1) and g is defined by g (z ) = z p +
∞ X
bp+k z p+k .
k=n
Then the Hadamard product (or convolution) f ∗ g of the functions f and g is defined by
(f ∗ g )(z ) := z p +
∞ X
ap+k bp+k z p+k =: (g ∗ f )(z ).
k=n
For parameters a ∈ R,
(Z− 0 := {0, −1, −2, . . .}).
c ∈ R \ Z− 0
Saitoh [14] introduced a linear operator:
Lp (a, c ) : Ap −→ Ap defined by
Lp (a, c )f (z ) = φp (a, c ; z ) ∗ f (z ) (z ∈ U; f ∈ Ap ), where
φp (a, c ; z ) =
∞ X (a)k k=0
(c )k
z k +p
(1.2)
and (λ)k is the Pochhammer symbol defined by
(λ)k :=
1
λ(λ + 1) · · · (λ + k − 1)
(k = 0), (k ∈ N).
We now introduce the following family of linear operators Ipλ,n (a, c ) analogous to Lp (a, c ):
Ipλ,n (a, c ) : Ap (n) −→ Ap (n), which is defined as
Ipλ,n (a, c )f (z ) := φpĎ (a, c ; z ) ∗ f (z )
a, c ∈ R \ Z− 0 ; λ > −p ; f ∈ Ap (n); z ∈ U ,
(1.3)
Ď
where φp (a, c ; z ) is the function defined in terms of the Hadamard product (or convolution) by the following condition:
φp (a, c ; z ) ∗ φpĎ (a, c ; z ) =
zp
(1 − z )λ+p
.
(1.4)
We can easily find from (1.2)–(1.4) that
Ipλ,n (a, c )f (z ) = z p +
∞ X (λ + p)k (c )k k=n
k!(a)k
ak+p z k+p
(z ∈ U; λ > −p).
(1.5)
Clearly, Ipλ,1 (a, c ) is the well-known Cho–Kwon–Srivastava operator (see [2]). It is also readily verified from (1.5) that z Ipλ,n (a + 1, c )f
0
(z ) = aIpλ,n (a, c )f (z ) − (a − p)Ipλ,n (a + 1, c )f (z ),
(1.6)
and z Ipλ,n (a, c )f
0
1 λ (z ) = (λ + p)Ipλ+ ,n (a, c )f (z ) − λIp,n (a, c )f (z ).
Also by definition and specializing the parameters λ, a, c, we obtain
Ip1,n (p + 1, 1)f (z ) = f (z ),
Ip1,n (p, 1)f (z ) =
zf 0 (z ) p
,
and
Ipλ,1 (a, a)f (z ) = D λ+p−1 f (z ) (λ > −p), where D λ+p−1 is the well-known Ruscheweyh derivative of (λ + p − 1)-th order. For two functions f and g, analytic in U, we say that the function f is subordinate to g in U, and write f (z ) ≺ g (z )
(z ∈ U),
(1.7)
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1971
if there exists a Schwarz function w, which is analytic in U with w(0) = 0
|w(z )| < 1 (z ∈ U)
and
such that f (z ) = g (w(z ))
(z ∈ U).
Indeed, it is known that f (z ) ≺ g (z )
(z ∈ U) H⇒ f (0) = g (0) and f (U) ⊂ g (U).
Furthermore, if the function g is univalent in U, then we have the following equivalence: f (z ) ≺ g (z )
(z ∈ U) ⇐⇒ f (0) = g (0) and f (U) ⊂ g (U).
By making use of the linear operator Ipλ,n (a, c ) and the above-mentioned principle of subordination between analytic functions, we introduce and investigate the following subclass of the class Ap (n) of p-valent analytic functions. α,β
Definition 1. A function f ∈ Ap (n) is said to be in the class Bp,λ (a, c ; n; A, B) if it satisfies the following subordination condition:
(1 − α)
Ipλ,n (a, c )f (z )
!β
1 Ipλ+ ,n (a, c )f (z )
+α
zp
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
zp
!β ≺
1 + Az 1 + Bz
(z ∈ U),
(1.8)
where (and throughout this paper unless otherwise mentioned) the parameters α, β, p, λ, a, c , n, A and B are constrained as follows:
α ∈ C; R(β) > 0; a, c ∈ R \ Z− 0 ; λ > −p; − 1 5 B 5 1; R 3 A 6= B and p, n ∈ N, and the powers are understood as principle values. For simplicity, we write 1,β
B1,0 (1, 1; 1; 1, −1) =: B (β). Clearly, the class B (β) is a subclass of the familiar class of Bazilevič functions of type β . α,β If we set λ = 0 and a = c = p = 1 in the class Bp,λ (a, c ; n; A, B), then it reduces to the class Bn (α, β; A, B), which was studied by Liu [5]. In particular, Zhu [18] determined the sufficient conditions such that Bn (α, β; A, 0) ⊂ Sp∗,n (%). In recent years, Aghalary [1], Patel [11], Patel et al. [12], Sokól and Trojnar-Spelina [16], and Zeng et al. [17] obtained many interesting results associated with the Cho–Kwon–Srivastava operator. In the present paper, we aim at proving such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion α,β theorems, inequality properties and sufficient conditions for multivalent starlikeness of the class Bp,λ (a, c ; n; A, B). The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained. 2. Preliminary results In order to establish our main results, we need the following definition and lemmas. Definition 2 (See [10]). Denote by Q the set of all functions f that are analytic and injective on U − E (f ), where
n
o
E (f ) = ε ∈ ∂ U : lim f (z ) = ∞ , z →ε
and such that f 0 (ε) 6= 0 for ε ∈ ∂ U − E (f ). Lemma 1 (see [9]). Let the function h be analytic and convex (univalent) in U with h(0) = 1. Suppose also that the function k given by k(z ) = 1 + cn z n + cn+1 z n+1 + · · · is analytic in U. If k(z ) +
zk0 (z )
ζ
≺ h(z ) (R(ζ ) > 0; ζ 6= 0; z ∈ U),
then k(z ) ≺ χ (z ) =
ζ n
ζ
z− n
z
Z
ζ
t n −1 h(t )dt ≺ h(z ) 0
and χ(z ) is the best dominant of (2.1).
(z ∈ U),
(2.1)
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Lemma 2 (See [15]). Let q be a convex univalent function in U and let σ , η ∈ C with
R 1+
zq00 (z )
σ > max 0, −R . η
q0 (z )
If the function p is analytic in U and
σ p(z ) + ηz p0 (z ) ≺ σ q(z ) + ηzq0 (z ), then p(z ) ≺ q(z ) and q is the best dominant. Lemma 3 (See [10]). Let q be convex univalent in U and κ ∈ C. Further assume that R(κ) > 0. If p(z ) ∈ H [q(0), 1] ∩ Q ,
and p(z ) + κ z p0 (z ) is univalent in U, then q(z ) + κ zq0 (z ) ≺ p(z ) + κ z p0 (z ) implies q(z ) ≺ p(z ) and q is the best subdominant. Lemma 4 (Jack’s Lemma [4]). Let ω be a non-constant analytic function in U with w(0) = 0. If |ω| attains its maximum value on the circle |z | = r < 1 at z0 , then z0 ω0 (z0 ) = kω(z0 ), where k = 1 is a real number. Lemma 5 (See [6]). Let F be analytic and convex in U. If f , g ∈ A and
f , g ≺ F,
then
λf + (1 − λ)g ≺ F (0 5 λ 5 1). Lemma 6 (See [3]). Let κ, ϑ ∈ C. Suppose also that m is convex and univalent in U with
m(0) = 1 and R(κ m(z ) + ϑ) > 0 (z ∈ U). If u is analytic in U with u(0) = 1, then the following subordination:
u(z ) +
z u0 (z )
κ u(z ) + ϑ
≺ m(z ) (z ∈ U)
implies that
u(z ) ≺ m(z ) (z ∈ U). Lemma 7 (See [13]). Let f (z ) = 1 +
∞ X
ak z k
k=1
be analytic in U and g (z ) = 1 +
∞ X
bk z k
k=1
be analytic and convex in U. If f ≺ g, then
|ak | 5 |b1 |
(k ∈ N).
Lemma 8 (See [7]). Let 0 6= δ ∈ R, νδ > 0, 0 5 ρ < 1, p ∈ H [1, n] and
p(z ) ≺ 1 + Kz
K :=
νM nδ + ν
,
where
(1 − ρ) |δ| 1 + nνδ q M := Mn (δ, ν, ρ) = |1 − δ + ρδ| + 1 + 1 +
.
nδ 2
ν
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1973
If q ∈ H [1, n] satisfies the following subordination condition:
p(z )[1 − δ + δ((1 − ρ)q(z ) + ρ)] ≺ 1 + Mz , then
R(q(z )) > 0 (z ∈ U). 3. Main results We begin by presenting our first subordination property given by Theorem 1 below. α,β
Theorem 1. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. Then
Ipλ,n (a, c )f (z )
!β ≺
zp
(λ + p)β nα
1
Z
1 + Azu 1 + Bzu
0
u
(λ+p)β nα −1 du
≺
1 + Az 1 + Bz
(z ∈ U).
(3.1)
Proof. Define the function P by
Ipλ,n (a, c )f (z )
P(z ) :=
!β (z ∈ U).
zp
(3.2)
Then P is analytic in U with P(0) = 1. By taking the derivatives in the both sides in equality (3.2) and using (1.7), we get
Ipλ,n (a, c )f (z )
(1 − α)
!β +α
zp
= P(z ) +
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
α z P0 (z ) 1 + Az ≺ (λ + p)β 1 + Bz
!β
zp
(z ∈ U).
(3.3)
An application of Lemma 1 to (3.3) yields
Ipλ,n (a, c )f (z )
!β
Z (λ + p)β − (λ+p)β z (λ+p)β −1 1 + At n α ≺ z t nα dt nα 1 + Bt 0 Z 1 + Az (λ + p)β 1 (λ+p)β −1 1 + Azu t nα du ≺ = nα 1 + Bzu 1 + Bz 0
zp
The proof of Theorem 1 is thus completed.
(z ∈ U).
(3.4)
Theorem 2. Let q be univalent in U, 0 6= α ∈ C. Suppose also that q satisfies
R 1+
zq00 (z )
q0 (z )
(λ + p)β > max 0, − R . α
(3.5)
If f ∈ Ap (n) satisfies the following subordination:
Ipλ,n (a, c )f (z )
(1 − α)
!β
zp
+α
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
zp
!β ≺ q(z ) +
α zq0 (z ) , (λ + p)β
(3.6)
then
Ipλ,n (a, c )f (z ) zp
!β ≺ q(z ),
and q is the best dominant. Proof. Let the function P be defined by (3.2). We know that (3.3) holds true. Combining (3.3) and (3.6), we find that
P(z ) +
α z P0 (z ) α zq0 (z ) ≺ q(z ) + . (λ + p)β (λ + p)β
By Lemma 2 and (3.7), we easily get the assertion of Theorem 2.
(3.7)
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Az Taking q(z ) = 11+ in Theorem 2, we get the following result. +Bz Az Corollary 1. Let α ∈ C and −1 5 B < A 5 1. Suppose also that 11+ satisfies the condition (3.5). If f ∈ Ap (n) satisfies the +Bz following subordination:
Ipλ,n (a, c )f (z )
(1 − α) ≺
!β +α
zp
1 + Az
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β
zp
α(A − B)z , (λ + p)β(1 + Bz )2
+
1 + Bz
1 Ipλ+ ,n (a, c )f (z )
then
Ipλ,n (a, c )f (z )
!β ≺
zp
1 + Az 1 + Bz
,
Az and 11+ is the best dominant. +Bz
If f is subordinate to F , then F is superordinate to f . We now derive the following superordination result for the class
α,β
Bp,λ (a, c ; n; A, B). Theorem 3. Let q be convex univalent in U, α ∈ C with R(α) > 0. Also let
Ipλ,n (a, c )f (z )
!β ∈ H [q(0), 1] ∩ Q
zp and
Ipλ,n (a, c )f (z )
(1 − α)
!β +α
zp
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β
zp
be univalent in U. If
Ipλ,n (a, c )f (z ) α zq0 (z ) q(z ) + ≺ (1 − α) (λ + p)β zp
!β +α
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β ,
zp
then
Ipλ,n (a, c )f (z )
q(z ) ≺
!β
zp
,
and q is the best subdominant. Proof. Let the function P be defined by (3.2). Then
Ipλ,n (a, c )f (z ) α zq0 (z ) q(z ) + ≺ (1 − α) (λ + p)β zp
= P(z ) +
!β +α
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
Taking q(z ) =
Ipλ,n (a, c )f (z )
!β
zp
α z P0 (z ) . (λ + p)β
An application of Lemma 3 yields the assertion of Theorem 3. 1+Az 1+Bz
!
in Theorem 3, we get the following corollary.
Corollary 2. Let q be convex univalent in U and −1 5 B < A 5 1, α ∈ C with R(α) > 0. Also let
Ipλ,n (a, c )f (z )
!β ∈ H [q(0), 1] ∩ Q
zp and
(1 − α)
Ipλ,n (a, c )f (z ) zp
!β +α
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z ) zp
!β
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
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be univalent in U. If
Ipλ,n (a, c )f (z ) α(A − B)z + ≺ ( 1 − α) 1 + Bz (λ + p)β(1 + Bz )2 zp
1 + Az
!β +α
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
zp
!β ,
then 1 + Az 1 + Bz
≺
Ipλ,n (a, c )f (z )
!β ,
zp
Az and 11+ is the best subdominant. +Bz
Combining the above results of subordination and superordination. we easily get the following ‘‘Sandwich-type result’’. Corollary 3. Let q1 be convex univalent and let q2 be univalent in U, α ∈ C with R(α) > 0. Let q2 satisfies (3.5). If 0 6=
Ipλ,n (a, c )f (z )
!β ∈ H [q1 (0), 1] ∩ Q ,
zp
and
Ipλ,n (a, c )f (z )
(1 − α)
!β
1 Ipλ+ ,n (a, c )f (z )
+α
zp
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β
zp
is univalent in U, also
Ipλ,n (a, c )f (z ) α zq01 (z ) ≺ (1 − α) q1 (z ) + (λ + p)β zp
≺ q2 (z ) +
!β +α
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
!β
zp
α zq02 (z ) , (λ + p)β
then
Ipλ,n (a, c )f (z )
q1 (z ) ≺
!β
zp
≺ q2 (z ),
and q1 and q2 are, respectively, the best subordinant and dominant. Theorem 4. Let f ∈ Ap (n), ξ ∈ C \ {0} and 0 5 γ < 1. Also let the function ϕ be defined by
z
ϕ(z ) :=
1 Ipλ+ ,n (a,c )f (z )
1 Ipλ+ ,n (a,c )f (z )
Ipλ,n (a,c )f (z )
Ipλ,n (a,c )f (z )
Ipλ,n (a,c )f (z ) zp
Ipλ,n (a,c )f (z )
!0
β
β
zp
−1 (z ∈ U).
(3.8)
−1
If ϕ satisfies one of the following conditions:
1 < R(ξ ) |ξ |2 R(ϕ(z )) 6= 0 > 1 R(ξ ) |ξ |2
(R(ξ ) > 0), (R(ξ ) = 0),
(3.9)
(R(ξ ) < 0),
or
1 > − |ξ |2 =(ξ ) =(ϕ(z )) 6= 0 1 < − 2 =(ξ ) |ξ |
(=(ξ ) > 0), (=(ξ ) = 0), (=(ξ ) < 0),
(3.10)
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
then
ξ ! !β λ 1 I ( a , c ) f ( z ) Ipλ+ ( a , c ) f ( z ) p,n ,n − 1 < 1 − γ . λ (a, c )f (z ) p I z p,n
(3.11)
Proof. We define the function φ by
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
ξ
!β
zp
− 1 = (1 − γ )φ(z )
(3.12)
(0 5 γ < 1; ξ ∈ C \ {0}; z ∈ U) . It is easy to see that the function φ is analytic in U with φ(0) = 0. Differentiating both sides of (3.12) with respect to z logarithmically, we get
z φ 0 (z )
φ(z )
z
=ξ
1 Ipλ+ ,n (a,c )f (z )
1 Ipλ+ ,n (a,c )f (z )
Ipλ,n (a,c )f (z )
Ipλ,n (a,c )f (z ) zp
Ipλ,n (a,c )f (z )
Ipλ,n (a,c )f (z )
!0
β
β
zp
−1 (z ∈ U; ξ ∈ C \ {0}).
(3.13)
−1
We now consider the function ϕ defined by
ϕ :=
ξ z φ 0 (z ) (z ∈ U; ξ ∈ C \ {0}). |ξ |2 φ(z )
(3.14)
Assume that there exists a point z0 ∈ U such that max|z |5|z0 | |φ(z )| = |φ(z0 )| = 1, by Lemma 4, we know that z0 φ 0 (z0 ) = kφ(z0 )
(k = 1).
(3.15)
It follows from (3.14) and (3.15) that
ξ z φ 0 (z0 ) R(ϕ(z0 )) = R |ξ |2 φ(z0 )
!
1 R(ξ ) = |ξ |2 k k R ξ = R (ξ ) = 0 = |ξ |2 |ξ |2 1 5 R(ξ ) |ξ |2
(R(ξ ) > 0), (R(ξ ) = 0),
(3.16)
(R(ξ ) < 0),
and
ξ z φ 0 (z0 ) =(ϕ(z0 )) = = |ξ |2 φ(z0 )
!
1 5 − |ξ |2 =(ξ ) k k = = ξ = − 2 = (ξ ) = 0 2 |ξ | |ξ | 1 = − 2 =(ξ ) |ξ |
(=(ξ ) > 0), (=(ξ ) = 0),
(3.17)
(=(ξ ) < 0).
But the inequalities in (3.16) and (3.17) contradict, respectively, the inequalities in (3.9) and (3.10). Therefore, we can conclude that
|φ(z )| < 1 (z ∈ U), which implies that
ξ ! !β 1 λ Ipλ+ ( a , c ) f ( z ) I ( a , c ) f ( z ) ,n p,n − 1 = (1 − γ ) |φ(z )| < 1 − γ . λ p Ip,n (a, c )f (z ) z We thus complete the proof of Theorem 4.
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1977
From Theorem 4, we easily get the following result for the class B (β) of Bazilevič functions of type β . Corollary 4. Let f ∈ A, λ = 0, p = a = c = ξ = 1 and γ = 0. Also let the function ϕ be defined by (3.8). If ϕ satisfies one of the following conditions:
R(ϕ(z )) < 1 or
=(ϕ(z )) 6= 0,
then f ∈ B (β). 0,β
α,β
Theorem 5. Let R(α) > 0, β > 0 and f ∈ Bp,λ (a, c ; n; 1 − 2ρ, −1) (0 5 ρ < 1). Then f ∈ Bp,λ (a, c ; n; 1 − 2ρ, −1) for |z | < R(α, β, λ, p), where
−α +
R(α, β, λ, p) =
α 2 + (λ + p)2 β 2 . (λ + p)β
p
(3.18)
The bound R(α, β, λ, p) is the best possible. Proof. Suppose that
Ipλ,n (a, c )f (z )
!β = ρ + (1 − ρ)h(z ) (z ∈ U; 0 5 ρ < 1),
zp
(3.19)
where h is analytic and has a positive real part in U. By taking the derivatives in the both sides in equality (3.19) and using (1.7), we get
Ipλ,n (a, c )f (z )
R (1 − α)
!β
1 Ipλ+ ,n (a, c )f (z )
+α
zp
α zh0 (z ) = (1 − ρ)R h(z ) + (λ + p)β
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β −ρ
zp
! α zh0 (z ) = (1 − ρ)R h(z ) − . (λ + p)β
(3.20)
By making use of the following well-known estimate (see [8]):
0 zh (z ) 5
2r 1 − r2
R(h(z )) (|z | = r < 1)
in (3.20), we obtain that
Ipλ,n (a, c )f (z )
R (1 − α)
= (1 − ρ)
!β +α
zp
!
Ipλ,n (a, c )f (z )
2α r
1−
1 Ipλ+ ,n (a, c )f (z )
(λ + p)β(1 − r 2 )
Ipλ,n (a, c )f (z ) zp
!β
− ρ
R (h(z )) > 0
(3.21)
for r < R(α, β, λ, p), where R(α, β, λ, p) is given by (3.18). To show that the bound R(α, β, λ, p) is the best possible, we consider the function f ∈ Ap (n) defined by
Ipλ,n (a, c )f (z )
!β
zp
= ρ + (1 − ρ)
1+z
(z ∈ U).
1−z
By noting that
R (1 − α)
= (1 − ρ)R
Ipλ,n (a, c )f (z ) zp
1+z 1−z
+
!β
1 Ipλ+ ,n (a, c )f (z )
+α 2α
(λ + p)β
Ipλ,n (a, c )f (z )
·
z
(1 − z )2
!
Ipλ,n (a, c )f (z ) zp
!β −ρ
=0
for z = R(α, β, λ, p), we conclude that the bound is the best possible. Theorem 5 is thus proved.
(3.22)
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
α,β
Theorem 6. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. Then f (z ) =
z
p
1 + Aω(z )
β1 !
p
∗ z +
1 + Bω(z )
!
∞ X
k!(a)k
k=n
(λ + p)k (c )k
z
,
k+p
(3.23)
where ω is analytic in U with
ω(0) = 0 and
|ω(z )| < 1 (z ∈ U). α,β
Proof. Suppose that f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. It follows from (3.1) that
Ipλ,n (a, c )f (z )
!β =
zp
1 + Aω(z ) 1 + Bω(z )
,
(3.24)
where ω is analytic in U with
ω(0) = 0 and
|ω(z )| < 1 (z ∈ U).
By virtue of (3.24), we easily find that
Ipλ,n (a, c )f (z ) = z p
1 + Aω(z )
β1
1 + Bω(z )
.
(3.25)
Combining (1.3), (1.5) and (3.25), we have p
z +
∞ X (λ + p)k (c )k
k!(a)k
k =n
! z
∗ f (z ) = z
k+p
p
1 + Aω(z )
β1
1 + Bω(z )
.
(3.26)
The assertion (3.23) of Theorem 6 can now easily be derived from (3.26). Theorem 7. Let f ∈ 1 z
1 + Beiθ
α,β Bp,λ (a, c ; A, B)
β1
zp +
∞ P k=n
with R(α) > 0. Then
(λ+p)k (c )k k+p z k!(a)k
∗ f (z ) − z p 1 + Aeiθ
β1
6= 0 (z ∈ U; 0 < θ < 2π ).
(3.27)
α,β
Proof. Suppose that f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. We know that (3.1) holds true, which implies that
Ipλ,n (a, c )f (z )
!β 6=
zp
1 + Aeiθ 1 + Beiθ
(z ∈ U; 0 < θ < 2π ).
(3.28)
It is easy to see that the condition (3.28) can be written as follows: 1 z
Ipλ,n (a, c )f (z )
1 + Be
iθ
β1
−z
p
1 + Ae
iθ
β1
6= 0 (z ∈ U; 0 < θ < 2π ).
(3.29)
Combining (1.3), (1.5) and (3.29), we easily get the convolution property (3.27) asserted by Theorem 7.
Theorem 8. Let α2 = α1 = 0 and −1 5 B1 5 B2 < A2 5 A1 5 1. Then α ,β
α ,β
2 Bp,λ (a, c ; n; A2 , B2 ) ⊂ Bp,λ1 (a, c ; n; A1 , B1 ).
(3.30)
α ,β
2 Proof. Suppose that f ∈ Bp,λ (a, c ; n; A2 , B2 ). We know that
(1 − α2 )
Ipλ,n (a, c )f (z )
!β + α2
zp
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β ≺
zp
1 + A2 z 1 + B2 z
.
Since −1 5 B1 5 B2 < A2 5 A1 5 1, we easily find that
(1 − α2 ) α ,β
Ipλ,n (a, c )f (z ) zp
!β + α2
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z ) zp
!β ≺
1 + A2 z 1 + B2 z
2 that is f ∈ Bp,λ (a, c ; n; A1 , B1 ). Thus the assertion of Theorem 8 holds for α2 = α1 = 0.
≺
1 + A1 z 1 + B1 z
,
(3.31)
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1979
0,β
If α2 > α1 = 0, by Theorem 1 and (3.31), we know that f ∈ Bp,λ (a, c ; n; A1 , B1 ), that is
Ipλ,n (a, c )f (z )
!β ≺
zp
1 + A1 z 1 + B1 z
.
(3.32)
At the same time, we have
Ipλ,n (a, c )f (z )
(1 − α1 )
!β + α1
zp
α1 = 1− α2
Ipλ,n (a, c )f (z )
zp
1 Ipλ+ ,n (a, c )f (z )
+ α2
!β
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z ) α1 (1 − α2 ) + α2 zp
Ipλ,n (a, c )f (z )
zp
!β
zp
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
!
!β
!β .
(3.33)
Moreover, since 05
α1 <1 α2
and h1 (z ) :=
1 + A1 z 1 + B1 z
(z ∈ U)
is analytic and convex in U. Combining (3.31)–(3.33) and Lemma 5, we find that
(1 − α1 )
Ipλ,n (a, c )f (z )
!β + α1
zp
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
zp
α ,β
!β ≺
1 + A1 z 1 + B1 z
,
1 that is f ∈ Bp,λ (a, c ; n; A1 , B1 ), which implies that the assertion (3.30) of Theorem 8 holds.
Let P denote the class of functions of the following form:
t(z ) = 1 +
∞ X
(n ∈ N),
tk z k
k=n
which are analytic and convex in U and satisfy the following condition:
R(t(z )) > 0 (z ∈ U). By making use of the principle of subordination between analytic functions, we introduce the subclasses Sp∗,n (µ; φ) and Cp,n (µ; φ) of the class Ap (n):
Sp∗,n (µ; φ) := f ∈ Ap (n) :
1
p−µ
zf 0 (z ) f (z )
− µ ≺ φ(z )
(φ ∈ P ; 0 5 µ < p; z ∈ U)
and
Cp,n (µ; φ) := f ∈ Ap (n) :
1 p−µ
1+
zf 00 (z ) f 0 (z )
− µ ≺ φ(z )
(φ ∈ P ; 0 5 µ < p; z ∈ U) .
Next, by using the operator defined by (1.5), we define the following two subclasses Spλ,n (a, c ; µ; φ) and Cpλ,n (a, c ; µ; φ) of the class Ap (n):
Spλ,n (a, c ; µ; φ) := f ∈ Ap (n) : Ipλ,n (a, c )f ∈ Sp∗,n (µ; φ)
and
Cpλ,n (a, c ; µ; φ) := f ∈ Ap (n) : Ipλ,n (a, c )f ∈ Cp,n (µ; φ) .
Clearly, we know that f ∈ Cpλ,n (a, c , µ; φ) ⇐⇒
zf 0 p
∈ Spλ,n (a, c , µ; φ).
(3.34)
We now derive some inclusion relationships for the classes Spλ,n (a, c ; µ; φ) and Cpλ,n (a, c ; µ; φ). By similarly applying the method of proof of Proposition 1 obtained by Cho et al. [2], we easily get Theorem 9. But the conditions of Theorem 9 are much weaker than the conditions of the corresponding result obtained by Cho et al. [2].
1980
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Theorem 9. Let 0 5 µ < p, λ > −p and φ ∈ P with
λ+µ a+µ−p R (φ(z )) > max 0, − , − p−µ p−µ
(z ∈ U).
(3.35)
Then 1 λ λ Spλ+ ,n (a, c ; µ; φ) ⊂ Sp,n (a, c ; µ; φ) ⊂ Sp,n (a + 1, c ; µ; φ).
Theorem 10. Let 0 5 µ < p, λ > −p and φ ∈ P with (3.35) holds. Then 1 λ λ Cpλ+ ,n (a, c ; µ; φ) ⊂ Cp,n (a, c ; µ; φ) ⊂ Cp,n (a + 1, c ; µ; φ).
Proof. By virtue of (3.34) and Theorem 9, we observe that 1 f ∈ Cpλ+ ,n (a, c ; µ; φ)
⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒
1 Ipλ+ Cp,n (µ; φ) ,n (a, c )f ∈ 0 1 z Ipλ+ ( a , c ) f ,n ∈ Sp∗,n (µ; φ)
p
1 Ipλ+ ,n (a, c )
zf 0 p zf 0
zf 0 p
∈ Sp∗,n (µ; φ)
1 ∈ Spλ+ ,n (a, c ; µ; φ)
⇐⇒
∈ Spλ,n (a, c ; µ; φ) 0 zf λ ∈ Sp∗,n (µ; φ) Ip,n (a, c ) p 0 z Ipλ,n (a, c )f ∈ Sp∗,n (µ; φ)
⇐⇒ ⇐⇒
Ipλ,n (a, c )f ∈ Cp,n (µ; φ) f ∈ Cpλ,n (a, c ; µ; φ),
H⇒ ⇐⇒
(3.36)
p
p
and f ∈ Cpλ,n (a, c ; µ; φ)
⇐⇒
zf 0 p zf 0
∈ Spλ,n (a, c ; µ; φ)
⇐⇒
∈ Spλ,n (a + 1, c ; µ; φ) p 0 z Ipλ,n (a + 1, c )f ∗
⇐⇒ ⇐⇒
Ipλ,n (a + 1, c )f ∈ Cp,n (µ; φ) f ∈ Cpλ,n (a + 1, c ; µ; φ).
H⇒
(3.37)
∈ Sp,n (µ; φ)
p
From (3.36) and (3.37), we conclude that the assertion of Theorem 10 holds true. Taking φ(z ) =
1+Az 1+Bz
in Theorems 9 and 10, we get the following results.
Corollary 5. Let 0 5 µ < p, λ > −p and −1 5 B < A 5 1. Then
1 + Az
1 Spλ+ a, c ; µ; ,n
1 + Az 1 + Az ⊂ Spλ,n a, c ; µ; ⊂ Spλ,n a + 1, c ; µ; 1 + Bz 1 + Bz
1 + Az 1 + Az ⊂ Cpλ,n a, c ; µ; ⊂ Cpλ,n a + 1, c ; µ; . 1 + Bz 1 + Bz
1 + Bz
and
1 Cpλ+ a, c ; µ; ,n
1 + Az 1 + Bz α,β
Theorem 11. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 B < A 5 1. Then
(λ + p)β nα
1
Z 0
1 − Au 1 − Bu
(λ+p)β u nα −1 du
Ipλ,n (a, c )f (z ) zp
!β <
(λ + p)β nα
1
Z 0
1 + Au 1 + Bu
u
(λ+p)β nα −1 du
.
(3.38)
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1981
The extremal function of (3.38) is defined by
Ipλ,n (a, c )Fα,β,A,B (z )
=z
p
(λ + p)β nα
1
Z 0
1 + Azu 1 + Bzu
(λ+p)β u nα −1 du
β1
.
(3.39)
α,β
Proof. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. From Theorem 1, we know that (3.1) holds, which implies that
Ipλ,n (a, c )f (z )
R
!β
zp
Z (λ + p)β 1 1 + Azu (λ+p)β −1 n α < sup R u du nα z ∈U 0 1 + Bzu Z (λ+p)β (λ + p)β 1 1 + Azu 5 sup R u nα −1 du nα 1 + Bzu z ∈ U 0 Z (λ + p)β 1 1 + Au (λ+p)β −1 < u nα du, nα 0 1 + Bu
(3.40)
and
Ipλ,n (a, c )f (z )
R
!β
Z (λ + p)β 1 1 + Azu (λ+p)β −1 u nα du z ∈U nα 0 1 + Bzu Z 1 (λ+p)β (λ + p)β 1 + Azu = inf R u nα −1 du z ∈ U nα 1 + Bzu 0 Z 1 1 − Au (λ+p)β −1 (λ + p)β du. u nα > nα 0 1 − Bu > inf R
zp
(3.41)
Combining (3.40) and (3.41), we get (3.38). By noting that the function Ipλ,n (a, c )Fα,β,A,B (z ) defined by (3.39) belongs to the α,β
class Bp,λ (a, c ; n; A, B), we obtain that equality (3.38) is sharp. The proof of Theorem 11 is evidently completed.
By similarly applying the method of proof of Theorem 11, we easily get the following result. α,β
Corollary 6. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 A < B 5 1. Then
(λ + p)β nα
1
Z 0
1 + Au 1 + Bu
(λ+p)β u nα −1 du
Ipλ,n (a, c )f (z ) zp
!β <
(λ + p)β nα
1
Z 0
1 − Au 1 − Bu
u
(λ+p)β nα −1 du
.
(3.42)
The extremal function of (3.42) is defined by (3.39). α,β
In view of Theorem 11 and Corollary 6, we easily derive the following distortion theorems for the class Bp,λ (a, c ; n; A, B). α,β
Corollary 7. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 B < A 5 1. Then for |z | = r < 1, we have
r
p
(λ + p)β nα
1
Z
1 − Aur 1 − Bur
0
(λ+p)β u nα −1 du
β1
< Ipλ,n (a, c )f (z ) < r p
(λ + p)β nα
1
Z
1 + Aur 1 + Bur
0
(λ+p)β u nα −1 du
β1
.
(3.43)
.
(3.44)
The extremal function of (3.43) is defined by (3.39). α,β
Corollary 8. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 A < B 5 1. Then for |z | = r < 1, we have
r
p
(λ + p)β nα
1
Z 0
1 + Aur 1 + Bur
(λ+p)β u nα −1 du
β1
< Ipλ,n (a, c )f (z ) < r p
The extremal function of (3.44) is defined by (3.39). By noting that
1 1 1 (R(υ)) 2 5 R υ 2 5 |υ| 2
(υ ∈ C; R(υ) = 0) .
(λ + p)β nα
1
Z 0
1 − Aur 1 − Bur
(λ+p)β u nα −1 du
β1
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Form Theorem 11 and Corollary 6, we easily get the following results. α,β
Corollary 9. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 B < A 5 1. Then
(λ + p)β nα
1
Z
1 − Au 1 − Bu
0
(λ+p)β u nα −1 du
12
Ipλ,n (a, c )f (z )
! β2 <
zp
(λ + p)β nα
Z
1
1 + Au 1 + Bu
0
(λ+p)β u nα −1 du
12
.
α,β
Corollary 10. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 A < B 5 1. Then
(λ + p)β nα
1
Z
1 + Au 1 + Bu
0
(λ+p)β u nα −1 du
12
Ipλ,n (a, c )f (z )
! β2
zp
<
(λ + p)β nα
Z 0
1
1 − Au 1 − Bu
(λ+p)β u nα −1 du
12
.
Theorem 12. Let f (z ) = z p +
∞ X
α,β
ap+k z p+k ∈ Bp,λ (a, c ; n; A, B).
(3.45)
k=n
Then
ap+n 5 (λ + p)n!(a)n (λ + p)n (c )n
A−B (λ + p)β + nα .
(3.46)
The inequality (3.46) is sharp, with the extremal function defined by (3.39). Proof. Combining (1.8) and (3.45), we obtain
(1 − α)
Ipλ,n (a, c )f (z )
!β +α
zp
=1+ 1+
1 Ipλ+ ,n (a, c )f (z )
nα
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β
zp
(λ + p)n (c )n 1 + Az β ap+n z n + · · · ≺ . n!(a)n 1 + Bz
(λ + p)β
(3.47)
An application of Lemma 7 to (3.47) yields
nα (λ + p)n (c )n 1+ β a p+n 5 |A − B| . (λ + p)β n!(a)n
(3.48)
Thus, from (3.48), we easily arrive at (3.46) asserted by Theorem 12. Theorem 13. Let 0 6= α ∈ R, β ∈ R,
α,β
β α
> 0, λ > −p and 0 5 ρ < 1. If f ∈ Bp,λ (a, c ; n; A, 0) with (1 − ρ) |α| 1 + (λ+nαp)β A = An (α, β, λ, ρ, p) = r 2 , nα |1 − α + ρα| + 1 + 1 + (λ+p)β
then
Ipλ,n (a, c )f ∈ Sp∗,n (ρ p − (1 − ρ)λ). α,β
Proof. Suppose that f ∈ Bp,λ (a, c ; n; A, 0). By definition, we have
(1 − α)
Ipλ,n (a, c )f (z )
!β +α
zp
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
zp
!β ≺ 1 + Az (z ∈ U).
(3.49)
Let the function P be defined by (3.2). We then find from (3.1) and (3.49) that
P(z ) ≺
(λ + p)β − (λ+p)β nα z nα
Z
z
(1 + At )t
(λ+p)β nα −1 dt
=1+
0
(λ + p)β A z. nα + (λ + p)β
We now suppose that 1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
= (1 − ρ)q(z ) + ρ
(0 5 ρ < 1;
z ∈ U) .
(3.50)
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1983
Then q ∈ H [1, n]. It follows from (3.49) and (3.50) that
P(z ){(1 − α) + α[(1 − ρ)q(z ) + ρ]} ≺ 1 + Az
(z ∈ U).
(3.51)
An application of Lemma 8 to (3.51) yields
R(q(z )) > 0 (z ∈ U).
(3.52)
Combining (3.50) and (3.52), we find that 1 Ipλ+ ,n (a, c )f (z )
R
!
Ipλ,n (a, c )f (z )
= (1 − ρ)R(q(z )) + ρ > ρ (z ∈ U).
(3.53)
The assertion of Theorem 13 can now easily be derived from (1.7) and (3.53).
α,β
Theorem 14. Let f ∈ Bp,λ (a, c ; n; A, 0) with
β > 0,
(λ + p)β R (α) > 0 and |α| n + R α
A > 0,
> A(λ + p)β.
Then
h i z Iλ (a, c )f 0 (z ) A(λ + p) |α| n + R (λ+αp)β + (λ + p)β p,n h i . − p < Ipλ,n (a, c )f (z ) |α| |α| n + R (λ+αp)β − A(λ + p)β Proof. Let the function P be defined by (3.2). It follows from (3.3) that
P(z ) +
α z P0 (z ) = 1 + A (z ), (λ + p)β
(3.54)
where
(z ) =
∞ X
kz
k
(n ∈ N)
k=n
is analytic in U with | (z )| < 1 (z ∈ U). From (3.54), we easily get
P(z ) = 1 + A
(λ + p)β α
1
Z
t
(λ+p)β −1 α
(tz )dt = 1 + A
0
∞ (λ + p)β X 1 (λ+ p)β α k=n k + α
kz
k
.
(3.55)
It follows from (3.55) that
(z P(z ))0 = 1 + A
∞ (λ + p)β X k+1 (λ+p)β α k=n k + α
kz
∞ (λ + p)β X 1 = 1+A (λ+p)β α k=n k + α
k
(λ + p)β kz + A α k
(λ + p)β (z ) − α
1
Z
t
(λ+p)β −1 α
(tz )dt .
(3.56)
0
We next find from (3.55) and (3.56) that
(λ + p)β z P (z ) = A α 0
(λ + p)β (z ) − α
1
Z
t
(λ+p)β −1 α
(tz )dt .
(3.57)
0
Combining (3.55) and (3.57), we get
h i 0 A(λ + p)β |α| n + R (λ+p)β + (λ + p)β z P (z ) α h i . P(z ) < (λ+p)β |α| |α| n + R − A (λ + p )β α Thus, from (3.2) and (3.58), we easily arrive at the assertion of Theorem 14.
(3.58)
Remark. By specializing the parameters α, β, p, λ, a, c and n in Theorems 13 and 14, we can get some criteria for starlikeness. Here, we choose to omit the details involved.
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Acknowledgements The present investigation was supported by the Scientific Research Fund of Hunan Provincial Education Department under Grant 08C118 of People’s Republic of China, and also, by SAGA Grant: STGL-012-2006, Academy of Science, Malaysia. The authors would like to thank the referees for their careful reading and making some valuable comments which have essentially improved the presentation of this paper. The first-named author would also like to thank Professors Chun-Yi Gao, Yue-Ping Jiang, Qing-Guo Li and Ming-Sheng Liu for their support and encouragement. References [1] R. Aghalary, On subclasses of p-valent analytic functions defined by integral operators, Kyungpook Math. J. 47 (2007) 393–401. [2] N.E. Cho, O.S. Kwon, H.M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl. 292 (2004) 470–483. [3] P. Eenigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Briot–Bouquet differential subordination, in: General Mathematics 3, in: International Series of Numerical Mathematics, vol. 64, Birkhäuser Verlag, Basel, 1983, pp. 339–348; see also Rev. Roumaine Math. Pures Appl. 29 (1984) 567–573. [4] I.S. Jack, Functions starlike and convex of order α , J. London Math. Soc. 3 (1971) 469–474. [5] M.-S. Liu, On certain class of analytic functions defined by differential subordination, Acta Math. Sci. Ser. B Engl. Ed. 22 (2002) 388–392. [6] M.-S. Liu, On certain subclass of analytic functions, J. South China Normal Univ. 4 (2002) 15–20 (in Chinese). [7] M.-S. Liu, On the starlikeness for certain subclass of analytic functions and a problem of Ruscheweyh, Adv. Math. (China) 34 (2005) 416–424. [8] T.H. Macgregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14 (1963) 514–520. [9] S.S. Miller, P.T. Mocanu, Differential Subordination: Theory and Applications, in: Series in Pure and Applied Mathematics, vol. 225, Marcel Dekker, New York, 2000. [10] S.S. Miller, P.T. Mocanu, Subordinats of differential superordinations, Complex Var. 48 (2003) 815–826. [11] J. Patel, On certain subclasses of multivalent functions involving Cho–Kwon–Srivastava operator, Ann. Univ. Mariae Curie-Skłodowska Sect. A 60 (2006) 75–86. [12] J. Patel, N.E. Cho, H.M. Srivastava, Certain subclasses of multivalent functions associated with a family of linear operators, Math. Comput. Modelling 43 (2006) 320–338. [13] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (Ser. 2) 48 (1943) 48–82. [14] H. Saitoh, A linear operator and its applications of first order differential subordinations, Math. Japon. 44 (1996) 31–38. [15] T.N. Shanmugam, V. Ravichandran, S. Sivasubramanian, Differential Sandwich theorems for subclasses of analytic functions, Australian J. Math. Anal. Appl. 3 (2006) 1–11. Article 8 (electronic). [16] J. Sokól, L. Trojnar-Spelina, Convolution properties for certain classes of multivalent functions, J. Math. Anal. Appl. 337 (2008) 1190–1197. [17] T. Zeng, C.-Y. Gao, Z.-G. Wang, R. Aghalary, Certain subclass of multivalent functions involving the Cho–Kwon–Srivastava operator, J. Math. Appl. 30 (2008) 161–170. [18] Y.-C. Zhu, Some starlikeness criterions for analytic functions, J. Math. Anal. Appl. 335 (2007) 1452–1459.
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Some properties of certain multivalent analytic functions involving the Cho–Kwon–Srivastava operator Zhi-Gang Wang a,b,∗ , R. Aghalary c , M. Darus d , R.W. Ibrahim d a
School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha 410004, Hunan, PR China
b
School of Mathematics and Econometrics, Hunan University, Changsha 410082, Hunan, PR China
c
Department of Mathematics, University of Urmia, Urmia, Iran
d
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor Darul Ehsan, Malaysia
article
a b s t r a c t
info
Article history: Received 8 July 2008 Received in revised form 22 November 2008 Accepted 25 November 2008
By making use of the principle of subordination between analytic functions and the Cho–Kwon–Srivastava operator, we introduce a certain subclass of multivalent analytic functions. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties and sufficient conditions for multivalent starlikeness are proved. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Analytic functions Multivalent functions Differential subordination Superordination Jack’s lemma Hadamard product (or Convolution) Cho–Kwon–Srivastava operator
1. Introduction Let Ap (n) denote the class of functions of the form: f (z ) = z p +
∞ X
ap+k z p+k
(p, n ∈ N := {1, 2, 3, . . .}),
(1.1)
k=n
which are analytic in the open unit disk
U := {z : z ∈ C and
|z | < 1}.
For simplicity, we write
Ap (1) =: Ap and A1 (1) =: A. A function f ∈ Ap (n) is said to be in the class Sp∗,n (%) of p-valent starlike functions of order % in U, if it satisfies the following inequality:
R
zf 0 (z ) f (z )
> % (0 5 % < p ; z ∈ U).
∗ Corresponding author at: School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha 410004, Hunan, PR China. E-mail addresses: [email protected], [email protected] (Z.-G. Wang), [email protected] (R. Aghalary), [email protected] (M. Darus), [email protected] (R.W. Ibrahim). 0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.11.003
1970
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Let H [a, n] be the class of analytic functions of the form: F (z ) = a + an z n + an+1 z n+1 + · · ·
(z ∈ U).
Let f , g ∈ Ap (n), where f is given by (1.1) and g is defined by g (z ) = z p +
∞ X
bp+k z p+k .
k=n
Then the Hadamard product (or convolution) f ∗ g of the functions f and g is defined by
(f ∗ g )(z ) := z p +
∞ X
ap+k bp+k z p+k =: (g ∗ f )(z ).
k=n
For parameters a ∈ R,
(Z− 0 := {0, −1, −2, . . .}).
c ∈ R \ Z− 0
Saitoh [14] introduced a linear operator:
Lp (a, c ) : Ap −→ Ap defined by
Lp (a, c )f (z ) = φp (a, c ; z ) ∗ f (z ) (z ∈ U; f ∈ Ap ), where
φp (a, c ; z ) =
∞ X (a)k k=0
(c )k
z k +p
(1.2)
and (λ)k is the Pochhammer symbol defined by
(λ)k :=
1
λ(λ + 1) · · · (λ + k − 1)
(k = 0), (k ∈ N).
We now introduce the following family of linear operators Ipλ,n (a, c ) analogous to Lp (a, c ):
Ipλ,n (a, c ) : Ap (n) −→ Ap (n), which is defined as
Ipλ,n (a, c )f (z ) := φpĎ (a, c ; z ) ∗ f (z )
a, c ∈ R \ Z− 0 ; λ > −p ; f ∈ Ap (n); z ∈ U ,
(1.3)
Ď
where φp (a, c ; z ) is the function defined in terms of the Hadamard product (or convolution) by the following condition:
φp (a, c ; z ) ∗ φpĎ (a, c ; z ) =
zp
(1 − z )λ+p
.
(1.4)
We can easily find from (1.2)–(1.4) that
Ipλ,n (a, c )f (z ) = z p +
∞ X (λ + p)k (c )k k=n
k!(a)k
ak+p z k+p
(z ∈ U; λ > −p).
(1.5)
Clearly, Ipλ,1 (a, c ) is the well-known Cho–Kwon–Srivastava operator (see [2]). It is also readily verified from (1.5) that z Ipλ,n (a + 1, c )f
0
(z ) = aIpλ,n (a, c )f (z ) − (a − p)Ipλ,n (a + 1, c )f (z ),
(1.6)
and z Ipλ,n (a, c )f
0
1 λ (z ) = (λ + p)Ipλ+ ,n (a, c )f (z ) − λIp,n (a, c )f (z ).
Also by definition and specializing the parameters λ, a, c, we obtain
Ip1,n (p + 1, 1)f (z ) = f (z ),
Ip1,n (p, 1)f (z ) =
zf 0 (z ) p
,
and
Ipλ,1 (a, a)f (z ) = D λ+p−1 f (z ) (λ > −p), where D λ+p−1 is the well-known Ruscheweyh derivative of (λ + p − 1)-th order. For two functions f and g, analytic in U, we say that the function f is subordinate to g in U, and write f (z ) ≺ g (z )
(z ∈ U),
(1.7)
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1971
if there exists a Schwarz function w, which is analytic in U with w(0) = 0
|w(z )| < 1 (z ∈ U)
and
such that f (z ) = g (w(z ))
(z ∈ U).
Indeed, it is known that f (z ) ≺ g (z )
(z ∈ U) H⇒ f (0) = g (0) and f (U) ⊂ g (U).
Furthermore, if the function g is univalent in U, then we have the following equivalence: f (z ) ≺ g (z )
(z ∈ U) ⇐⇒ f (0) = g (0) and f (U) ⊂ g (U).
By making use of the linear operator Ipλ,n (a, c ) and the above-mentioned principle of subordination between analytic functions, we introduce and investigate the following subclass of the class Ap (n) of p-valent analytic functions. α,β
Definition 1. A function f ∈ Ap (n) is said to be in the class Bp,λ (a, c ; n; A, B) if it satisfies the following subordination condition:
(1 − α)
Ipλ,n (a, c )f (z )
!β
1 Ipλ+ ,n (a, c )f (z )
+α
zp
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
zp
!β ≺
1 + Az 1 + Bz
(z ∈ U),
(1.8)
where (and throughout this paper unless otherwise mentioned) the parameters α, β, p, λ, a, c , n, A and B are constrained as follows:
α ∈ C; R(β) > 0; a, c ∈ R \ Z− 0 ; λ > −p; − 1 5 B 5 1; R 3 A 6= B and p, n ∈ N, and the powers are understood as principle values. For simplicity, we write 1,β
B1,0 (1, 1; 1; 1, −1) =: B (β). Clearly, the class B (β) is a subclass of the familiar class of Bazilevič functions of type β . α,β If we set λ = 0 and a = c = p = 1 in the class Bp,λ (a, c ; n; A, B), then it reduces to the class Bn (α, β; A, B), which was studied by Liu [5]. In particular, Zhu [18] determined the sufficient conditions such that Bn (α, β; A, 0) ⊂ Sp∗,n (%). In recent years, Aghalary [1], Patel [11], Patel et al. [12], Sokól and Trojnar-Spelina [16], and Zeng et al. [17] obtained many interesting results associated with the Cho–Kwon–Srivastava operator. In the present paper, we aim at proving such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion α,β theorems, inequality properties and sufficient conditions for multivalent starlikeness of the class Bp,λ (a, c ; n; A, B). The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained. 2. Preliminary results In order to establish our main results, we need the following definition and lemmas. Definition 2 (See [10]). Denote by Q the set of all functions f that are analytic and injective on U − E (f ), where
n
o
E (f ) = ε ∈ ∂ U : lim f (z ) = ∞ , z →ε
and such that f 0 (ε) 6= 0 for ε ∈ ∂ U − E (f ). Lemma 1 (see [9]). Let the function h be analytic and convex (univalent) in U with h(0) = 1. Suppose also that the function k given by k(z ) = 1 + cn z n + cn+1 z n+1 + · · · is analytic in U. If k(z ) +
zk0 (z )
ζ
≺ h(z ) (R(ζ ) > 0; ζ 6= 0; z ∈ U),
then k(z ) ≺ χ (z ) =
ζ n
ζ
z− n
z
Z
ζ
t n −1 h(t )dt ≺ h(z ) 0
and χ(z ) is the best dominant of (2.1).
(z ∈ U),
(2.1)
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Lemma 2 (See [15]). Let q be a convex univalent function in U and let σ , η ∈ C with
R 1+
zq00 (z )
σ > max 0, −R . η
q0 (z )
If the function p is analytic in U and
σ p(z ) + ηz p0 (z ) ≺ σ q(z ) + ηzq0 (z ), then p(z ) ≺ q(z ) and q is the best dominant. Lemma 3 (See [10]). Let q be convex univalent in U and κ ∈ C. Further assume that R(κ) > 0. If p(z ) ∈ H [q(0), 1] ∩ Q ,
and p(z ) + κ z p0 (z ) is univalent in U, then q(z ) + κ zq0 (z ) ≺ p(z ) + κ z p0 (z ) implies q(z ) ≺ p(z ) and q is the best subdominant. Lemma 4 (Jack’s Lemma [4]). Let ω be a non-constant analytic function in U with w(0) = 0. If |ω| attains its maximum value on the circle |z | = r < 1 at z0 , then z0 ω0 (z0 ) = kω(z0 ), where k = 1 is a real number. Lemma 5 (See [6]). Let F be analytic and convex in U. If f , g ∈ A and
f , g ≺ F,
then
λf + (1 − λ)g ≺ F (0 5 λ 5 1). Lemma 6 (See [3]). Let κ, ϑ ∈ C. Suppose also that m is convex and univalent in U with
m(0) = 1 and R(κ m(z ) + ϑ) > 0 (z ∈ U). If u is analytic in U with u(0) = 1, then the following subordination:
u(z ) +
z u0 (z )
κ u(z ) + ϑ
≺ m(z ) (z ∈ U)
implies that
u(z ) ≺ m(z ) (z ∈ U). Lemma 7 (See [13]). Let f (z ) = 1 +
∞ X
ak z k
k=1
be analytic in U and g (z ) = 1 +
∞ X
bk z k
k=1
be analytic and convex in U. If f ≺ g, then
|ak | 5 |b1 |
(k ∈ N).
Lemma 8 (See [7]). Let 0 6= δ ∈ R, νδ > 0, 0 5 ρ < 1, p ∈ H [1, n] and
p(z ) ≺ 1 + Kz
K :=
νM nδ + ν
,
where
(1 − ρ) |δ| 1 + nνδ q M := Mn (δ, ν, ρ) = |1 − δ + ρδ| + 1 + 1 +
.
nδ 2
ν
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1973
If q ∈ H [1, n] satisfies the following subordination condition:
p(z )[1 − δ + δ((1 − ρ)q(z ) + ρ)] ≺ 1 + Mz , then
R(q(z )) > 0 (z ∈ U). 3. Main results We begin by presenting our first subordination property given by Theorem 1 below. α,β
Theorem 1. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. Then
Ipλ,n (a, c )f (z )
!β ≺
zp
(λ + p)β nα
1
Z
1 + Azu 1 + Bzu
0
u
(λ+p)β nα −1 du
≺
1 + Az 1 + Bz
(z ∈ U).
(3.1)
Proof. Define the function P by
Ipλ,n (a, c )f (z )
P(z ) :=
!β (z ∈ U).
zp
(3.2)
Then P is analytic in U with P(0) = 1. By taking the derivatives in the both sides in equality (3.2) and using (1.7), we get
Ipλ,n (a, c )f (z )
(1 − α)
!β +α
zp
= P(z ) +
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
α z P0 (z ) 1 + Az ≺ (λ + p)β 1 + Bz
!β
zp
(z ∈ U).
(3.3)
An application of Lemma 1 to (3.3) yields
Ipλ,n (a, c )f (z )
!β
Z (λ + p)β − (λ+p)β z (λ+p)β −1 1 + At n α ≺ z t nα dt nα 1 + Bt 0 Z 1 + Az (λ + p)β 1 (λ+p)β −1 1 + Azu t nα du ≺ = nα 1 + Bzu 1 + Bz 0
zp
The proof of Theorem 1 is thus completed.
(z ∈ U).
(3.4)
Theorem 2. Let q be univalent in U, 0 6= α ∈ C. Suppose also that q satisfies
R 1+
zq00 (z )
q0 (z )
(λ + p)β > max 0, − R . α
(3.5)
If f ∈ Ap (n) satisfies the following subordination:
Ipλ,n (a, c )f (z )
(1 − α)
!β
zp
+α
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
zp
!β ≺ q(z ) +
α zq0 (z ) , (λ + p)β
(3.6)
then
Ipλ,n (a, c )f (z ) zp
!β ≺ q(z ),
and q is the best dominant. Proof. Let the function P be defined by (3.2). We know that (3.3) holds true. Combining (3.3) and (3.6), we find that
P(z ) +
α z P0 (z ) α zq0 (z ) ≺ q(z ) + . (λ + p)β (λ + p)β
By Lemma 2 and (3.7), we easily get the assertion of Theorem 2.
(3.7)
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Az Taking q(z ) = 11+ in Theorem 2, we get the following result. +Bz Az Corollary 1. Let α ∈ C and −1 5 B < A 5 1. Suppose also that 11+ satisfies the condition (3.5). If f ∈ Ap (n) satisfies the +Bz following subordination:
Ipλ,n (a, c )f (z )
(1 − α) ≺
!β +α
zp
1 + Az
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β
zp
α(A − B)z , (λ + p)β(1 + Bz )2
+
1 + Bz
1 Ipλ+ ,n (a, c )f (z )
then
Ipλ,n (a, c )f (z )
!β ≺
zp
1 + Az 1 + Bz
,
Az and 11+ is the best dominant. +Bz
If f is subordinate to F , then F is superordinate to f . We now derive the following superordination result for the class
α,β
Bp,λ (a, c ; n; A, B). Theorem 3. Let q be convex univalent in U, α ∈ C with R(α) > 0. Also let
Ipλ,n (a, c )f (z )
!β ∈ H [q(0), 1] ∩ Q
zp and
Ipλ,n (a, c )f (z )
(1 − α)
!β +α
zp
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β
zp
be univalent in U. If
Ipλ,n (a, c )f (z ) α zq0 (z ) q(z ) + ≺ (1 − α) (λ + p)β zp
!β +α
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β ,
zp
then
Ipλ,n (a, c )f (z )
q(z ) ≺
!β
zp
,
and q is the best subdominant. Proof. Let the function P be defined by (3.2). Then
Ipλ,n (a, c )f (z ) α zq0 (z ) q(z ) + ≺ (1 − α) (λ + p)β zp
= P(z ) +
!β +α
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
Taking q(z ) =
Ipλ,n (a, c )f (z )
!β
zp
α z P0 (z ) . (λ + p)β
An application of Lemma 3 yields the assertion of Theorem 3. 1+Az 1+Bz
!
in Theorem 3, we get the following corollary.
Corollary 2. Let q be convex univalent in U and −1 5 B < A 5 1, α ∈ C with R(α) > 0. Also let
Ipλ,n (a, c )f (z )
!β ∈ H [q(0), 1] ∩ Q
zp and
(1 − α)
Ipλ,n (a, c )f (z ) zp
!β +α
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z ) zp
!β
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1975
be univalent in U. If
Ipλ,n (a, c )f (z ) α(A − B)z + ≺ ( 1 − α) 1 + Bz (λ + p)β(1 + Bz )2 zp
1 + Az
!β +α
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
zp
!β ,
then 1 + Az 1 + Bz
≺
Ipλ,n (a, c )f (z )
!β ,
zp
Az and 11+ is the best subdominant. +Bz
Combining the above results of subordination and superordination. we easily get the following ‘‘Sandwich-type result’’. Corollary 3. Let q1 be convex univalent and let q2 be univalent in U, α ∈ C with R(α) > 0. Let q2 satisfies (3.5). If 0 6=
Ipλ,n (a, c )f (z )
!β ∈ H [q1 (0), 1] ∩ Q ,
zp
and
Ipλ,n (a, c )f (z )
(1 − α)
!β
1 Ipλ+ ,n (a, c )f (z )
+α
zp
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β
zp
is univalent in U, also
Ipλ,n (a, c )f (z ) α zq01 (z ) ≺ (1 − α) q1 (z ) + (λ + p)β zp
≺ q2 (z ) +
!β +α
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
!β
zp
α zq02 (z ) , (λ + p)β
then
Ipλ,n (a, c )f (z )
q1 (z ) ≺
!β
zp
≺ q2 (z ),
and q1 and q2 are, respectively, the best subordinant and dominant. Theorem 4. Let f ∈ Ap (n), ξ ∈ C \ {0} and 0 5 γ < 1. Also let the function ϕ be defined by
z
ϕ(z ) :=
1 Ipλ+ ,n (a,c )f (z )
1 Ipλ+ ,n (a,c )f (z )
Ipλ,n (a,c )f (z )
Ipλ,n (a,c )f (z )
Ipλ,n (a,c )f (z ) zp
Ipλ,n (a,c )f (z )
!0
β
β
zp
−1 (z ∈ U).
(3.8)
−1
If ϕ satisfies one of the following conditions:
1 < R(ξ ) |ξ |2 R(ϕ(z )) 6= 0 > 1 R(ξ ) |ξ |2
(R(ξ ) > 0), (R(ξ ) = 0),
(3.9)
(R(ξ ) < 0),
or
1 > − |ξ |2 =(ξ ) =(ϕ(z )) 6= 0 1 < − 2 =(ξ ) |ξ |
(=(ξ ) > 0), (=(ξ ) = 0), (=(ξ ) < 0),
(3.10)
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
then
ξ ! !β λ 1 I ( a , c ) f ( z ) Ipλ+ ( a , c ) f ( z ) p,n ,n − 1 < 1 − γ . λ (a, c )f (z ) p I z p,n
(3.11)
Proof. We define the function φ by
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
ξ
!β
zp
− 1 = (1 − γ )φ(z )
(3.12)
(0 5 γ < 1; ξ ∈ C \ {0}; z ∈ U) . It is easy to see that the function φ is analytic in U with φ(0) = 0. Differentiating both sides of (3.12) with respect to z logarithmically, we get
z φ 0 (z )
φ(z )
z
=ξ
1 Ipλ+ ,n (a,c )f (z )
1 Ipλ+ ,n (a,c )f (z )
Ipλ,n (a,c )f (z )
Ipλ,n (a,c )f (z ) zp
Ipλ,n (a,c )f (z )
Ipλ,n (a,c )f (z )
!0
β
β
zp
−1 (z ∈ U; ξ ∈ C \ {0}).
(3.13)
−1
We now consider the function ϕ defined by
ϕ :=
ξ z φ 0 (z ) (z ∈ U; ξ ∈ C \ {0}). |ξ |2 φ(z )
(3.14)
Assume that there exists a point z0 ∈ U such that max|z |5|z0 | |φ(z )| = |φ(z0 )| = 1, by Lemma 4, we know that z0 φ 0 (z0 ) = kφ(z0 )
(k = 1).
(3.15)
It follows from (3.14) and (3.15) that
ξ z φ 0 (z0 ) R(ϕ(z0 )) = R |ξ |2 φ(z0 )
!
1 R(ξ ) = |ξ |2 k k R ξ = R (ξ ) = 0 = |ξ |2 |ξ |2 1 5 R(ξ ) |ξ |2
(R(ξ ) > 0), (R(ξ ) = 0),
(3.16)
(R(ξ ) < 0),
and
ξ z φ 0 (z0 ) =(ϕ(z0 )) = = |ξ |2 φ(z0 )
!
1 5 − |ξ |2 =(ξ ) k k = = ξ = − 2 = (ξ ) = 0 2 |ξ | |ξ | 1 = − 2 =(ξ ) |ξ |
(=(ξ ) > 0), (=(ξ ) = 0),
(3.17)
(=(ξ ) < 0).
But the inequalities in (3.16) and (3.17) contradict, respectively, the inequalities in (3.9) and (3.10). Therefore, we can conclude that
|φ(z )| < 1 (z ∈ U), which implies that
ξ ! !β 1 λ Ipλ+ ( a , c ) f ( z ) I ( a , c ) f ( z ) ,n p,n − 1 = (1 − γ ) |φ(z )| < 1 − γ . λ p Ip,n (a, c )f (z ) z We thus complete the proof of Theorem 4.
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1977
From Theorem 4, we easily get the following result for the class B (β) of Bazilevič functions of type β . Corollary 4. Let f ∈ A, λ = 0, p = a = c = ξ = 1 and γ = 0. Also let the function ϕ be defined by (3.8). If ϕ satisfies one of the following conditions:
R(ϕ(z )) < 1 or
=(ϕ(z )) 6= 0,
then f ∈ B (β). 0,β
α,β
Theorem 5. Let R(α) > 0, β > 0 and f ∈ Bp,λ (a, c ; n; 1 − 2ρ, −1) (0 5 ρ < 1). Then f ∈ Bp,λ (a, c ; n; 1 − 2ρ, −1) for |z | < R(α, β, λ, p), where
−α +
R(α, β, λ, p) =
α 2 + (λ + p)2 β 2 . (λ + p)β
p
(3.18)
The bound R(α, β, λ, p) is the best possible. Proof. Suppose that
Ipλ,n (a, c )f (z )
!β = ρ + (1 − ρ)h(z ) (z ∈ U; 0 5 ρ < 1),
zp
(3.19)
where h is analytic and has a positive real part in U. By taking the derivatives in the both sides in equality (3.19) and using (1.7), we get
Ipλ,n (a, c )f (z )
R (1 − α)
!β
1 Ipλ+ ,n (a, c )f (z )
+α
zp
α zh0 (z ) = (1 − ρ)R h(z ) + (λ + p)β
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β −ρ
zp
! α zh0 (z ) = (1 − ρ)R h(z ) − . (λ + p)β
(3.20)
By making use of the following well-known estimate (see [8]):
0 zh (z ) 5
2r 1 − r2
R(h(z )) (|z | = r < 1)
in (3.20), we obtain that
Ipλ,n (a, c )f (z )
R (1 − α)
= (1 − ρ)
!β +α
zp
!
Ipλ,n (a, c )f (z )
2α r
1−
1 Ipλ+ ,n (a, c )f (z )
(λ + p)β(1 − r 2 )
Ipλ,n (a, c )f (z ) zp
!β
− ρ
R (h(z )) > 0
(3.21)
for r < R(α, β, λ, p), where R(α, β, λ, p) is given by (3.18). To show that the bound R(α, β, λ, p) is the best possible, we consider the function f ∈ Ap (n) defined by
Ipλ,n (a, c )f (z )
!β
zp
= ρ + (1 − ρ)
1+z
(z ∈ U).
1−z
By noting that
R (1 − α)
= (1 − ρ)R
Ipλ,n (a, c )f (z ) zp
1+z 1−z
+
!β
1 Ipλ+ ,n (a, c )f (z )
+α 2α
(λ + p)β
Ipλ,n (a, c )f (z )
·
z
(1 − z )2
!
Ipλ,n (a, c )f (z ) zp
!β −ρ
=0
for z = R(α, β, λ, p), we conclude that the bound is the best possible. Theorem 5 is thus proved.
(3.22)
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
α,β
Theorem 6. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. Then f (z ) =
z
p
1 + Aω(z )
β1 !
p
∗ z +
1 + Bω(z )
!
∞ X
k!(a)k
k=n
(λ + p)k (c )k
z
,
k+p
(3.23)
where ω is analytic in U with
ω(0) = 0 and
|ω(z )| < 1 (z ∈ U). α,β
Proof. Suppose that f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. It follows from (3.1) that
Ipλ,n (a, c )f (z )
!β =
zp
1 + Aω(z ) 1 + Bω(z )
,
(3.24)
where ω is analytic in U with
ω(0) = 0 and
|ω(z )| < 1 (z ∈ U).
By virtue of (3.24), we easily find that
Ipλ,n (a, c )f (z ) = z p
1 + Aω(z )
β1
1 + Bω(z )
.
(3.25)
Combining (1.3), (1.5) and (3.25), we have p
z +
∞ X (λ + p)k (c )k
k!(a)k
k =n
! z
∗ f (z ) = z
k+p
p
1 + Aω(z )
β1
1 + Bω(z )
.
(3.26)
The assertion (3.23) of Theorem 6 can now easily be derived from (3.26). Theorem 7. Let f ∈ 1 z
1 + Beiθ
α,β Bp,λ (a, c ; A, B)
β1
zp +
∞ P k=n
with R(α) > 0. Then
(λ+p)k (c )k k+p z k!(a)k
∗ f (z ) − z p 1 + Aeiθ
β1
6= 0 (z ∈ U; 0 < θ < 2π ).
(3.27)
α,β
Proof. Suppose that f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. We know that (3.1) holds true, which implies that
Ipλ,n (a, c )f (z )
!β 6=
zp
1 + Aeiθ 1 + Beiθ
(z ∈ U; 0 < θ < 2π ).
(3.28)
It is easy to see that the condition (3.28) can be written as follows: 1 z
Ipλ,n (a, c )f (z )
1 + Be
iθ
β1
−z
p
1 + Ae
iθ
β1
6= 0 (z ∈ U; 0 < θ < 2π ).
(3.29)
Combining (1.3), (1.5) and (3.29), we easily get the convolution property (3.27) asserted by Theorem 7.
Theorem 8. Let α2 = α1 = 0 and −1 5 B1 5 B2 < A2 5 A1 5 1. Then α ,β
α ,β
2 Bp,λ (a, c ; n; A2 , B2 ) ⊂ Bp,λ1 (a, c ; n; A1 , B1 ).
(3.30)
α ,β
2 Proof. Suppose that f ∈ Bp,λ (a, c ; n; A2 , B2 ). We know that
(1 − α2 )
Ipλ,n (a, c )f (z )
!β + α2
zp
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β ≺
zp
1 + A2 z 1 + B2 z
.
Since −1 5 B1 5 B2 < A2 5 A1 5 1, we easily find that
(1 − α2 ) α ,β
Ipλ,n (a, c )f (z ) zp
!β + α2
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z ) zp
!β ≺
1 + A2 z 1 + B2 z
2 that is f ∈ Bp,λ (a, c ; n; A1 , B1 ). Thus the assertion of Theorem 8 holds for α2 = α1 = 0.
≺
1 + A1 z 1 + B1 z
,
(3.31)
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1979
0,β
If α2 > α1 = 0, by Theorem 1 and (3.31), we know that f ∈ Bp,λ (a, c ; n; A1 , B1 ), that is
Ipλ,n (a, c )f (z )
!β ≺
zp
1 + A1 z 1 + B1 z
.
(3.32)
At the same time, we have
Ipλ,n (a, c )f (z )
(1 − α1 )
!β + α1
zp
α1 = 1− α2
Ipλ,n (a, c )f (z )
zp
1 Ipλ+ ,n (a, c )f (z )
+ α2
!β
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z ) α1 (1 − α2 ) + α2 zp
Ipλ,n (a, c )f (z )
zp
!β
zp
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
!
!β
!β .
(3.33)
Moreover, since 05
α1 <1 α2
and h1 (z ) :=
1 + A1 z 1 + B1 z
(z ∈ U)
is analytic and convex in U. Combining (3.31)–(3.33) and Lemma 5, we find that
(1 − α1 )
Ipλ,n (a, c )f (z )
!β + α1
zp
1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
zp
α ,β
!β ≺
1 + A1 z 1 + B1 z
,
1 that is f ∈ Bp,λ (a, c ; n; A1 , B1 ), which implies that the assertion (3.30) of Theorem 8 holds.
Let P denote the class of functions of the following form:
t(z ) = 1 +
∞ X
(n ∈ N),
tk z k
k=n
which are analytic and convex in U and satisfy the following condition:
R(t(z )) > 0 (z ∈ U). By making use of the principle of subordination between analytic functions, we introduce the subclasses Sp∗,n (µ; φ) and Cp,n (µ; φ) of the class Ap (n):
Sp∗,n (µ; φ) := f ∈ Ap (n) :
1
p−µ
zf 0 (z ) f (z )
− µ ≺ φ(z )
(φ ∈ P ; 0 5 µ < p; z ∈ U)
and
Cp,n (µ; φ) := f ∈ Ap (n) :
1 p−µ
1+
zf 00 (z ) f 0 (z )
− µ ≺ φ(z )
(φ ∈ P ; 0 5 µ < p; z ∈ U) .
Next, by using the operator defined by (1.5), we define the following two subclasses Spλ,n (a, c ; µ; φ) and Cpλ,n (a, c ; µ; φ) of the class Ap (n):
Spλ,n (a, c ; µ; φ) := f ∈ Ap (n) : Ipλ,n (a, c )f ∈ Sp∗,n (µ; φ)
and
Cpλ,n (a, c ; µ; φ) := f ∈ Ap (n) : Ipλ,n (a, c )f ∈ Cp,n (µ; φ) .
Clearly, we know that f ∈ Cpλ,n (a, c , µ; φ) ⇐⇒
zf 0 p
∈ Spλ,n (a, c , µ; φ).
(3.34)
We now derive some inclusion relationships for the classes Spλ,n (a, c ; µ; φ) and Cpλ,n (a, c ; µ; φ). By similarly applying the method of proof of Proposition 1 obtained by Cho et al. [2], we easily get Theorem 9. But the conditions of Theorem 9 are much weaker than the conditions of the corresponding result obtained by Cho et al. [2].
1980
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Theorem 9. Let 0 5 µ < p, λ > −p and φ ∈ P with
λ+µ a+µ−p R (φ(z )) > max 0, − , − p−µ p−µ
(z ∈ U).
(3.35)
Then 1 λ λ Spλ+ ,n (a, c ; µ; φ) ⊂ Sp,n (a, c ; µ; φ) ⊂ Sp,n (a + 1, c ; µ; φ).
Theorem 10. Let 0 5 µ < p, λ > −p and φ ∈ P with (3.35) holds. Then 1 λ λ Cpλ+ ,n (a, c ; µ; φ) ⊂ Cp,n (a, c ; µ; φ) ⊂ Cp,n (a + 1, c ; µ; φ).
Proof. By virtue of (3.34) and Theorem 9, we observe that 1 f ∈ Cpλ+ ,n (a, c ; µ; φ)
⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒
1 Ipλ+ Cp,n (µ; φ) ,n (a, c )f ∈ 0 1 z Ipλ+ ( a , c ) f ,n ∈ Sp∗,n (µ; φ)
p
1 Ipλ+ ,n (a, c )
zf 0 p zf 0
zf 0 p
∈ Sp∗,n (µ; φ)
1 ∈ Spλ+ ,n (a, c ; µ; φ)
⇐⇒
∈ Spλ,n (a, c ; µ; φ) 0 zf λ ∈ Sp∗,n (µ; φ) Ip,n (a, c ) p 0 z Ipλ,n (a, c )f ∈ Sp∗,n (µ; φ)
⇐⇒ ⇐⇒
Ipλ,n (a, c )f ∈ Cp,n (µ; φ) f ∈ Cpλ,n (a, c ; µ; φ),
H⇒ ⇐⇒
(3.36)
p
p
and f ∈ Cpλ,n (a, c ; µ; φ)
⇐⇒
zf 0 p zf 0
∈ Spλ,n (a, c ; µ; φ)
⇐⇒
∈ Spλ,n (a + 1, c ; µ; φ) p 0 z Ipλ,n (a + 1, c )f ∗
⇐⇒ ⇐⇒
Ipλ,n (a + 1, c )f ∈ Cp,n (µ; φ) f ∈ Cpλ,n (a + 1, c ; µ; φ).
H⇒
(3.37)
∈ Sp,n (µ; φ)
p
From (3.36) and (3.37), we conclude that the assertion of Theorem 10 holds true. Taking φ(z ) =
1+Az 1+Bz
in Theorems 9 and 10, we get the following results.
Corollary 5. Let 0 5 µ < p, λ > −p and −1 5 B < A 5 1. Then
1 + Az
1 Spλ+ a, c ; µ; ,n
1 + Az 1 + Az ⊂ Spλ,n a, c ; µ; ⊂ Spλ,n a + 1, c ; µ; 1 + Bz 1 + Bz
1 + Az 1 + Az ⊂ Cpλ,n a, c ; µ; ⊂ Cpλ,n a + 1, c ; µ; . 1 + Bz 1 + Bz
1 + Bz
and
1 Cpλ+ a, c ; µ; ,n
1 + Az 1 + Bz α,β
Theorem 11. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 B < A 5 1. Then
(λ + p)β nα
1
Z 0
1 − Au 1 − Bu
(λ+p)β u nα −1 du
Ipλ,n (a, c )f (z ) zp
!β <
(λ + p)β nα
1
Z 0
1 + Au 1 + Bu
u
(λ+p)β nα −1 du
.
(3.38)
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1981
The extremal function of (3.38) is defined by
Ipλ,n (a, c )Fα,β,A,B (z )
=z
p
(λ + p)β nα
1
Z 0
1 + Azu 1 + Bzu
(λ+p)β u nα −1 du
β1
.
(3.39)
α,β
Proof. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0. From Theorem 1, we know that (3.1) holds, which implies that
Ipλ,n (a, c )f (z )
R
!β
zp
Z (λ + p)β 1 1 + Azu (λ+p)β −1 n α < sup R u du nα z ∈U 0 1 + Bzu Z (λ+p)β (λ + p)β 1 1 + Azu 5 sup R u nα −1 du nα 1 + Bzu z ∈ U 0 Z (λ + p)β 1 1 + Au (λ+p)β −1 < u nα du, nα 0 1 + Bu
(3.40)
and
Ipλ,n (a, c )f (z )
R
!β
Z (λ + p)β 1 1 + Azu (λ+p)β −1 u nα du z ∈U nα 0 1 + Bzu Z 1 (λ+p)β (λ + p)β 1 + Azu = inf R u nα −1 du z ∈ U nα 1 + Bzu 0 Z 1 1 − Au (λ+p)β −1 (λ + p)β du. u nα > nα 0 1 − Bu > inf R
zp
(3.41)
Combining (3.40) and (3.41), we get (3.38). By noting that the function Ipλ,n (a, c )Fα,β,A,B (z ) defined by (3.39) belongs to the α,β
class Bp,λ (a, c ; n; A, B), we obtain that equality (3.38) is sharp. The proof of Theorem 11 is evidently completed.
By similarly applying the method of proof of Theorem 11, we easily get the following result. α,β
Corollary 6. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 A < B 5 1. Then
(λ + p)β nα
1
Z 0
1 + Au 1 + Bu
(λ+p)β u nα −1 du
Ipλ,n (a, c )f (z ) zp
!β <
(λ + p)β nα
1
Z 0
1 − Au 1 − Bu
u
(λ+p)β nα −1 du
.
(3.42)
The extremal function of (3.42) is defined by (3.39). α,β
In view of Theorem 11 and Corollary 6, we easily derive the following distortion theorems for the class Bp,λ (a, c ; n; A, B). α,β
Corollary 7. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 B < A 5 1. Then for |z | = r < 1, we have
r
p
(λ + p)β nα
1
Z
1 − Aur 1 − Bur
0
(λ+p)β u nα −1 du
β1
< Ipλ,n (a, c )f (z ) < r p
(λ + p)β nα
1
Z
1 + Aur 1 + Bur
0
(λ+p)β u nα −1 du
β1
.
(3.43)
.
(3.44)
The extremal function of (3.43) is defined by (3.39). α,β
Corollary 8. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 A < B 5 1. Then for |z | = r < 1, we have
r
p
(λ + p)β nα
1
Z 0
1 + Aur 1 + Bur
(λ+p)β u nα −1 du
β1
< Ipλ,n (a, c )f (z ) < r p
The extremal function of (3.44) is defined by (3.39). By noting that
1 1 1 (R(υ)) 2 5 R υ 2 5 |υ| 2
(υ ∈ C; R(υ) = 0) .
(λ + p)β nα
1
Z 0
1 − Aur 1 − Bur
(λ+p)β u nα −1 du
β1
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Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Form Theorem 11 and Corollary 6, we easily get the following results. α,β
Corollary 9. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 B < A 5 1. Then
(λ + p)β nα
1
Z
1 − Au 1 − Bu
0
(λ+p)β u nα −1 du
12
Ipλ,n (a, c )f (z )
! β2 <
zp
(λ + p)β nα
Z
1
1 + Au 1 + Bu
0
(λ+p)β u nα −1 du
12
.
α,β
Corollary 10. Let f ∈ Bp,λ (a, c ; n; A, B) with R(α) > 0 and −1 5 A < B 5 1. Then
(λ + p)β nα
1
Z
1 + Au 1 + Bu
0
(λ+p)β u nα −1 du
12
Ipλ,n (a, c )f (z )
! β2
zp
<
(λ + p)β nα
Z 0
1
1 − Au 1 − Bu
(λ+p)β u nα −1 du
12
.
Theorem 12. Let f (z ) = z p +
∞ X
α,β
ap+k z p+k ∈ Bp,λ (a, c ; n; A, B).
(3.45)
k=n
Then
ap+n 5 (λ + p)n!(a)n (λ + p)n (c )n
A−B (λ + p)β + nα .
(3.46)
The inequality (3.46) is sharp, with the extremal function defined by (3.39). Proof. Combining (1.8) and (3.45), we obtain
(1 − α)
Ipλ,n (a, c )f (z )
!β +α
zp
=1+ 1+
1 Ipλ+ ,n (a, c )f (z )
nα
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
!β
zp
(λ + p)n (c )n 1 + Az β ap+n z n + · · · ≺ . n!(a)n 1 + Bz
(λ + p)β
(3.47)
An application of Lemma 7 to (3.47) yields
nα (λ + p)n (c )n 1+ β a p+n 5 |A − B| . (λ + p)β n!(a)n
(3.48)
Thus, from (3.48), we easily arrive at (3.46) asserted by Theorem 12. Theorem 13. Let 0 6= α ∈ R, β ∈ R,
α,β
β α
> 0, λ > −p and 0 5 ρ < 1. If f ∈ Bp,λ (a, c ; n; A, 0) with (1 − ρ) |α| 1 + (λ+nαp)β A = An (α, β, λ, ρ, p) = r 2 , nα |1 − α + ρα| + 1 + 1 + (λ+p)β
then
Ipλ,n (a, c )f ∈ Sp∗,n (ρ p − (1 − ρ)λ). α,β
Proof. Suppose that f ∈ Bp,λ (a, c ; n; A, 0). By definition, we have
(1 − α)
Ipλ,n (a, c )f (z )
!β +α
zp
1 Ipλ+ ,n (a, c )f (z )
!
Ipλ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
zp
!β ≺ 1 + Az (z ∈ U).
(3.49)
Let the function P be defined by (3.2). We then find from (3.1) and (3.49) that
P(z ) ≺
(λ + p)β − (λ+p)β nα z nα
Z
z
(1 + At )t
(λ+p)β nα −1 dt
=1+
0
(λ + p)β A z. nα + (λ + p)β
We now suppose that 1 Ipλ+ ,n (a, c )f (z )
Ipλ,n (a, c )f (z )
= (1 − ρ)q(z ) + ρ
(0 5 ρ < 1;
z ∈ U) .
(3.50)
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
1983
Then q ∈ H [1, n]. It follows from (3.49) and (3.50) that
P(z ){(1 − α) + α[(1 − ρ)q(z ) + ρ]} ≺ 1 + Az
(z ∈ U).
(3.51)
An application of Lemma 8 to (3.51) yields
R(q(z )) > 0 (z ∈ U).
(3.52)
Combining (3.50) and (3.52), we find that 1 Ipλ+ ,n (a, c )f (z )
R
!
Ipλ,n (a, c )f (z )
= (1 − ρ)R(q(z )) + ρ > ρ (z ∈ U).
(3.53)
The assertion of Theorem 13 can now easily be derived from (1.7) and (3.53).
α,β
Theorem 14. Let f ∈ Bp,λ (a, c ; n; A, 0) with
β > 0,
(λ + p)β R (α) > 0 and |α| n + R α
A > 0,
> A(λ + p)β.
Then
h i z Iλ (a, c )f 0 (z ) A(λ + p) |α| n + R (λ+αp)β + (λ + p)β p,n h i . − p < Ipλ,n (a, c )f (z ) |α| |α| n + R (λ+αp)β − A(λ + p)β Proof. Let the function P be defined by (3.2). It follows from (3.3) that
P(z ) +
α z P0 (z ) = 1 + A (z ), (λ + p)β
(3.54)
where
(z ) =
∞ X
kz
k
(n ∈ N)
k=n
is analytic in U with | (z )| < 1 (z ∈ U). From (3.54), we easily get
P(z ) = 1 + A
(λ + p)β α
1
Z
t
(λ+p)β −1 α
(tz )dt = 1 + A
0
∞ (λ + p)β X 1 (λ+ p)β α k=n k + α
kz
k
.
(3.55)
It follows from (3.55) that
(z P(z ))0 = 1 + A
∞ (λ + p)β X k+1 (λ+p)β α k=n k + α
kz
∞ (λ + p)β X 1 = 1+A (λ+p)β α k=n k + α
k
(λ + p)β kz + A α k
(λ + p)β (z ) − α
1
Z
t
(λ+p)β −1 α
(tz )dt .
(3.56)
0
We next find from (3.55) and (3.56) that
(λ + p)β z P (z ) = A α 0
(λ + p)β (z ) − α
1
Z
t
(λ+p)β −1 α
(tz )dt .
(3.57)
0
Combining (3.55) and (3.57), we get
h i 0 A(λ + p)β |α| n + R (λ+p)β + (λ + p)β z P (z ) α h i . P(z ) < (λ+p)β |α| |α| n + R − A (λ + p )β α Thus, from (3.2) and (3.58), we easily arrive at the assertion of Theorem 14.
(3.58)
Remark. By specializing the parameters α, β, p, λ, a, c and n in Theorems 13 and 14, we can get some criteria for starlikeness. Here, we choose to omit the details involved.
1984
Z.-G. Wang et al. / Mathematical and Computer Modelling 49 (2009) 1969–1984
Acknowledgements The present investigation was supported by the Scientific Research Fund of Hunan Provincial Education Department under Grant 08C118 of People’s Republic of China, and also, by SAGA Grant: STGL-012-2006, Academy of Science, Malaysia. The authors would like to thank the referees for their careful reading and making some valuable comments which have essentially improved the presentation of this paper. The first-named author would also like to thank Professors Chun-Yi Gao, Yue-Ping Jiang, Qing-Guo Li and Ming-Sheng Liu for their support and encouragement. References [1] R. Aghalary, On subclasses of p-valent analytic functions defined by integral operators, Kyungpook Math. J. 47 (2007) 393–401. [2] N.E. Cho, O.S. Kwon, H.M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl. 292 (2004) 470–483. [3] P. Eenigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Briot–Bouquet differential subordination, in: General Mathematics 3, in: International Series of Numerical Mathematics, vol. 64, Birkhäuser Verlag, Basel, 1983, pp. 339–348; see also Rev. Roumaine Math. Pures Appl. 29 (1984) 567–573. [4] I.S. Jack, Functions starlike and convex of order α , J. London Math. Soc. 3 (1971) 469–474. [5] M.-S. Liu, On certain class of analytic functions defined by differential subordination, Acta Math. Sci. Ser. B Engl. Ed. 22 (2002) 388–392. [6] M.-S. Liu, On certain subclass of analytic functions, J. South China Normal Univ. 4 (2002) 15–20 (in Chinese). [7] M.-S. Liu, On the starlikeness for certain subclass of analytic functions and a problem of Ruscheweyh, Adv. Math. (China) 34 (2005) 416–424. [8] T.H. Macgregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14 (1963) 514–520. [9] S.S. Miller, P.T. Mocanu, Differential Subordination: Theory and Applications, in: Series in Pure and Applied Mathematics, vol. 225, Marcel Dekker, New York, 2000. [10] S.S. Miller, P.T. Mocanu, Subordinats of differential superordinations, Complex Var. 48 (2003) 815–826. [11] J. Patel, On certain subclasses of multivalent functions involving Cho–Kwon–Srivastava operator, Ann. Univ. Mariae Curie-Skłodowska Sect. A 60 (2006) 75–86. [12] J. Patel, N.E. Cho, H.M. Srivastava, Certain subclasses of multivalent functions associated with a family of linear operators, Math. Comput. Modelling 43 (2006) 320–338. [13] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (Ser. 2) 48 (1943) 48–82. [14] H. Saitoh, A linear operator and its applications of first order differential subordinations, Math. Japon. 44 (1996) 31–38. [15] T.N. Shanmugam, V. Ravichandran, S. Sivasubramanian, Differential Sandwich theorems for subclasses of analytic functions, Australian J. Math. Anal. Appl. 3 (2006) 1–11. Article 8 (electronic). [16] J. Sokól, L. Trojnar-Spelina, Convolution properties for certain classes of multivalent functions, J. Math. Anal. Appl. 337 (2008) 1190–1197. [17] T. Zeng, C.-Y. Gao, Z.-G. Wang, R. Aghalary, Certain subclass of multivalent functions involving the Cho–Kwon–Srivastava operator, J. Math. Appl. 30 (2008) 161–170. [18] Y.-C. Zhu, Some starlikeness criterions for analytic functions, J. Math. Anal. Appl. 335 (2007) 1452–1459.