The purpose of this paper is to introduce two novel subclasses Γλ n, α, β and Γ∗ λ n, α, β of meromorphic p-valent functions by using the linear operator D n.. Then we prove the necessar
Trang 1Volume 2011, Article ID 401913, 16 pages
doi:10.1155/2011/401913
Research Article
λ
Amin Saif and Adem Kılıc¸man
Department of Mathematics, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia
Correspondence should be addressed to Adem Kılıc¸man,akilicman@putra.upm.edu.my
Received 26 July 2010; Accepted 28 February 2011
Academic Editor: Jong Kim
Copyrightq 2011 A Saif and A Kılıc¸man This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
The purpose of this paper is to introduce two novel subclasses Γλ n, α, β and Γ∗
λ n, α, β of meromorphic p-valent functions by using the linear operator D n Then we prove the necessary and sufficient conditions for a function f in order to be in the new classes Further we study several important properties such as coefficients inequalities, inclusion properties, the growth
and distortion theorems, the radii of meromorphically p-valent starlikeness, convexity, and
subordination properties We also prove that the results are sharp for a certain subclass of functions
1 Introduction
LetΣpdenote the class of functions of the form
fz z −p ∞
kp1
a k z k
a k ≥ 0; p ∈ N {1, 2, }, 1.1
which are meromorphic and p-valent in the punctured unit disc U∗ {z ∈ C : 0 < |z| < 1}
U − {0} For the functions f in the class Σ p , we define a linear operator D n λby the following form:
D λ fz 1 pλfz λzfz, λ ≥ 0,
D0
λ fz fz,
D1
λ fz D λ fz,
D2
λ fz D λ
D1
λ fz,
1.2
Trang 2and in general for n 0, 1, 2, , we can write
D n
λ fz z1p ∞
kp1
1 pλ kλn a k z k ,
n ∈ N0 N ∪ {0}; p ∈ N. 1.3
Then we can observe easily that for f ∈ Σp,
zλ
D n
λ fz D n1
λ fz −1 pλD n
λ fz, p ∈ N; n ∈ N0
Recall 1,2 p is said to be meromorphically starlike of order α if it is
satisfying the following condition:
Re
−zfz
fz
for some α0 α < 1 Similarly recall 3 pis said to be meromorphically
convex of order α if it is satisfying the following condition:
Re
−1 −zf fz z
> α, z ∈ U∗ for some α 0 ≤ α < 1. 1.6 LetΣp α be a subclass of Σ pconsisting the functions which satisfy the following inequality:
Re
−z
D n
λ fz
D n
λ fz > pα, z ∈ U∗; α ≥ 0. 1.7
In the following definitions, we will define subclassesΓλ n, α, β and Γ∗
λ n, α, β by using the linear operator D λ n
Definition 1.1 For fixed parameters α ≥ 0, 0 ≤ β < 1, the meromorphically p-valent function
fz ∈ Σ p α will be in the class Γ λ n, α, β if it satisfies the following inequality:
Re
−z
D n
λ fz
p
D n
λ fz ≥ α
z
D n
λ fz
p
D n
λ fz 1
β, n ∈ N0. 1.8
Definition 1.2 For fixed parameters α ≥ 1/2 β; 0 ≤ β < 1, the meromorphically p-valent function fz ∈ Σ p α will be in the class Γ∗
λ n, α, β if it satisfies the following inequality:
z
D n
λ fz
p
D n
λ fz α αβ
≤Re
−z
D n
λ fz
p
D n
λ fz α − αβ, ∀n ∈ N0. 1.9
Trang 3Meromorphically multivalent functions have been extensively studied by several authors, see
punctured unit disk was considered
In 17
following differential subordinations:
z
Ip r, λfzj1
p − jIp r, λfzj ≺
a aB A − Bβz
and studied the related coefficients inequalities with β complex number
This paper is organized as follows It consists of four sections Sections 2 and 3 investigate the various important properties and characteristics of the classesΓλ n, α, β and
Γ∗
λ n, α, β by giving the necessary and sufficient conditions Further we study the growth and distortion theorems and determine the radii of meromorphically p-valent starlikeness
of order μ 0 ≤ μ < p and meromorphically p-valent convexity of order μ 0 ≤ μ < p In
Section4we give some results related to the subordination properties
2 Properties of the Class Γλn, α, β
We begin by giving the necessary and sufficient conditions for functions f in order to be in the classΓλ n, α, β.
Lemma 2.1 see 2
R a
⎧
⎪
⎪
⎪
⎪
a − α β
1− β
α1 α ,
1 − a21 − α2 − 21− β1 − a, for a ≥ 1 1− β
α1 α .
2.1
Then
{w : |w − a| ≤ R a} ⊆w : Rew ≥ α|w − 1| β. 2.2
Theorem 2.2 Let f ∈ Σ p Then f is in the class Γ λ n, α, β if and only if
∞
kp1
p
α β k1 αkλ pλ 1n a k ≤ p1− β
α ≥ 0; 0 ≤ β < 1; p ∈ N; n ∈ N0
.
2.3
Trang 4Proof Suppose that f∈ Γλ n, α, β Then by the inequalities 1.3 and 1.8, we get that
Re
−z
D n
λ fz
p
D n
λ fz ≥ α
z
D n
λ fz
p
D n
λ fz 1
That is,
Re
1−∞kp1k/p
kλ pλ 1n a k z kp
1∞kp1kλ pλ 1n a k z kp
≥ α
∞
kp1
k/p
1kλ pλ 1n a k z kp
1∞kp1kλ pλ 1n a k z kp
β
≥ Re
α ·
∞
kp1
k/p
1kλ pλ 1n a k z kp
1∞
kp1
kλ pλ 1n a k z kp β
Re
β ∞
kp1
α
k/p
1 βkλ pλ 1n a k z kp
1∞
kp1
kλ pλ 1n a k z kp ,
2.5
that is,
Re
p
1− β−∞kp1k kα pα pβkλ pλ 1n a k z kp
1∞kp1kλ pλ 1n a k z kp ≥ 0. 2.6
Taking z to be real and putting z → 1−through real values, then the inequality2.6 yields
p
1− β−∞kp1k kα pα pβkλ pλ 1n a k
1∞kp1kλ pλ 1n a k ≥ 0, 2.7
which leads us at once to2.3
In order to prove the converse, suppose that the inequality 2.3 holds true In Lemma 2.1, since 1 ≤ 1 1 − β/α1 α, put a 1 Then for p ∈ N and n ∈ N0, let
w np −zD n
λ fz/pD n
λ fz If we let z ∈ ∂U∗ {z ∈ C : |z| 1}, we get from the
inequalities1.3 and 2.3 that |w np − 1| ≤ R1 Thus by Lemma2.1above, we ge that
Re
−z
D n
λ fz
p
D n
λ fz − 1 Re
w np≥ α w np− 1 β α −zD λ n fz
p
D n
λ fz − 1
β
α
z
D n
λ fz
p
D n
λ fz 1
β,
α ≥ 0; 0 ≤ β < 1; p ∈ N; n ∈ N0
.
2.8
Therefore by the maximum modulus theorem, we obtain f ∈ Γλ n, α, β.
Trang 5Corollary 2.3 If f ∈ Γ λ n, α, β, then
1− β
p
α β k1 αkλ pλ 1n ,
α ≥ 0; 0 ≤ β < 1; p ∈ N; n ∈ N0
. 2.9
The result is sharp for the function fz given by
fz z −p ∞
kp1
p
1− β
p
α β k1 αkλ pλ 1n z
k ,
α ≥ 0; 0 ≤ β < 1; p ∈ N; n ∈ N0
.
2.10
Theorem 2.4 The class Γ λ n, α, β is closed under convex linear combinations.
Proof Suppose the function
fz z −p ∞
kp1
a k z k,j
a k,j ≥ 0; j 1, 2; p ∈ N, 2.11
be in the classΓλ n, α, β It is sufficient to show that the function hz defined by
hz 1 − δf1z δf2z 0 ≤ δ ≤ 1, 2.12
is also in the classΓλ n, α, β Since
hz z −p ∞
kp1
1 − δa k,1 δa k,2 k,j , 0 ≤ δ ≤ 1, 2.13
and by Theorem2.2, we get that
∞
kp1
p
α β k1 αkλ pλ 1n 1 − δa k,1 δa k,2
∞
kp1
1 − δp
α β k1 αkλ pλ 1n a k,1
∞
kp1
δ
p
α β k1 αkλ pλ 1n a k,2
≤ 1 − δp1− β δp1− β p1− β,
α ≥ 0; 0 ≤ β < 1; p ∈ N; n ∈ N0
.
2.14
Hence f∈ Γλ n, α, β.
The following are the growth and distortion theorems for the classΓλ n, α, β.
Trang 6Theorem 2.5 If f ∈ Γ λ n, α, β, then
p m − 1!
p − 1! −
1− β
2α β 1p 1n−1 ·
p!2 −n
p − m!r 2p r −pm≤ f m z
≤
p m − 1!
p − 1!
1− β
2α β 1p 1n−1 ·
p!2 −n
p − m!r 2p r −pm
0 < |z| r < 1; α ≥ 0; 0 ≤ β < 1; p ∈ N; n, m ∈ N0; p > m
.
2.15
The result is sharp for the function f given by
fz z −p ∞
kp1
1− β
2α β 12p 2n z
p ,
n ∈ N0; p ∈ N. 2.16
Proof From Theorem2.2, we get that
p
2α β 12p 2n
p 1!
∞
kp1
k!a k ≤ ∞
kp1
p
α β k1 αkλ pλ 1n a k
≤ p1− β,
2.17
that is,
∞
kp1
k!a k≤ p
1− βp 1!
p
2α β 12p 2n
1− βp!2 −n
2α β 1p 1n−1 . 2.18
By the differentiating the function f in the form 1.1 m times with respect to z, we get that
f m z −1 m
p m − 1!
p − 1! z −pm ∞
kp1
k!
k − m! a k z k−m ,
m ∈ N0; p ∈ N 2.19
and Theorem2.5follows easily from2.18 and 2.19 Finally, it is easy to see that the bounds
in2.15 are attained for the function f given by 2.18
Next we determine the radii of meromorphically p-valent starlikeness of order μ0 ≤
μ < p and meromorphically p-valent convexity of order μ 0 ≤ μ < p for the class Γ λ n, α, β.
Theorem 2.6 If f ∈ Γ λ n, α, β, then f is meromorphically p-valent starlike of order μ0 ≤ μ < 1
in the disk |z| < r1, that is,
Re
−zfz
fz
> μ
0≤ μ < p; |z| < r1; p ∈ N, 2.20
Trang 7r1 inf
k≥p1
p − μ
p
α β k1 αkλ pλ 1n
p
k μ1− β
1/kp
Proof By the form1.1, we get that
zfz/fz p
zfz/fz− p 2μ
∞
kp1
k pa k z k
2
p − μz −p∞kp1k − p 2μa k z k
≤
∞
kp1
k p|z| k
2
p − μa k |z| −p∞
kp1
k − p 2μa k |z| k
∞
kp1
k pa k |z| kp
2
p − μ∞kp1k − p 2μa k |z| kp .
2.22
Then the following incurability
zfz/fz p
zfz/fz− p 2μ
≤1,
0≤ μ < p; p ∈ N 2.23
also holds if
∞
kp1
k μ
p − μ a k |z| kp ≤ 1,
0≤ μ < p; p ∈ N. 2.24
Then by Corollary2.3the inequality2.24 will be true if
k μ
p − μ |z| kp≤
p
α β k1 αkλ pλ 1n
p
0≤ μ < p; p ∈ N, 2.25
that is,
|z| kp≤
p − μp
α β k1 αkλ pλ 1n
p
k μ1− β ,
0≤ μ < p; p ∈ N. 2.26
Therefore the inequality2.26 leads us to the disc |z| < r1, where r1 is given by the form
2.21
Theorem 2.7 If f ∈ Γ λ n, α, β, then f is meromorphically p-valent convex of order μ 0 ≤ μ < 1
in the disk |z| < r2, that is,
Re
−1 −zf fz z
> μ
0≤ μ < p; |z| < r2; p ∈ N, 2.27
Trang 8r2 inf
k≥p1
p − μα β k1 αkλ pλ 1n
k
k μ1− β
1/kp
Proof By the form1.1, we get that
1zfz/fz p
1zfz/fz− p 2μ
∞
kp1 k
k pa k z k
2p
p − μz −p∞
kp1 k
k − p 2μa k z k
≤
∞
kp1 k
k p|z| k 2p
p − μa k |z| −p∞kp1 k
k − p 2μa k |z| k
∞
kp1 k
k pa k |z| kp 2p
p − μ∞
kp1 k
k − p 2μa k |z| kp
2.29
Then the following incurability:
1zfz/fz p
1zfz/fz− p 2μ
≤1,
0≤ μ < p; p ∈ N 2.30
will hold if
∞
kp1
k
k μ
p
p − μ a k |z| kp ≤ 1,
0≤ μ < p; p ∈ N. 2.31
Then by Corollary2.3the inequality2.31 will be true if
k
k μ
p
p − μ |z| kp ≤
p
α β k1 αkλ pλ 1n
p
0≤ μ < p; p ∈ N, 2.32
that is,
|z| kp≤
p − μα β k1 αkλ pλ 1n
k
k μ1− β ,
0≤ μ < p; p ∈ N. 2.33
Therefore the inequality2.33 leads us to the disc |z| < r2, where r2 is given by the form
2.28
3 Properties of the Class Γ∗
λn, α, β
We first give the necessary and sufficient conditions for functions f in order to be in the class
Γ∗
λ n, α, β.
Trang 9Lemma 3.1 see 2
R a
⎧
⎨
⎩
a − δ, for a ≤ 2μ δ,
2
μ
a − μ − δ, for a ≥ 2μ δ. 3.1 Then
{w : |w − a| ≤ R a} ⊆w : w − μ δ ≤ Rew μ − δ. 3.2
Lemma 3.2 Let α ≥ 0 and 0 ≤ β < 1
R a
⎧
⎨
⎩
a − αβ, for a ≤ 2α αβ,
2
α
a − α − αβ, for a ≥ 2α αβ. 3.3 Then
{w : |w − a| ≤ R a} ⊆w : w − α αβ ≤ Rew α − αβ. 3.4
Proof Since α ≥ 0 and 0 ≤ β < 1, then α > αβ Then in Lemma3.1, put μ α and δ αβ.
Theorem 3.3 Let f ∈ Σ p Then f is in the class Γ∗
λ n, α, β if and only if
∞
kp1
k pαβkλ pλ 1n a k ≤ p1− αβ α ≥ 1
2 β; 0≤ β < 1; p ∈ N; n ∈ N0
.
3.5
Proof Suppose that f∈ Γ∗
λ n, α, β Then by the inequality 1.9, we get that
z
D n
λ fz
p
D n
λ fz α αβ
≤Re
−z
D n
λ fz
p
D n
λ fz α − αβ. 3.6
That is,
Re
z
D n
λ fz
p
D n
λ fz α αβ ≤
z
D n
λ fz
p
D n
λ fz α αβ
≤ Re
−z
D n
λ fz
p
D n
λ fz α − αβ,
3.7
Trang 10that is,
Re
2z
D n
λ fz
p
D n
Hence by the inequality1.3,
Re
−2p1− αβ∞kp12
k pαβkλ pλ 1n a k z kp
p ∞kp1 p
kλ pλ 1n a k z kp ≤ 0. 3.9
Taking z to be real and putting z → 1−through real values, then the inequality3.9 yields
−2p1− αβ∞kp12
k pαβkλ pλ 1n a k
p ∞kp1 p
kλ pλ 1n a k ≤ 0, 3.10
which leads us at once to3.5
In order to prove the converse, consider that the inequality 3.5 holds true In Lemma 3.2 above, since α > αβ and α ≥ 1/2 β, that is, 1 ≤ 2α αβ, we can put
a 1 Then for p ∈ N and n ∈ N0, let w np −zD n
λ fz/pD n
λ fz Now, if we let
z ∈ ∂U∗ {z ∈ C : |z| 1}, we get from the inequalities 1.3 and 3.5 that |w np − 1| ≤ R1 Thus by Lemma3.2above, we ge that
z
D n
λ fz
p
D n
λ fz α αβ
−
z
D n
λ fz
p
D n
λ fz −
α αβ
w − α αβ
≤ Rew α − αβ Re{w} α − αβ
−z
D n
λ fz
p
D n
λ fz α − αβ,
α ≥ 1
2 β; 0≤ β < 1; p ∈ N; n ∈ N0
.
3.11
Therefore by the maximum modulus theorem, we obtain f ∈ Γ∗
λ n, α, β.
Corollary 3.4 If f ∈ Γ∗
λ n, α, β, then
1− αβ
k pαβkλ pλ 1n
α ≥ 1
2 β; 0≤ β < 1; p ∈ N; n ∈ N0
Trang 11
The result is sharp for the function fz given by
fz z −p ∞
kp1
p
1− αβ
k pαβkλ pλ 1n z
k
α ≥ 1
2 β; 0≤ β < 1; p ∈ N; n ∈ N0
.
3.13
Theorem 3.5 The class Γ∗
λ n, α, β is closed under convex linear combinations.
Proof This proof is similar as the proof of Theorem2.4
The following are the growth and distortion theorems for the classΓ∗
λ n, α, β.
Theorem 3.6 If f ∈ Γ∗
λ n, α, β, then
p m − 1!
p − 1! −
1− αβ
1 αβp 1n−1 ·
p!2 −n
p − m!r 2p r −pm≤ f m z
≤
p m − 1!
p − 1!
1− αβ
1 αβp 1n−1·
p!2 −n
p − m!r 2p r −pm
0 < |z| r < 1; α ≥ 1
2 β; 0≤ β < 1; p ∈ N; n, m ∈ N0; p > m
.
3.14
The result is sharp for the function f given by
fz z −p ∞
kp1
1− αβ
1 αβ2p 2n z
p ,
n ∈ N0; p ∈ N. 3.15
Next we determine the radii of meromorphically p-valent starlikeness of order μ 0 ≤ μ < p and meromorphically p-valent convexity of order μ 0 ≤ μ < p for the class Γ∗
λ n, α, β.
Theorem 3.7 If f ∈ Γ∗
λ n, α, β, then f is meromorphically p-valent starlike of order μ 0 ≤ μ < 1
in the disk |z| < r1, that is,
Re
−zfz
fz
> μ
0≤ μ < p; |z| < r1; p ∈ N, 3.16
where
r1 inf
k≥p1
p − μk pαβkλ pλ 1n
p
k μ1− αβ
1/kp
Proof This proof is similar to the proof of Theorem2.6
Trang 12Theorem 3.8 If f ∈ Γ∗
λ n, α, β, then f is meromorphically p-valent convex of order μ 0 ≤ μ < 1
in the disk |z| < r2, that is,
Re
−1 −zf fz z
> μ
0≤ μ < p; |z| < r2; p ∈ N, 3.18
where
r2 inf
k≥p1
p − μk pαβkλ pλ 1n
k
k μ1− αβ
1/kp
Proof This proof is similar to the proof of Theorem2.7
4 Subordination Properties
If f and g are analytic functions in U, we say that f is subordinate to g, written symbolically
as follows:
if there exists a function w which is analytic in U with
such that
Indeed it is known that
fz ≺ gz z ∈ U ⇒ f0 g0, fU ⊂ gU. 4.4
In particular, if the function g is univalent in U we have the following equivalencesee 18
fz ≺ gz z ∈ U ⇐⇒ f0 g0, fU ⊂ gU. 4.5
Let φ : C2 → C be a function and let h be univalent in U If J is analytic function in U and
satisfied the differential subordination φJz, Jz ≺ hz then J is called a solution of the
differential subordination φJz, Jz ≺ hz The univalent function q is called a dominant of
the solution of the differential subordination, J ≺ q