Volume 2011, Article ID 401913,16pages doi:10.1155/2011/401913
Research Article
On Certain Subclasses of Meromorphically
p
-Valent
Functions Associated by the Linear Operator
D
n
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,[email protected]
Received 26 July 2010; Accepted 28 February 2011
Academic Editor: Jong Kim
Copyrightq2011 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 Γλn, α, β and Γ∗λn, α, β of meromorphicp-valent functions by using the linear operatorDnλ. Then we prove the necessary and sufficient conditions for a functionf 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
akzk ak≥0; p∈N{1,2, . . .},
1.1
which are meromorphic andp-valent in the punctured unit discU∗{z∈C: 0<|z|<1} U− {0}. For the functionsfin the classΣp, we define a linear operatorDnλby the following form:
Dλfz 1pλfz λzfz, λ≥0, D0
λfz fz, D1
λfz Dλfz,
D2
λfz Dλ
D1
λfz
,
and in general forn0,1,2, . . ., we can write
Dn λfz
1 zp
∞
kp1
1pλkλnakzk, n∈N0N∪ {0}; p∈N
. 1.3
Then we can observe easily that forf∈Σp,
zλDn λfz
Dn1
λ fz−
1pλDnλfz, p∈N; n∈N0
. 1.4
Recall 1,2that a functionf ∈ Σpis said to be meromorphically starlike of orderαif it is satisfying the following condition:
Re
−zffzz
> α, z∈U∗, 1.5
for someα0 α <1. Similarly recall 3a functionf ∈Σpis said to be meromorphically convex of orderαif it is satisfying the following condition:
Re
−1−zf
z
fz
> α, z∈U∗for someα0≤α <1. 1.6
LetΣpαbe a subclass ofΣpconsisting the functions which satisfy the following inequality:
Re −z
Dn λfz
Dn λfz
> pα, z∈U∗; α≥0. 1.7
In the following definitions, we will define subclassesΓλn, α, βandΓ∗λn, α, βby using the linear operatorDλn.
Definition 1.1. For fixed parametersα≥0, 0≤β <1, the meromorphicallyp-valent function fz∈Σpαwill be in the classΓλn, α, βif it satisfies the following inequality:
Re −z
Dn λfz
pDn λfz
≥αz
Dn λfz
pDn λfz
1β, n∈N0. 1.8
Definition 1.2. For fixed parametersα≥ 1/2β; 0 ≤β < 1, the meromorphicallyp-valent functionfz∈Σpαwill be in the classΓ∗λn, α, βif it satisfies the following inequality:
zDn λfz
pDn λfz
ααβ≤Re −z
Dn λfz
pDn λfz
Meromorphically multivalent functions have been extensively studied by several authors, see for example, Aouf 4–6, Joshi and Srivastava 7, Mogra 8,9, Owa et al. 10, Srivastava et al. 11, Raina and Srivastava 12, Uralegaddi and Ganigi 13, Uralegaddi and Somanatha
14, and Yang 15. Similarly, in 16, some new subclasses of meromorphic functions in the punctured unit disk was considered.
In 17, similar results were proved by using thep-valent functions that satisfy the following differential subordinations:
zIpr, λfzj1
p−jIpr, λfzj ≺
aaB A−Bβz
a1Bz 1.10
and studied the related coefficients inequalities withβcomplex 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 classesΓλn, α, βand Γ∗
λn, α, βby giving the necessary and sufficient conditions. Further we study the growth and distortion theorems and determine the radii of meromorphically p-valent starlikeness of orderμ0 ≤ μ < pand meromorphicallyp-valent convexity of orderμ0 ≤ μ < p. In Section4we give some results related to the subordination properties.
2. Properties of the Class
Γ
λn, α, β
We begin by giving the necessary and sufficient conditions for functionsf in order to be in the classΓλn, α, β.
Lemma 2.1see 2. Let
Ra
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
a−αβ
1α, fora≤1 1−β α1α,
1−a21−α2−21−β1−a, fora≥1 1−β
α1α.
2.1
Then
{w:|w−a| ≤Ra} ⊆w: Rew≥α|w−1|β. 2.2
Theorem 2.2. Letf ∈Σp. Thenfis in the classΓλn, α, βif and only if
∞
kp1
pαβk1αkλpλ1nak≤p1−β
α≥0; 0≤β <1; p∈N; n∈N0
.
Proof. Suppose thatf∈Γλn, α, β. Then by the inequalities1.3and1.8, we get that
Re −z
Dn λfz
pDn λfz
≥αz
Dn λfz
pDn λfz
1
β. 2.4
That is,
Re 1−
∞
kp1
k/pkλpλ1nakzkp 1∞kp1kλpλ1nakzkp
≥α
∞
kp1
k/p1kλpλ1nakzkp 1∞kp1kλpλ1nakzkp
β
≥Re α·
∞
kp1
k/p1kλpλ1nakzkp 1∞kp1kλpλ1nakzkp
β
Re β
∞
kp1
αk/p1βkλpλ1nakzkp 1∞kp1kλpλ1nakzkp
, 2.5 that is, Re p
1−β−∞kp1kkαpαpβkλpλ1nakzkp 1∞kp1kλpλ1nakzkp
≥0. 2.6
Takingzto be real and puttingz → 1−through real values, then the inequality2.6yields p1−β−∞kp1kkαpαpβkλpλ1nak
1∞kp1kλpλ1nak ≥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/α1α, put a 1. Then forp ∈ N andn ∈ N0, let
wnp −zDn
λfz/pDnλfz. If we let z ∈ ∂U∗ {z ∈ C : |z| 1}, we get from the inequalities1.3and2.3that|wnp−1| ≤R1. Thus by Lemma2.1above, we ge that
Re −z
Dn λfz
pDn λfz
−1
Rewnp≥αwnp−1βα
−
zDn λfz
pDn λfz −1 β α
zDn λfz
pDn λfz 1 β,
α≥0; 0≤β <1; p∈N; n∈N0
.
2.8
Corollary 2.3. Iff ∈Γλn, α, β, then
ak≤ p
1−β
pαβk1αkλpλ1n,
α≥0; 0≤β <1; p∈N; n∈N0
. 2.9
The result is sharp for the functionfzgiven by
fz z−p ∞ kp1
p1−β
pαβk1αkλpλ1nz
k, α≥0; 0≤β <1; p∈N; n∈N
0
.
2.10
Theorem 2.4. The classΓλn, α, βis closed under convex linear combinations.
Proof. Suppose the function
fz z−p ∞ kp1
akzk,j ak,j ≥0; j1,2; p∈N, 2.11
be in the classΓλn, α, β. It is sufficient to show that the functionhzdefined by
hz 1−δf1z δf2z 0≤δ≤1, 2.12
is also in the classΓλn, α, β. Since
hz z−p ∞ kp1
1−δak,1δak,2zk,j, 0≤δ≤1, 2.13
and by Theorem2.2, we get that
∞
kp1
pαβk1αkλpλ1n 1−δak,1δak,2
∞ kp1
1−δpαβk1αkλpλ1nak,1
∞ kp1
δpαβk1αkλpλ1nak,2
≤1−δp1−βδp1−βp1−β, α≥0; 0≤β <1; p∈N; n∈N0
.
2.14
Hencef∈Γλn, α, β.
Theorem 2.5. Iff ∈Γλn, α, β, then
pm−1!
p−1! −
1−β
2αβ1p1n−1 · p!2−n
p−m!r 2p
r−pm≤fmz
≤ pm−1
!
p−1!
1−β
2αβ1p1n−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 functionfgiven by
fz z−p ∞ kp1
1−β
2αβ12p2nz
p, n∈N
0; p∈N
. 2.16
Proof. From Theorem2.2, we get that
p2αβ12p2n
p1!
∞
kp1
k!ak ≤
∞
kp1
pαβk1αkλpλ1nak
≤p1−β,
2.17
that is,
∞
kp1
k!ak≤ p
1−βp1! p2αβ12p2n
1−βp!2−n
2αβ1p1n−1. 2.18
By the differentiating the functionfin the form1.1mtimes with respect toz, we get that
fmz −1m
pm−1!
p−1! z
−pm ∞
kp1
k! k−m!akz
k−m, m∈N
0; p∈N
2.19
and Theorem2.5follows easily from2.18and2.19. Finally, it is easy to see that the bounds in2.15are attained for the functionfgiven by2.18.
Next we determine the radii of meromorphicallyp-valent starlikeness of orderμ0≤ μ < pand meromorphicallyp-valent convexity of orderμ0≤μ < pfor the classΓλn, α, β. Theorem 2.6. Iff ∈Γλn, α, β, thenfis meromorphicallyp-valent starlike of orderμ0≤μ <1
in the disk|z|< r1, that is,
Re
−zffzz
> μ 0≤μ < p; |z|< r1; p∈N
where
r1kinf
≥p1
p−μpαβk1αkλpλ1n pkμ1−β
1/kp
. 2.21
Proof. By the form1.1, we get that
zfz/fzp
zfz/fz−p2μ
∞
kp1
kpakzk
2p−μz−p∞kp1k−p2μakzk
≤
∞
kp1
kp|z|k
2p−μak|z|−p∞kp1
k−p2μak|z|k
∞
kp1
kpak|z|kp
2p−μ∞kp1k−p2μak|z|kp.
2.22
Then the following incurability
zfz/fzp
zfz/fz−p2μ
≤1,
0≤μ < p; p∈N 2.23
also holds if
∞
kp1
kμ
p−μak|z|
kp≤1, 0≤μ < p; p∈N.
2.24
Then by Corollary2.3the inequality2.24will be true if
kμ
p−μ|z| kp≤
pαβk1αkλpλ1n
p1−β ,
0≤μ < p; p∈N, 2.25
that is,
|z|kp≤
p−μpαβk1αkλpλ1n
pkμ1−β ,
0≤μ < p; p∈N. 2.26
Therefore the inequality2.26leads us to the disc|z| < r1, where r1 is given by the form
2.21.
Theorem 2.7. Iff ∈Γλn, α, β, thenfis meromorphicallyp-valent convex of orderμ0≤μ <1
in the disk|z|< r2, that is,
Re
−1−zf
z
fz
> μ 0≤μ < p; |z|< r2; p∈N
where
r2 inf
k≥p1
p−μαβk1αkλpλ1n kkμ1−β
1/kp
. 2.28
Proof. By the form1.1, we get that
1zfz/fzp 1zfz/fz−p2μ
∞
kp1k
kpakzk
2pp−μz−p∞kp1kk−p2μakzk
≤
∞
kp1k
kp|z|k
2pp−μak|z|−p∞kp1kk−p2μak|z|k
∞
kp1k
kpak|z|kp
2pp−μ∞kp1kk−p2μak|z|kp .
2.29
Then the following incurability:
1zfz/fzp 1zfz/fz−p2μ
≤1,
0≤μ < p; p∈N 2.30
will hold if
∞
kp1
kkμ pp−μak|z|
kp≤1, 0≤μ < p; p∈N.
2.31
Then by Corollary2.3the inequality2.31will be true if kkμ
pp−μ|z| kp≤
pαβk1αkλpλ1n
p1−β ,
0≤μ < p; p∈N, 2.32
that is,
|z|kp≤
p−μαβk1αkλpλ1n
kkμ1−β ,
0≤μ < p; p∈N. 2.33
Therefore the inequality2.33leads 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 functionsfin order to be in the class Γ∗
Lemma 3.1see 2. Letμ > δand
Ra
⎧ ⎨ ⎩
a−δ, fora≤2μδ,
2
μa−μ−δ, fora≥2μδ.
3.1
Then
{w:|w−a| ≤Ra} ⊆w:w−μδ≤Rewμ−δ. 3.2
Lemma 3.2. Letα≥0and0≤β <1
Ra
⎧ ⎨ ⎩
a−αβ, fora≤2ααβ,
2
αa−α−αβ, fora≥2ααβ.
3.3
Then
{w:|w−a| ≤Ra} ⊆w:w−ααβ≤Rewα−αβ. 3.4
Proof. Sinceα≥0 and 0≤β <1, thenα > αβ. Then in Lemma3.1, putμαandδαβ. Theorem 3.3. Letf ∈Σp. Thenfis in the classΓ∗λn, α, βif and only if
∞
kp1
kpαβkλpλ1nak≤p1−αβ α≥ 1
2β; 0≤β <1; p∈N; n∈N0
.
3.5
Proof. Suppose thatf∈Γ∗λn, α, β. Then by the inequality1.9, we get that
zDn λfz
pDn λfz
ααβ≤Re −z
Dn λfz
pDn λfz
α−αβ. 3.6
That is,
Re z
Dn λfz
pDn λfz
ααβ
≤z
Dn λfz
pDn λfz
ααβ
≤Re −z
Dn λfz
pDn λfz
α−αβ,
that is,
Re 2z
Dn λfz
pDn λfz
2αβ
≤0. 3.8
Hence by the inequality1.3,
Re −2p
1−αβ∞kp12kpαβkλpλ1nakzkp p∞kp1p
kλpλ1nakzkp
≤0. 3.9
Takingzto be real and puttingz → 1−through real values, then the inequality3.9yields
−2p1−αβ∞kp12kpαβkλpλ1nak p∞kp1p
kλpλ1nak ≤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 wnp −zDλnfz/pDnλfz. Now, if we let z∈ ∂U∗ {z ∈C:|z|1}, we get from the inequalities1.3and3.5that|wnp−1| ≤ R1.
Thus by Lemma3.2above, we ge that
zDn λfz
pDn λfz
ααβ
−z
Dn λfz
pDn λfz
−ααβ
w−ααβ
≤Rewα−αβRe{w}α−αβ
−z
Dn λfz
pDn λfz
α−αβ,
α≥ 1
2β; 0≤β <1; p∈N; n∈N0
.
3.11
Therefore by the maximum modulus theorem, we obtainf ∈Γ∗λn, α, β. Corollary 3.4. Iff ∈Γ∗λn, α, β, then
ak≤ p
1−αβ
kpαβkλpλ1n
α≥ 1
2β; 0≤β <1; p∈N; n∈N0
The result is sharp for the functionfzgiven by
fz z−p ∞ kp1
p1−αβ
kpαβkλpλ1nz
k α≥ 1
2β; 0≤β <1; p∈N; n∈N0
.
3.13
Theorem 3.5. The classΓ∗λn, α, βis 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∗λn, α, β. Theorem 3.6. Iff ∈Γ∗λn, α, β, then
pm−1!
p−1! −
1−αβ
1αβp1n−1 · p!2−n
p−m!r 2p
r−pm ≤fmz
≤ pm−1
!
p−1!
1−αβ
1αβp1n−1 · p!2−n
p−m!r 2p
r−pm
0<|z|r <1; α≥ 1
2β; 0≤β <1; p∈N; n, m∈N0; p > m
.
3.14
The result is sharp for the functionfgiven by
fz z−p ∞ kp1
1−αβ
1αβ2p2nz
p, n∈N
0; p∈N
. 3.15
Next we determine the radii of meromorphicallyp-valent starlikeness of orderμ0≤μ < p and meromorphicallyp-valent convexity of orderμ0≤μ < pfor the classΓ∗λn, α, β. Theorem 3.7. Iff ∈Γ∗λn, α, β, thenfis meromorphicallyp-valent starlike of orderμ0≤μ <1
in the disk|z|< r1, that is,
Re
−zffzz
> μ 0≤μ < p; |z|< r1; p∈N
, 3.16
where
r1kinf
≥p1
p−μkpαβkλpλ1n pkμ1−αβ
1/kp
. 3.17
Theorem 3.8. Iff ∈Γ∗λn, α, β, thenfis meromorphicallyp-valent convex of orderμ0≤μ <1
in the disk|z|< r2, that is,
Re
−1−zf
z
fz
> μ 0≤μ < p; |z|< r2; p∈N
, 3.18
where
r2kinf
≥p1
p−μkpαβkλpλ1n kkμ1−αβ
1/kp
. 3.19
Proof. This proof is similar to the proof of Theorem2.7.
4. Subordination Properties
Iff andgare analytic functions inU, we say thatf issubordinatetog, written symbolically as follows:
f≺g inU or fz≺gz z∈U 4.1
if there exists a functionwwhich is analytic inUwith
w0 0, |wz|<1 z∈U, 4.2
such that
fz gwz z∈U. 4.3
Indeed it is known that
fz≺gz z∈U ⇒f0 g0, fU⊂gU. 4.4
In particular, if the functiongis univalent inUwe have the following equivalencesee 18:
fz≺gz z∈U⇐⇒f0 g0, fU⊂gU. 4.5
Lemma 4.1 see 19. Let qz/0 be univalent in U. Letθ and φ be analytic in a domain D containingqUwithφw/0whenw∈qU. Set
Qz zqzφqz, hz θqzQz. 4.6
Suppose that
iQzis starlike univalent inU, iiRe{zhz/Qz}>0forz∈U. IfJis analytic function inUand
θJz zJzφJz≺θqzzqzφqz, 4.7
thenJz≺qzandqis the best dominant.
Lemma 4.2 see 20. Let w, γ ∈ Cand φ is convex and univalent in U with φ0 1 and
Re{wφz γ}>0for allz∈U. Ifqis analytic inUwithq0 1and
qz wqz zqzγ ≺φz z∈U, 4.8
thenqz≺φzandφis the best dominant.
Theorem 4.3. Letqz/0be univalent inUsuch thatzqz/qzis starlike univalent inUand
Re
1
γqz
zqz
qz −
zqz
qz
>0, , γ ∈C, γ /0. 4.9
Iff∈Σpsatisfies the subordination
z
Dn λfz
Dn λfz
γ
1 z
Dn λfz
Dn λfz
−z
Dn λfz
Dn λfz
≺qz γzqqzz, 4.10
thenz Dnλfz/ Dnλfz≺qzandqis the best dominant.
Proof. Our aim is to apply Lemma4.1. Setting
Jz z
Dn λfz
Dn λfz
−p
∞
kp1k
kλpλ1nakzkp 1∞kp1kλpλ1nakzkp
, n∈N0; p∈N
, 4.11
θw wandφw γ/w,γ /0. It can be easily observed thatJis analytic inU,θis analytic inC,φis analytic inC/{0}andφw/0. By computation shows that
zJz
Jz 1
zDn λfz
Dn λfz
−z
Dn λfz
Dn λfz
which yields, by4.10, the following subordination:
Jz γzJz
Jz ≺qz
γzqz
qz , 4.13
that is,
θJz zJzφJz≺θqzzqzφqz. 4.14
Now by letting
Qz zqzφqz γzqz
qz ,
hz θqzQz qz γzqz
qz .
4.15
We findQistarlike univalent inUand that
Re
zhz
Qz
Re
1
γqz
zqz
qz −
zqz
qz
>0. 4.16
Hence by Lemma4.1,z Dnλfz/ Dnλfz≺qzandqis the best dominant. Corollary 4.4. Iff ∈Σpand assume that4.9holds, then
1z
Dn λfz
Dn λfz
≺
1Az 1Bz
A−Bz
1Az1Bz 4.17
implies thatz Dλnfz/ Dnλfz≺1Az/1Bz,−1≤B < A≤1and1Az/1Bz
is the best dominant.
Proof. By setting γ1 andqz 1Az/1Bzin Theorem4.3, then we can obtain the result.
Corollary 4.5. Iff ∈Σpand assume that4.9holds, then
1 z
Dn λfz
Dn λfz
≺eαzαz 4.18
implies thatz Dnλfz/ Dλnfz≺eαz,|α|< πandeαzis the best dominant.
Theorem 4.6. Letw, γ ∈C, andφbe convex and univalent inUwithφ0 1andRe{wφzγ}> 0for allz∈U. Iff∈Σpsatisfies the subordination
1γzDnλfz/Dλnfz−w/p1zDnλfz/Dnλfz w−γpDn
λfz
/zDn λfz
≺φz, 4.19
then−z Dnλfz/p Dλnfz≺φzandφis the best dominant.
Proof. Our aim is to apply Lemma4.2. Setting
qz −z
Dn λfz
pDn λfz
p
∞
kp1k
kλpλ1nakzkp p∞kp1p
kλpλ1nakzkp,
n∈N0; p∈N
. 4.20
It can be easily observed thatqis analytic inUandq0 1. Computation shows that zqz
qz 1
zDn λfz
Dn λfz
− z
Dn λfz
Dn λfz
4.21
which yields, by4.19, the following subordination:
qz wqz zqzγ ≺φz, z∈U. 4.22
Hence by Lemma4.2,−z Dnλfz/ pDnλfz≺φzandφis the best dominant.
Acknowledgments
The authors express their sincere thanks to the referees for their very constructive comments and suggestions. The authors also acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme 05-01-09-0720RU.
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