Volume 2008, Article ID 280183, 5 pagesdoi:10.1155/2008/280183 Research Article The Radius of Starlikeness of the Certain Classes of p-Valent Functions Defined by Multiplier Transformati
Trang 1Volume 2008, Article ID 280183, 5 pages
doi:10.1155/2008/280183
Research Article
The Radius of Starlikeness of the Certain
Classes of p-Valent Functions Defined by
Multiplier Transformations
Mugur Acu, 1 Yas¸ar Polato ˜glu, 2 and Emel Yavuz 2
1 Department of Mathematics, ”Lucian Blaga” University of Sibiu, 5-7 Ion Ratiu Street,
Sibiu 550012, Romania
2 Department of Mathematics and Computer Science, TC ˙Istanbul K ¨ult ¨ur University,
˙Istanbul 34156, Turkey
Correspondence should be addressed to Emel Yavuz, e.yavuz@iku.edu.tr
Received 12 November 2007; Accepted 02 January 2008
Recommended by Narendra Kumar K Govil
The aim of this paper is to give the radius of starlikeness of the certain classes of p-valent functions
defined by multiplier transformations The results are obtained by using techniques of Robertson
1953,1963 which was used by Bernardi 1970, Libera 1971, Livingstone 1966, and Goel 1972 Copyright q 2008 Mugur Acu et al 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.
1 Introduction
LetH be the class of analytic functions in the open unit disc D {z ∈ C | |z| < 1} and Ha, n
be the subclasses ofH consisting of the functions of the form fz a a n z n a n1z n1· · · LetAp, n denote the class of functions fz normalized by
f z z p ∞
k np
a k z k
p, n∈ N :1, 2, 3,
1.1 which are analytic in the open unit discD In particular, we set
If f z and gz are analytic in D, we say that fz is subordinate to gz, written symbolically
as
Trang 2If there exists a Schwarz function wz which is analytic in D with w0 0, |wz| < 1 such that f z gwz, z ∈ D.
For two analytic functions fz and Fz, we say that Fz is superordinate to fz if
f z is subordinate to Fz.
For integer n ≥ 1, let Ωn denote the class of functions wz which are regular in D and satisfy the conditions w0 0, |wz| < 1, and wz z n φ z for all z ∈ D, where φz
is regular and analytic inD and satisfies|φz| < 1 for every z ∈ D Also, let P{p, n denote the class of functions pz p ∞
k n p k z k which are regular inD and satisfy the conditions
p 0 p, Re pz > 0 for all z ∈ D We note that if pz ∈ Pp, n, then
p z p1− wz
1 wz
1− z n φ z
for some functions wz ∈ Ωn and every z ∈ D.
Definition 1.1 Let f z ∈ Ap, n for m ∈ N0 N ∪ {0}, λ ≥ 0, l > 0, one defines the multiplier
transformationsIp m, λ, l on Ap, n by the following infinite series:
Ip m, λ, lfz : z p ∞
k pn
p λk − p l
p l
m
It follows that
Ip 0, λ, lfz fz,
p lI p 2, λ, lfz p 1 − λ lIp 1, λ, lfz λzI p 1, λ, lfz,
Ip
m1, λ, lIp
m2, λ, l
f z Ip
m2, λ, l
Ip
m1, λ, l
f z
1.6
for all integers m1, m2
Remark 1.2 This multiplier transformation was introduced by C˘atas¸1 For p 1, l 0, λ ≥ 0,
the operatorDm
λ : I1m, λ, 0 was introduced by Al-Oboudi 2 which reduces to the S˘al˘agean differantial operator 3 For λ 1, the operator I m
l : I1m, 1, l was studied recently by Cho
and Srivastava4 and Cho and Kim 5 The operator Im: I1m, 1, 1 was studied by
Urale-gaddi and Somanatha6 and the operator Ip m, l : I p m, 1, l was investigated recently by
Sivaprasad Kumar et al.7
Definition 1.3see 1 Let ϕz be analytic in D and ϕ0 1 A function fz ∈ Ap, n is said
to be in the classAp m, λ, l, n; ϕ if it satisfies the following subordination:
Ip m 1, λ, lfz
Definition 1.4 The radius of starlikeness of the classAp m, λ, l, n, ϕ is defined by the following For each f z ∈ A p m, λ, l, n; ϕ, let rf be the supremum of all numbers r such that
fDr is starlike with respect to the origin Then the radius of starlikeness for Ap m, λ, l, n; ϕ is
rst
Ap m, λ, l, n; ϕ inf
Trang 3Theorem 1.5 Let fz ∈ Ap, n and λ > 0, then fz belongs to the class A p m, λ, l, n; χ if and
only if F z, defined by
F z p l
λz p1−λl/λ
z
0
ζ p1−λl/λ−1 f ζdζ z p ∞
k pn
p l k − pλ
a k z k , 1.9
belongs to the classAp m 1, λ, l, n; χ.
This theorem was proved by C˘atas¸1
2 Main result
Theorem 2.1 The radius of starlikeness of the class A p m, λ, l, n, φ is
rst
⎛
λ p n λ2p n2 p lp l − 2λp
⎞
⎟
1/n
This radius is sharp because the extremal function is
f∗z λ
p l
z p
c p c − pz n
1 z n2p/n1 , c p 1 − λ l
Proof If we take c p1 − λ l/λ, then the function Fz inTheorem 1.5can be written in the form
F z p l
λz c
z
0
If we take the logarithmic derivative from2.3 and after simple calculations, we get
z F
z
F z
z c f z − cz
0ζ c−1f ζdζ
z
Since Fz is starlike, hence there exists a function wz ∈ Ωn such that
z F
z
F z
z c f z − cz
0ζ c−1f ζdζ
z
0ζ c−1f ζdζ p
1− wz
Solving for f z,
f z c p c − pwz
1 wzz c
z
0
Trang 4Taking the logarithmic derivative from2.6, we get
z f
z
f z p
1− wz
1 wz b − 1
zwz
1 wz1 bwz , 2.7 where b c − p/c p To show that fz is starlike in |z| < r0, we must show that
Re
z f
z
f z
for|z| < r0 This condition is equivalent to
1 − bRe
zwz
1 wz1 bwz
≤ Re
p1− wz
1 wz
On the other hand, we have the following relations:
Re
p1− wz
1 wz
p1−w z2
1 wz2,
1 − bRe
zwz
1 wz1 bwz
≤ 1 − bzwz
1 wz1 bwz ,
zwz ≤ n|z| n
1− |z| 2n
1−w z2
2.10
Golusin inequality, 8 Therefore, the inequality 2.9 will be satisfied if
n 1 − b|z| n
1 wz1 bwz1−w z
2
1− |z| 2n ≤ p1−w z2
1 wz2. 2.11 Simplifying and writing|z| r, we obtain
n 1 − br n
1− r 2n ≤ p
11 bwz wz . 2.12 Since|wz| ≤ |z| n r n , p|1 bwz/1 wz| ≥ p1 br n /1 r n so that 2.12 will be satisfied if
n 1 − br n
1− r 2n < p1 br n
The inequality2.13 can be written in the following form:
p − 1 − bp nr n − bpr 2n > 0, 2.14
which gives the required root r0of the theorem
To see that the result is sharp, consider the function Fz z p / 1 z n2p/n For this function, we have
f∗z λ
p l
z p
c p c − pz n
1 z n2p/n1 ,
z f
∗z
f∗z
p − 1 − bp nz n − pbz 2n
2.15
So that zf
∗z/f∗z 0 for |z| r0 Thus, f z is not starlike in any circle |z| < r if r > r0
Trang 5Remark 2.2 If we give special values to m, λ, l, n, we obtain the radius of starlikeness for the
corresponding integral operators
Acknowledgment
This paper was supported by GAR 20/2007
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