Paper23 Acta27 2011 [612109]

Acta Universitatis Apulensis
ISSN: 1582-5329No. 27/2011
pp. 225-228
A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY
MULTIPLIER TRANSFORMATION
Laura Stanciu and Daniel Breaz
Abstract. In this paper, we consider the multiplier transformation
Ip(n, λ)f(z) =zp+∞/summationdisplay
k=p+1/parenleftbiggk+λ
p+λ/parenrightbiggn
akzk
where p∈N, n∈N∪0, λ≥0 and we provide the sufficient conditions for functions
to be in the class B(n, µ, α, λ ).
2000 Mathematics Subject Classification. 30C45.
Key words and phrases. Univalent function, Starlike function, Convex function,
Multiplier transformation.
1. Introduction and Preliminaries
LetApdenote the class of functions of the form
f(z) =zp+∞/summationdisplay
k=p+1akzk, p ∈N={1,2, …}
which are analytic in the open unit disk
U={z∈C:|z|<1}.
LetSpdenote the subclass of functions that are univalent in U.
A function f∈Apis said to be p-valent starlike of order α(0≤α < p ) inU,if it
satisfies the following inequality:
Re/parenleftbiggzf/prime(z)
f(z)/parenrightbigg
> α, z ∈U.
We denote by S∗
p(α) the class of all such functions.
A function f∈Apis said to be p-valent convex of order α(0≤α < p ) inU,if and
only if
Re/parenleftbiggzf/prime/prime(z)
f/prime(z)+ 1/parenrightbigg
> α, z ∈U
225

Laura Stanciu and Daniel Breaz – A Subclass of Analytic Functions defined…
for some α,(0≤α <1).
We denote by Kp(α) the class of all those functions f∈Apwhich are multivalently
convex of order αinUand denote by R(α) the class of functions in Apwhich satisfy
Ref/prime(z)> α, z ∈U.
It is well known that Kp(α)⊂S∗
p(α)⊂Sp.
Iffandgare analytic functions in U,we say that fis subordinate to g,written
f≺gifw(0) = 0 ,|w(z)|<1,for all z∈U.Ifgis univalent then f≺gif and only
iff(0) = g(0) and f(U)⊆g(U).
The following multiplier transformation was given by Sukhwinder Singh, Sushma
Gupta and Sukhjit Singh [1].
Definition 1. ([1]). Forf∈Ap, p∈N, n∈N∪0, λ≥0,the operator Ip(n, λ)f(z)
is defined by the following infinite series
Ip(n, λ)f(z) =zp+∞/summationdisplay
k=p+1/parenleftbiggk+λ
p+λ/parenrightbiggn
akzk. (1)
It is easily verified from (1) that
(p+λ)Ip(n+ 1, λ)f(z) =p(1−λ)Ip(n, λ)f(z) +λz(Ip(n, λ)f(z))/prime. (2)
Remark 1. Ifp= 1we have I1(n, λ)f(z) =I(n, λ)and(λ+ 1)I(n+ 1, λ)f(z) =
(1−λ)I(n, λ)f(z) +λz(I(n, λ)f(z))/prime,forz∈U.
Remark 2. Iff∈An, f(z) =z+/summationtext∞
k=p+1akzk,then
I(n, λ)f(z) =z+∞/summationdisplay
k=p+1/parenleftbiggk+λ
p+λ/parenrightbiggn
akzk,
forz∈U.
In the proof of our main result we need the following lemma.
Lemma 1. ([2]).Letube analytic in Uwithu(0) = 1 and suppose that
Re/parenleftbigg
1 +zu/prime(z)
u(z)/parenrightbigg
>3α−1
2α, z∈U. (3)
Then Reu(z)> α forz∈Uand1
2≤α <1.
2. Main results
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Laura Stanciu and Daniel Breaz – A Subclass of Analytic Functions defined…
Definition 2. We say that a function f∈Apis in the class B(n, µ, α, λ ), n∈
N, µ≥0, α∈[0,1).
If/vextendsingle/vextendsingle/vextendsingle/vextendsingleI(n+ 1, λ)
z/parenleftbiggz
I(n, λ)f(z)/parenrightbiggµ
−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1−α, z∈U. (4)
In this paper we provide a sufficient condition for functions to be in the class
B(n, µ, α, λ ).
Theorem 1. For the functions f∈Ap, n∈N, µ≥0,1
2≤α <1.
If
(λ+ 1)
λI(n+ 2, λ)f(z)
I(n+ 1, λ)f(z)−µ(λ+ 1)
λI(n+ 1, λ)f(z)
I(n, λ)f(z)+1
λ(µ−1)≺1 +βz, z∈U(5)
where
β=3α−1
2α
thenf∈B(n, µ, α, λ ).
Proof. If we consider
u(z) =I(n+ 1, λ)f(z)
z/parenleftbiggz
I(n, λ)f(z)/parenrightbiggµ
,
then u(z) is analytic in Uwith u(0) = 1 .A simple differentation yields
zu/prime(z)
u(z)=(λ+ 1)
λI(n+ 2, λ)f(z)
I(n+ 1, λ)f(z)−µ(λ+ 1)
λI(n+ 1, λ)f(z)
I(n, λ)f(z)+(µ−1)
λ
Using (4) we get
Re/parenleftbigg
1 +zu/prime(z)
u(z)/parenrightbigg
>3α−1
2α.
From Lemma 1. we have
Re/parenleftbiggI(n+ 1, λ)f(z)
z/parenleftbiggz
I(n, λ)f(z)/parenrightbiggµ/parenrightbigg
> α.
Therefore, f∈B(n, µ, α, λ ),by Definition 2.
3. Applications of Theorem 1.
First of all, setting n= 1, µ= 1, α=1
2, λ= 1 in Theorem 1, we immediatly
arrive at the following application of Theorem 1. we have
Corollary 1. Iff∈A1and
Re/parenleftbiggzf/prime(z) + 3z2f/prime/prime(z) +z3f/prime/prime/prime(z)
zf/prime(z) +z2f/prime/prime(z)−zf/prime(z) +z2f/prime/prime(z)
zf/prime(z)/parenrightbigg
>−1
2
227

Laura Stanciu and Daniel Breaz – A Subclass of Analytic Functions defined…
thenf∈B/parenleftbig
1,1,1
2,1/parenrightbig
.
Setting n= 1, µ= 0, α=1
2, λ= 1 we obtain the following interesting consequence
of Theorem 1.
Corollary 2. Iff∈A1and
Re/parenleftbiggzf/prime(z) + 3z2f/prime/prime(z) +z3f/prime/prime/prime(z)
zf/prime(z) +z2f/prime/prime(z)/parenrightbigg
>−3
2
thenf∈B/parenleftbig
1,0,1
2,1/parenrightbig
.
Setting n= 0, µ= 1, α=1
2, λ= 1 we obtain another consequence of Theorem
1.
Corollary 3. Iff∈A1andRe/parenleftBig
zf/prime(z)+z2f/prime/prime(z)
zf/prime(z)−zf/prime(z)
f(z)/parenrightBig
>−1
2thenf∈B/parenleftbig
0,1,1
2,1/parenrightbig
.
Finally, setting n= 0, µ= 0, α=1
2, λ= 1 we obtain the next consequence of
Theorem 1.
Corollary 4. Iff∈A1andRe/parenleftBig
zf/prime(z)+z2f/prime/prime(z)
zf/prime(z)/parenrightBig
>3
2thenf∈B/parenleftbig
0,0,1
2,1/parenrightbig
.
Acknowledgments: This work was partially supported by the strategic project
POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 co-financed by the
European Social Fund-Investing in People.
References
[1]. Sukhwinder Singh, Sushma Gupta and Sukhjit Singh, On a subclass of analytic
functions , General Mathematics Vol. 16, No. 2 (2008), 37-47.
[2]. B. A. Frasin and Jay M. Jahangiri, A new and comprehensive class of analytic
functions , Analele Universit˜ at ¸ii din Oradea, Tom XV, 2008, 61-64.
Laura Filofteia Stanciu
University of Pite¸ sti
Department of Mathematics
Tˆ argul din Vale Str., No.1, 110040, Pite¸ sti
Arge¸ s, Romˆ ania.
E-mail address: laura stanciu 30@yahoo.com
Daniel Breaz
”1 Decembrie 1918”’ University of Alba Iulia
Department of Mathematics
Alba Iulia, Str. N. Iorga, 510000, No. 11-13, Romˆ ania.
E-mail address: dbreaz@uab.ro
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