Palestine Journal of Mathematics
Vol. 1(2) (2012) , 133–137 © Palestine Polytechnic University-PPU 2012
Some Properties for Certain Subclasses of k-fold Symmetric Functions with
Fractional Powers
S.P. Goyal and Rakesh Kumar
Communicated by Ayman Badawi
MSC2010 Classifications: Primary 30C45; Secondary 30C55.
Keywords: Starlike functions; Convex functions; Bazileviˇ c Functions.
The first-named author (S.P.G.) is thankful to CSIR, New Delhi, India for awarding Emeritus Scientistship, under scheme number
21(084)/10/EMR-II.
Abstract. In the present paper, we discuss some inclusion relations between certain subclasses of k-fold symmet-
ric functions with fractional powers. Their several interesting and important consequences along with some examples
are then discussed.
1 Introduction and preliminaries
Let H = H(U ) represent a space of analytic functions in the unit disk U = {z : z ∈C ; |z| < 1}. For a ∈ C and
n ∈ N, let
H[a,n]=
f ∈H(U ) : f (z)= a + a
n
z
n
+ a
n+1
z
n+1
+ ...,z ∈U
, (1.1)
with H
0
= H[0, 1].
We recall that a normalized function f analytic in U is called k-fold symmetric if its power series has the form
f (z)= z +
∞
n=1
a
nk+1
z
nk+1
(z ∈U ). (1.2)
and we denote this class by A
k
. Further let A
k,α
is the class of functions f (z) of the form
f (z)= z
α+1
+
∞
n=1
a
nk+α+1
z
nk+α+1
(0 ≤ α< 1), (1.3)
which are analytic in the open unit disk U
∗
= {z : z ∈C ;0 < |z| < 1}.
We note that A
k,0
= A
k
, the class of normalized k-fold symmetric functions. It is clear that the analytic function
f ∈ A
k,α
is normalized in the case when α = 0. Moreover, we have
f ∈ A
k,α
⇒ z
α
f (z) ∈ A
k
(0 ≤ α< 1)
Motivated by the works of Irmak et al. [1], Liu [3] and Zhao [5], we define the following new subclasses of A
k,α
as below
Re
μ
zf
′
(z)
f (z)
+ δ
1 +
zf
′′
(z)
f
′
(z)
+(μ + δ)(α + 1)
kz
k
1 − z
k
>β(z ∈U )
and
f (z)f
′
(z)
z
2α+1
(1 − z
k
)
2α+2
= 0, Re
f (z)
z
α+1
(1 − z
k
)
α+1
μ
f
′
(z)
z
α
(1 − z
k
)
α+1
δ
>γ (z ∈U )
μ,δ ∈ R and 0 ≤ β<α + 1, 0 ≤ γ< (α + 1)
δ
(0 ≤ α< 1).
We denote these classes by M
k,α
(μ,δ; β) and N
k,α
(μ,δ; γ ) respectively.
We note that, here and throughout this paper, the values of complex powers are taken to be their principal values.
In this investigation, we first focus on certain inequalities consisting of the differential operator:
J
k,α
(μ,δ; f )(z)= μ
zf
′
(z)
f (z)
+ δ
1 +
zf
′′
(z)
f
′
(z)
+(μ + δ)(α + 1)
kz
k
1 − z
k