ON SOME PROPERTIES OF A GENERALIZED CLASS OF
CLOSE-TO-STARLIKE FUNCTIONS
Ajab Bai Akbarally1 and Nor Siti Khadijah Arunah2
Faculty of Computer and Mathematical Sciences
1Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia
2Universiti Teknologi MARA Cawangan Johor, Kampus Segamat, 85000 Segamat, Johor,
Malaysia
1[email protected], 2[email protected]
ABSTRACT
In this paper, we consider a new class of close-to-starlike functions denoted byCS,, defined by the Carlson-Shaffer operatorL
,
. Let Sdenote the classof analytic univalent functions
f
defined by
2 ,
n n nz
a z z
f then fCS,
if
f
satisfy the condition
z E zg z
f
, 0 ,
Re L , where
2 1
1 ,
n
n n n n a z
z z f
L and g(z) is a starlike function. Properties of
the class CS, such as the coefficient bounds, growth and distortion theorems and radius results are investigated.
Keywords: close-to-starlike functions, analytic functions, starlike functions, Carlson-Shaffer operator, coefficient bound, growth and distortion theorems, radius results.
1. Introduction
Let A denote the class of functions of the form
2
n n nz
a z z
f (1.1)
that are analytic and normalized in the open unit disk E
z:z 1
and Sbe the subclass ofAconsisting of functions that are univalent inE.
Denote by P, the class of functions with positive real part, where functions in this class are of the form
1
1 ) (
n n nz
p z
et. al., 2018). Reade (1955) defined the class of close-to-starlike functions where functions in this class satisfy
z Ez z
f
, 0 Re
with
z belonging to S.Reade (1955) obtained that f(z) is close-to-starlike if and only if the inequality,
dre f
re f re
i i i
2
1
Re
holds for all12 and for all 0r1. The class of close-to-starlike functions is denoted by
CS .
Srivastava and Attiya (2007) introduced a family of linear operatorJ,b:AA by the Hadamard product of the Hurwitz-Lerch Zeta function with analytic functions as
f z G f
z J,b ,bwhere bC withb0,1,2,3,, μC, zEand G,bA isgiven by
2 ,
1 ; , ) 1 (
n
n n b
z a b n
b z
b z b b
G
Srivastava et al. (2013), then introduced a subclass of close-to-starlike functions using Srivastava-Attiya operator denoted by
CS
s,b where they gave the following definition.Definition 1.1 A function f is said to belong to the class
CS
s,b if and only if, there exists a functiong
S
such that
0 , .Re , z E
z g
z f
Jsb
In the special case whens0, the class
CS
s,bcan be reduced to the class CS, studiedearlier by Reade (1955).
This paper will define a new class of close-to-starlike functions using the operator defined by Carlson and Shaffer (1984), where the Carlson-Shaffer linear operator
A A : ) , (
L is given by
) ( ) ; , ( ) ( ) ,
( f z z f z L
, 0, 1, 2, ); , (
0 1
1 1 0
n
n n
n n
n n
n
z z
z z
The function,
(
,
;z) is known as the incomplete beta function. The term(
)
k is the Pochhammer symbol that can be expanded in Gamma functions as
. , 2 , 1
1 1
0 1 : )
( k k N
k k
k
k
Thus, Carlson-Shaffer linear operator can be written as,
,1
1a z E
n n n
n
z
z )f(z) , L(
2
n
(1.3)
We define our new class of functions as follows.
Definition 1.2 A function f is said to belong to the class of
CS
, if and only if there exists a function gSsuch thatRe
,
0, z E zg z
f
L
(1.4)
In the special case, when 1 and 1, the class
CS
, reduces to the class CS, studied by Reade (1955).The main objective of this paper is to find the coefficient inequalities and basic properties of the class
CS
,.2. Coefficient bounds
Theorem 2.1 Let f in the form of (1.1) be in the class
CS
*,. Then,
, \
11 1
2 N
n
n a
n n
n
(2.1)
Let
1 1 ) ( n n nz p zp and
1 ) ( n n nz b z z
g (2.3)
Then, by substituting (1.3) and (2.3) into (2.2), we have
, 1 1
2 1 1
b z p b b z z a z n n n k k n k n n n 2n n-1
1 -n ) ( ) ( (2.4)
Comparing coefficient of
z
non both sides of (2.4) gives
.
2 1 1
n n k k n k nn
b
b
p
a
2 n n-1
1 -n
)
(
)
(
(2.5) Also using the fact that
1
,
and
,
2
N
N
\
n
b
n
n
p
n nequation (2.5) becomes
. n n k n a n kn 2 1
1
1
2 N\
1 -n 1 -n ) ( ) (
So, we have that,
1,
2 N\
n n
an 1 -n 1 -n ) ( ) (
which will complete the proof of Theorem 2.1.
In its specific cases when
1 and
1, we obtain
1
,
2
N
\
n
n
a
nwhich is the result of Reade (1955).
3. Growth and distortion theorem
Using the coefficient bound we now derive the other properties for the class
CS
*,. Firstly we find the growth theorem for functions in the classCS
*,.Theorem 3.1 Let
f
CS
*,. Then,
, z .2 1 1 2 2 2 1 1 2 2 r E z n r r z f n r r n n n n n
n
Proof Let
f
CS
*,. By taking absolute values on both sides of (1.1), we have
2 n n nz
a
z
)
z
(
f
Hence, by using triangular inequalities, we have,
r
z
a
z
)
z
(
f
z
a
z
n n n n nn
2
2 n
n n n
n
n
r
f
(
z
)
r
a
r
a
r
. (3.1)By substituting (2.1) into (3.1), we have
2 1
1 2 2
2 1
1 2 2
n n
n
n n
n
f
(
z
)
r
r
n
n
r
r
.
Thus this completes the proof of Theorem 3.1.
We next derive the distortion theorem for functions in the class
CS
,. Theorem 3.2 Letf
CS
,. Then,
( ) 1
, z .1
2 1
1 3
2 1
1 3
r E
z n
r z
f n
r
n n
n
n n
n
Proof Let
f
CS
,. For distortion theorem, we differentiate (1.1) to obtain
1 21
nn n
z
na
z
f
By taking absolute values on both sides, we have
1 2
1
)
(
nn n
z
na
z
f
Hence, by using triangular inequalities, we have,
r
z
na
)
z
(
f
z
na
n
n n n
n
n
z
:
1
1
2
1 2
1
2
1 2
1 ( ) 1
1
n
n n n
n
nr f z na r
a
n (3.2)
By substituting (2.1) into (3.2), we have
2 1
1 3
2 1
1 3
1
1
n n
n
n n
n
f
(
z
)
r
n
n
r
.
Thus this completes the proof for Theorem 3.2.
4. Radius of convexity, starlikeness and close-to-convexity
Results on radius of convexity, starlikeness and close-to-convexity for the class CS*, will be obtained in this section.
A function fS is said to be convex of order (01) if
,
11 1
1
1 1
3 N\
n n n z n n n Proof
It is adequate to show that the values for
) ( ) ( 1 z f z f z
liein a circle centered at w1 with
radius 1. We have
1 1 1 1 ) ( ) ( 1 2 1 2 1 n n n n n n z a n z a n n z f z f z (4.2)
1
1
12 1 2 1
n n n n nn z na z
a n
n
Finally, we get
1 1 2 1
n n n z a n nIn view of Theorem 2.1, we have
,
1 1 1 1 1 2 1 \ N n n z n n n n n (4.3)Solving (4.3) for
z
we obtain
,
1.1 1
1
1 1
3 N\
n n n z n n n
This completes the proof of Theorem 4.1.
A function f S is said to be starlike of order (01) if it satisfies
. ) ( ) (
Re
z f z f z (4.4)
Theorem 4.2 Let the functions in the form of (1.1) be in the class of CS*, . Then, f is starlike of order
if,
,
1.1 1
1
1 1
2 N\
n n n z n n n
Proof We must show that,
1 1 1 1 ) ( ) ( 2 1 2 1 n n n n n n z a z a n z f z f z (4.5)
,
1 11 1
1 2 1
\ N
n n
z n
n n n
(4.6)
Solving (4.6) for
z
we obtain
,
11 1
1
1 1
2 N\
n
n n
z n
n n
This completes the proof of Theorem 4.2.
A function fS is said to be close-to-convex of order (01) if
( )
.Re f z
(4.7) Theorem 4.3 Let the functions in the form of (1.1) be in the class ofCS
*, . Then, f is close-to-convex of order
if,
,
n
.
n
z
n
n n
1
1
11
1 1
3
N
\
Proof We must show that
1 1
2
1
n
n n z
a n )
z (
f (4.8)
But in view of Theorem 2.1, the inequality (4.8) holds if
,
1
1
1
11 2 1
\
N
n
n
z
n
n n n
(4.9)Solving (4.9) for
z
we obtain
,
n
.
n
z
n
n
n
1
1
11
1 1
3
N
\
This completes the proof of Theorem 4.3.
5. Conclusion
In this paper, a certain class of analytic functions in the complex plane is discussed. The class of analytic univalent functions is denoted by
S
.This paper is specifically focused on a class of close-to-starlike functions defined using Carlson-Shaffer operator. This class is denoted by* ,
CS
.We find the coefficient bounds, growth and distortion theorems and radius propertiesfor the class defined.
References
Hurwitz-Goodman, A. W. (1983). Univalent Functions, vols I & II,(Mariner publishing Company Inc., Tampa Florida, 1983).
Peter, O., Akinduko, O., Ishola, C., Afolabi, O., & Ganiyu, A. (2018). Series Solution Of Typhoid Fever Model Using Differential Transform Method. Malaysian Journal of Computing, 3(1), 67-80.
Rathi, S., Cik Soh, S., & Akbarally, A. (2018). Coefficient Bounds For A Certain Subclass Of Close-To-Convex Functions Associated With The Koebe Function. Malaysian Journal of Computing, 3(2), 172-177.
Reade, M. O. (1955). On close-to-convex univalent functions. Michigan Math. J., 3, 59-62.
