DARUS, MASLINA

HAJI MOHD, MAISARAH

English

Research Publication Journals Repository

ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 1 (2012), no.1, 1521
ISSN 1857-8365, UDC: 517.544.8:517.546
STARLIKE FUNCTION WITH RESPECT TO A BOUNDARY POINT
DEFINED BY SUBORDINATION
MAISARAH HAJI MOHD AND MASLINA DARUS
Presented at the 8th International Symposium GEOMETRIC FUNCTION THEORY AND
APPLICATIONS, 27-31 August 2012, Ohrid, Republic of Macedonia.
Abstract. For a starlike univalent function ', the class of functions f that are
starlike with respect to a boundary point satisfying the subordination
2
zf 0(z)
f(z)
+
1 + z
1  z
 '(z)
is investigated. The integral representation, growth and distortion theorem are proved
by relating this functions with Ma and Minda starlike functions. Some earlier results
are shown to be special case of the results obtained.
1. Introduction and Motivation
Let D = fz : jzj < 1g be the open unit disk of the complex plane C and A be the
class of analytic function f normalized by f(0) = 0 and f 0(0) = 1. Let w0 be an interior
or a boundary point of a set D in C. The set D is starlike with respect to w0 if the line
segment joining w0 to every other point in D lies in the interior of D. If a function f 2 A
maps D onto a starlike domain with respect to origin, then f is a starlike function. The
class of starlike functions with respect to origin is denoted by S. Analytically,
S :=

f 2 A : Re
zf 0(z)
f(z)
> 0

:
Robertson [12] took a leap forward with the characterization of the class S and dened
the class Sb of starlike functions with respect to a boundary point. Geometrically, it is
the characterization of function f 2 Sb = ff(z) = 1 + d1z + d2z
2 +    jfunivalentg such
that f(D) is starlike with respect to the boundary point f(1) := limr!1  f(r) = 0 and
lies in a half plane. The analytic description given by Robertson was
Sb :=

f 2 Sb : Re

2
zf 0(z)
f(z)
+
1 + z
1  z

> 0

:
2010 Mathematics Subject Classication. 30C45, 30C50, 30C80.
Key words and phrases. Subordination, starlike with respect to boundary point.
15
16 M. HAJI MOHD AND M. DARUS
This was partially proved in[12]. It was only in 1984 that the characterization was vali-
dated by Lyzzaik [9]. Todorov [15] associated this class with functional f(z)=(1  z) and
obtained a structured formula and coecient estimates in the year 1986. Later, Silverman
and Silvia, [14] gave a full description of the class of univalent functions on D, the image
of which is star-shaped with respect to boundary point. Since then, this class of star-
like functions with respect to a boundary point gained notable interest among geometric
function theorist and also other researchers. See [1, 3, 4, 5, 7, 8].
On the other hand, Ma and Minda [10] gave a unied presentation of the class starlike
using the method of subordination. For two functions h and g in A, the function h is
subordinate to g, written h(z)  g(z); if there exists a function w 2 A, with w(0) = 0
and jw(z)j < 1 such that h(z) = g(w(z)): In particular, if the function g is univalent in
D, then h(z)  g(z) is equivalent to h(0) = g(0) and h(D)  g(D). A function h 2 A is
starlike if zh0(z)=h(z) is subordinated to (1 + z)=(1  z). Ma and Minda [10] introduced
the class
S(') =

h 2 A :
zh0(z)
h(z)
 '(z)

;
where ' is an analytic function with positive real part in D, '(D) is symmetric with respect
to the real axis and starlike with respect to '(0) = 1 and '0(0) > 0: A function f 2 S(')
is called Ma and Minda starlike (with respect to '). The class S() consisting of starlike
functions of order , 0   < 1 and the class S(A;B) of Janowski starlike functions are
special cases of S(') when '(z) := (1+(1 2)z)=(1 z) and '(z) := (1+Az)=(1+Bz)
for  1  B < A  1 respectively.
In the same direction and motivated mainly by [9, 10] and [12], we consider the fol-
lowing class.
Denition 1.1. Let f 2 Sb. Also let ' be an analytic function with positive real
part D, '(D) is symmetric with respect to the real axis and starlike with respect to
'(0) = 1 and '0(0) > 0. The function f 2 Sb (') if the subordination
(1.1) 2
zf 0(z)
f(z)
+
1 + z
1  z
 '(z); z 2 D
holds.
For '(z) = (1+Az)=(1+Bz), ( 1  B < A  1) denote the class Sb (') by S

b (A;B).
For 0   < 1, A = 1  2 and B =  1, denote Sb (A;B) by S

b (; ).
The class Sb (') dened by subordination is investigated to obtain representation,
estimates for f and f 0 and subordination conditions. We obtained some interesting result
in a wider context and our approach is mainly based on [10].
2. Representation for the class Sb (')
Theorem 2.1. The function f 2 Sb (') if and only if there exist p satisfying p  '
such that
f(z) = (1  z) exp

1
2
Z z
0
p()  1

d

:
Proof. Let f 2 Sb ('). Then dene p : D! C by
p(z) = 2
zf 0(z)
f(z)
+
1 + z
1  z
:
STARLIKE WITH RESPECT TO A BOUNDARY POINT DEFINED BY SUBORDINATION 17
Then f 2 Sb (') implies that p  '. Rewriting the above equation and integrating from
0 to z, it follows that
log

f(z)
1  z
2
=
Z z
0
p()  1

d:
The desired result follows from this. The converse follows easily. 
3. Estimates for f and f 0 in the class Sb (')
Theorem 3.1. Let h' be the analytic function with h'(0) = 0, h
0
'(0) = 1 satisfying
the equation zh0'(z)=h'(z) = '(z). If f 2 S

b (') then
(3.1)

 h'( r)
r
 1
2
j1  zj  jf(z)j 

h'(r)
r
 1
2
j1  zj; jzj = r:
Proof. Dene the function h 2 A by
(3.2) h(z) =
z
(1  z)2
f(z)2; z 2 D:
Since f is univalent and f(1) := limr!1  f(r) = 0, it is clear that f(z) 6= 0 in D.
Therefore, the function h is well-dened and analytic in D. A computation shows that
(3.3)
zh0(z)
h(z)
= 2
zf 0(z)
f(z)
+
1 + z
1  z
:
Hence we have the relation f 2 Sb (') if and only if h 2 S
('). Ma and Minda [10,
Corollary 1'] have shown that for h 2 S('),  h'( r)  jh(z)j  h'(r); for jzj = r:
Using this inequality for h in (3.2), gives
 h'( r) 
 z(1  z)2 f(z)2
  h'(r); jzj = r:
and hence the desired result follows. 
If f 2 Sb (A;B) and hence
h'(z) =
(
z(1 +Bz)
A B
B ; B 6= 0;
z exp(Az); B = 0,
then
j1  zj(1 Br)
A B
2B  jf(z)j  j1  zj2(1 +Br)
A B
2B for B 6= 0
j1  zj exp

 Ar
2

 jf(z)j  j1  zj exp

Ar
2

for B = 0:
If f 2 Sb () and
h'(z) =
z
(1  z)2 2
then
j1  zj
(1 + r)1 
 jf(z)j 
j1  zj
(1  r)1 
:
Theorem 3.2. Let '(z) = zh0'(z)=h'(z) and f 2 S

b ('). Then, for jzj = rarg f(z)1  z
  12 maxjzj=r arg h'(z)z :
Similar results for f 2 Sb (A;B) and f 2 S

b () could be easily obtained.
18 M. HAJI MOHD AND M. DARUS
Theorem 3.3. Let '(z) = zh0'(z)=h'(z) and
(3.4) min
jzj=r
j'(z)j = '( r) and max
jzj=r
j'(z)j = '(r):
Also let
H'1 =
j1  zj
2r

h'( r)
 r
 1
2

 
1 + z1  z
+ '( r)

and
H'2 =
j1  zj
2r

h'(r)
r
 1
2
1 + z1  z
+ '(r)

:
If f 2 Sb (') then
H'1  jf
0(z)j  H'2 :
Proof. By Denition 1.1, for f 2 Sb ('), we have
2
zf 0(z)
f(z)
+
1 + z
1  z
 '(z); z 2 D:
When (3.4) holds, the above subordination indicates that
'( r) 
2zf 0(z)f(z) + 1 + z1  z
  '(r) jzj = r:
Rewriting the above inequality and combining with Theorem 3.1, the desired results
follows. 
We could accordingly derive similar results for f 2 Sb (A;B) and f 2 S

b ().
4. Necessary and Sufficient Condition
Theorem 4.1. Let ' be a convex univalent function dened on D. The function
f 2 Sb (') if and only if for all jsj  1, jtj  1,
s
t

1  tz
1  sz
2
f(sz)
f(tz)
2

F (sz)
F (tz)
where F (z) = z exp
 R z
0
(('()  1)=)d

:
Proof. Ruscheweyh [13, Theorem 1] have shown that for ' a convex univalent function,
F as in the hypothesis and h 2 A, zh0(z)=h(z)  '(z) if and only if for all jsj  1, jtj  1
(4.1)
h(sz)
h(tz)

F (sz)
F (tz)
:
From the relation (3.3), we know that f 2 Sb (') if and only if h 2 S
('). Substituting
(3.2) in (4.1), we have
sz
(1 sz)2 f(sz)
2
tz
(1 tz)2 f(tz)
2

F (sz)
F (tz)
and hence the desired result follows. 
Theorem 4.2 is the special cases of Theorem 4.1 when s = 1 and t = 0. However, we
prove the below without the convexity assumption on ':
STARLIKE WITH RESPECT TO A BOUNDARY POINT DEFINED BY SUBORDINATION 19
Theorem 4.2. If f 2 Sb (') then
f(z)
1  z
2

h'(z)
z
;
where h'(z) = z exp
 R z
0 (('()  1)=)d

:
Proof. Clearly zh0'(z)=h'(z) = '(z). If h 2 S
('), then zh0(z)=h(z)  zh0'(z)=h'(z):
Therefore by [10, Theorem 1'] h(z)=z  h'(z)=z: Let h(z) be dened as in (3.2) and
hence we arrive to the desired conclusion. 
Similar results in this section for f 2 Sb (A;B) and f 2 S

b () could be easily obtained.
5. Coefficient Estimate for f 2 Sb (')
Theorem 5.1. Let '(z) = 1 + B1z + B2z
2 +    . If f 2 Sb ('), then the coecients
d1; d2; d3 satisfy the following inequalities:
jd1j 
B1
2
+ 1
jd2j 
B1
4
max

1;
B2B1 +
B1
2


+
B1
2
jd3j 
B1
6
H

6B21 + 16B2
8B1
;
B31 + 6B1B2 + 8B3
8B1

+
B1
4
max

1;
B2B1 +
B1
2


where H(q1; q2)
1 is as dened in [11] (see also [2, Lemma 3]) and
jd2   d
2
1j 
8>><
>>:
B1
4

B2
B1
  (2   1)B12

+ (2 + 1)B12 + 2;   1;
B1
4 + (2 + 1)
B1
2 + 2; 1    2;
B1
4

(2   1)B12  
B2
B1

+ (2 + 1)B12 + 2;   2.
where
1 =
1
B1

B2
B1
  1

+
1
2
2 =
1
B1

B2
B1
+ 1

+
1
2
:
Proof. Dene the function g(z) = 1 + g1z + g2z
2 +    by g(z) = f(z)=(1  z). Then, a
computation shows that
2
zg0(z)
g(z)
+ 1 = 2
zf 0(z)
f(z)
+
1 + z
1  z
:
Since f 2 Sb ('), there is an analytic function w(z) = w1z + w2z
2 +    such that
2
zg0(z)
g(z)
+ 1 = '(w(z)):
Comparing the coecients of z; z2 and z3, we see that
g1 =
B1w1
2
; g2 =
B1
4

w2 +

B2
B1
+
B1
2

w21

and
g3 =
B1
6

w3 +

6B21 + 16B2
8B1

w1w2 +

B31 + 6B1B2 + 8B3
8B1

w31

:
1The expression for H is too lengthy to be reproduced here. See [11] or [2] for the full expression.
20 M. HAJI MOHD AND M. DARUS
In view of the well known inequality jw1j  1, we have jg1j  B1=2: Applying [6, inequality
7, p.10] and [2, Lemma 3] (see also [11]), we get
jg2j 
B1
4
max

1;
B2B1 +
B1
2


and
jg3j 
B1
6
H

6B21 + 16B2
8B1
;
B31 + 6B1B2 + 8B3
8B1

respectively. Also, we see that by applying [2, Lemma 1] (see also [10]) to inequality
g2   g
2
1 =
B1
4

w2  

(2   1)
B1
2
 
B2
B1

w21

yields
jg2   g
2
1 j 
8>><
>>:
B1
4

B2
B1
  (2   1)B12

;   1;
B1
4 ; 1    2;
B1
4

(2   1)B12  
B2
B1

;   2
for 1 and 2 as in the hypothesis. Todorov, in [15] shows that for g(z) = 1 +
P1
1 gnz
n
the coecient gn = 1 + d1 + d2 +   + dn and hence from the above relation the desired
results are obtained. 
Remark 5.1. When '(z) = (1 + z)=(1  z), our results coincides with [15, Corollary
2.3].
Acknowledgment
The work here is supported by UKM-ST-06-FRGS00244-2010.
References
[1] A. S. Abdullah, R. M. Ali, V. Singh: On functions starlike with respect to a boundary point,
Ann. Univ. Mariae Curie-Skªodowska Sect. A, 50 (1996), 715.
[2] R. M. Ali, V. Ravichandran, N. Seenivasagan: Coecient bounds for p-valent functions,
Appl. Math. Comput., 187(1), (2007), 3546.
[3] M. P. Chên, S. Owa: Generalization of Robertson functions, Complex analysis and its applica-
tions, (1993), 159165.
[4] J. Dong, L. Liu: Distortion properties of a class of functions starlike with respect of a boundary
point, Heilongjiang Daxue Ziran Kexue Xuebao, 15(4), (1998), 16.
[5] M. Elin, S. Reich, D. Shoikhet: Dynamics of inequalities in geometric function theory, J.
Inequal. Appl., 6(6), (2001), 651664.
[6] F. R. Keogh, E. P. Merkes: A coecient inequality for certain classes of analytic functions,
Proc. Amer. Math. Soc., 20 (1969), 812.
[7] A. Lecko: On the class of functions starlike with respect to a boundary point, J. Math. Anal.
Appl., 261(2), (2001), 649664.
[8] A. Lecko, A. Lyzzaik: A note on univalent functions starlike with respect to a boundary point,
J. Math. Anal. Appl., 282(2), (2003), 846851.
[9] A. Lyzzaik: On a conjecture of M. S. Robertson, Proc. Amer. Math. Soc., 91(1), (1984), 108110.
[10] W. C. Ma, D. Minda: A unied treatment of some special classes of univalent functions, Pro-
ceedings of the Conference on Complex Analysis (Tianjin, 1992), 157169.
[11] D. V. Prokhorov, J. Szynal: Inverse coecients for (; )-convex functions, Ann. Univ.
Mariae Curie-Skªodowska Sect. A, 35 (1981), 125143.
[12] M. S. Robertson: Univalent functions starlike with respect to a boundary point, J. Math. Anal.
Appl., 81(2), (1981), 327345.
STARLIKE WITH RESPECT TO A BOUNDARY POINT DEFINED BY SUBORDINATION 21
[13] S. Ruscheweyh: A subordination theorem for F -like functions, J. London Math. Soc., 13(2)
(1976), 275280.
[14] H. Silverman, E. M. Silvia: Subclasses of univalent functions starlike with respect to a boundary
point, Houston J. Math. 16(2) (1990), 289299.
[15] P. G. Todorov: On the univalent functions starlike with respect to a boundary point, Proc. Amer.
Math. Soc., 97(4) (1986), 602604.
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi 43600, Selangor D. Ehsan, Malaysia
E-mail address: maisarah_hjmohd@yahoo.com
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi 43600, Selangor D. Ehsan, Malaysia
E-mail address: maslina@ukm.my (Corresponding author)


STARLIKE FUNCTION WITH RESPECT TO A BOUNDARY POINT DEFINED BY SUBORDINATION

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STARLIKE FUNCTION WITH RESPECT TO A BOUNDARY POINT DEFINED BY SUBORDINATION

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