J ournal of
Classical
A nalysis
Volume 1, Number 1 (2012), 85–90
doi:10.7153/jca-01-09
SUFFICIENT CONDITIONS FOR STARLIKENESS
M AMORU N UNOKAWA , S.P. G OYAL AND R AKESH K UMAR
Abstract. The purpose of the present paper is to consider some sufficient conditions for analytic
functions in the open unit disk to be starlike. Here we establish three theorems by using Jack’s
lemma and a simple result contained in Lemma 2.2. Our theorems provide improvements of the
results about sufficient conditions for starlike functions given earlier by Lewandowski et al. [2],
Li and Owa [3], Nunokawa et al. [5] and Ramesha et al. [6].
1. Introduction
Let A denote the class of functions f of the form
∞
f (z) = z + ∑ an zn
n=2
which are analytic in the open unit disc D = {z ∈ C : |z| < 1 }.
A function f belonging to the class A is said to be starlike of order α in D if
and only if
z f (z)
(1)
Re
> α (z ∈ D; 0 α < 1)
f (z)
We denote by S∗ (α ) the class of all functions in A which are starlike of order α
in D. Also we write S∗ (0) simply by S∗ .
Several results appeared previously about sufficient conditions of starlikeness. The
result discussed in Theorems A, B, C, D, E and F (given below) were obtained earlier
by Lewandowski et al. [2], Ramesha et al. [6], Li and Owa [3], and Nunokawa et al.
[5] respectively.
T HEOREM A. ([2]) Let f (z) ∈ A satisfy the condition
z f (z)
z f (z)
Re
1+ > 0 (z ∈ D),
f (z)
f (z)
then f (z) ∈ S∗ .
T HEOREM B. ([6]) Let f (z) ∈ A satisfy the condition
z f (z)
z f (z)
Re
1+α > 0 (z ∈ D; α 0),
f (z)
f (z)
then f (z) ∈ S∗ .
Mathematics subject classification (2010): 30C55.
Keywords and phrases: Analytic functions, starlike functions.
c
, Zagreb
Paper JCA-01-09
85
86
M AMORU N UNOKAWA , S.P. G OYAL AND R AKESH K UMAR
T HEOREM C. ([3]) Let f (z) ∈ A satisfy the condition
z f (z)
α
z f (z)
(z ∈ D; α 0),
Re
1+α >−
f (z)
f (z)
2
then f (z) ∈ S∗ .
T HEOREM D. ([3]) Let f (z) ∈ A satisfy the condition
z f (z)
α2
z f (z)
Re
1+α > − (1 − α ) (z ∈ D; 0 α < 2),
f (z)
f (z)
4
then f (z) ∈ S∗ α2 .
T HEOREM E. ([3]) Let f (z) ∈ A satisfy the condition
z f (z)
z f (z)
Re
1+ > 0 (z ∈ D),
f (z)
f (z)
then f (z) ∈ S∗ 12 .
T HEOREM F. ([5]) Let f (z) ∈ A satisfy the condition
z f (z) 2
z f (z)
z f (z)
α
1 + 3 Im
Re
1+α >−
f (z)
f (z)
2
f (z)
(z ∈ D; α 0),
then f (z) ∈ S∗ .
Obviously Theorems C, D and E are improvements of the Theorems A and B
while Theorem F is an improvement of Theorem C. The aim of this paper is to provide
improvements of all these theorems.
2. Main results
In order to establish our main results, we require the following lemmas:
L EMMA 2.1. (Jack’s lemma [1]) Let w(z) be a non-constant analytic function in
D with w(0) = 0 . If |w(z)| attains its maximum value on the circle |z| = r < 1 at z0 ,
then
z0 w (z0 ) = k w(z0 ),
where k 1 is a real number.
∞
L EMMA 2.2. Let p(z) = 1 + ∑ cn zn be analytic in D and suppose that there
exists a point z0 ∈ D such that
n=1
Re{p(z)} > 0
for |z| < |z0 |
(2)
and
Re{p(z0 )} = 0.
(3)
S UFFICIENT CONDITIONS FOR STARLIKENESS
Then we have
z0 p (z0 ) −
1
1 + |p(z0)|2 .
2
Proof. Let us put
ϕ (z) =
1 − p(z)
.
1 + p(z)
87
(4)
(5)
Then we have that
ϕ (0) = 0, |ϕ (z)| < 1 for |z| < |z0 | and |ϕ (z0 )| = 1.
From (2), (5) and Lemma 2.1, we find that
z0 ϕ (z0 )
−2z0 p (z0 )
−2z0 p (z0 )
=
1.
=
2
ϕ (z0 )
1 + |p(z0)|2
1 − {p(z0 )}
This shows that
z0 p (z0 ) −
and z0 p (z0 ) is a negative real number.
1
1 + |p(z0)|2 ,
2
T HEOREM 2.3. Let f (z) ∈ A satisfies f (z) f (z) = 0 in 0 < |z| < 1 and
2 z f (z) z f (z)
z f (z)
α
1 + 3 Re
(z ∈ D; α 0),
1+α >−
f (z)
f (z)
2
f (z) then f (z) ∈ S∗ .
Proof. Let
p(z) =
z f (z)
,
f (z)
(6)
then p(z) is analytic in D and p(0) = 1.
Suppose that there exists a point z0 ∈ D which satisfies the conditions (2) and (3)
of Lemma 2.2.
Now using (6), it follows that
z0 f (z0 )
z0 f (z0 )
1+α Re
f (z0 )
f (z0 )
= Re α z0 p (z0 ) + α (p(z0 ))2 − α p(z0 ) + p(z0 )
(7)
Since the function p(z) and the point z0 satisfy all conditions Lemma 2.2, therefore in view of (3) and (4), equation (7) gives
z0 f (z0 )
z0 f (z0 )
α
Re
1 + |p(z0)|2 − α |p(z0 )|2
1+α −
f (z0 )
f (z0 )
2
α
1 + 3|p(z0)|2
−
2
z0 f (z0 ) 2
α
. (by(4))
−
1 + 3
2
f (z0 ) 88
M AMORU N UNOKAWA , S.P. G OYAL AND R AKESH K UMAR
This is a contradiction and therefore proof of the Theorem 2.3 is completed.
The result discussed in Theorem 2.3 is an improvement of the result contained in
Theorem F.
R EMARK .. We have supposed the condition f (z) f (z) = 0 in 0 < |z| < 1 while Li
and Owa [3] did not put this condition in their result, but in fact they applied it during
the proof of their result and so, it is not a weak point of our paper.
T HEOREM 2.4. Let f (z) ∈ A satisfies f (z) f (z) = 0 in 0 < |z| < 1 and
z f (z)
z f (z)
1 z f (z) 2
Re
1+ >− f (z)
f (z)
2 f (z) then S∗
1
2
(z ∈ D),
.
Proof. Let
z f (z) 1
−
p(z) = 2
,
f (z)
2
p(0) = 1
Now using (8), we have
z0 f (z0 )
1
1 z0 f (z0 )
2
1+ = Re
z0 p (z0 ) + (1 + p(z0))
Re
f (z0 )
f (z0 )
2
4
where z0 ∈ D is a point which satisfies the conditions (2) and (3) of Lemma 2.2.
Under the conditions (3) and (4) of Lemma 2.2, it follows that
1
z0 f (z0 )
1
z0 f (z0 )
1
Re
1+ − 1 + |p(z0 )|2 − |p(z0 )|2 +
f (z0 )
f (z0 )
4
4
4
1
− |p(z0 )|2
2
1 z0 f (z0 ) 2
− 2 f (z0 ) which contradicts the hypothesis of Theorem 2.3 and therefore, we have
Re{p(z)} > 0 (z ∈ D)
or
Re
z f (z)
f (z)
>
1
2
(z ∈ D).
Theorem 2.4 is an improvement of Theorem E.
T HEOREM 2.5. Let f (z) ∈ A satisfies f (z) f (z) = 0 in 0 < |z| < 1 and
z f (z)
z f (z)
α
(z ∈ D; 0 α < 2)
Re
1+α > − (2 − α )(1 − α )
f (z)
f (z)
4
(8)
(9)
S UFFICIENT CONDITIONS FOR STARLIKENESS
then f (z) ∈ S∗
α 2
89
.
Proof. Let p(z) is defined by
α
α
z f (z) = 1−
p(z) + ,
f (z)
2
2
p(0) = 1.
Suppose that there exists a point z0 ∈ D such that it satisfies the conditions (2) and
(3) of Lemma 2.2. Now, we have
z0 f (z0 )
z0 f (z0 )
1+α Re
f (z0 )
f (z0 )
α
α 2
z0 p (z0 ) + α 1 −
(p(z0 ))2
= Re α 1 −
2
2
α 2
α3 α
+ (1 − α )
+ 1−
α + 1 − α p(z0 ) +
(10)
2
4
2
Using the conditions (3) and (4) in (10), we find that
z0 f (z0 )
z0 f (z0 )
1+α Re
f (z0 )
f (z0 )
α
α α 2
1 + |p(z0)|2 − α 1 −
1−
−
|p(z0 )|2
2
2
2
α3 α
+ (1 − α )
+
4
2
α
α α3 α
+ (1 − α )
1−
+
−
2
2
4
2
α
− (2 − α )(1 − α ).
4
This is a contradiction and therefore
z f (z)
α
>
(z ∈ D).
Re
f (z)
2
Thus proof of Theorem 2.4 is completed.
R EMARK . Theorem 2.4 improves Theorem D because
α2
α
(1 − α ) > − (1 − α )(2 − α ) when 0 α < 1,
4
4
α
α
− (1 − α ) = − (1 − α )(2 − α ) when α = 1,
4
4
2
α
α
− (1 − α ) > − (1 − α )(2 − α ) > 0 when 1 < α < 2.
4
4
0>−
Acknowledgement. The authors are thankful to the worthy referee for his valuable
suggestions regarding the improvement of the paper. The second author (S. P. G.) is
thankful to CSIR, New Delhi, India for awarding Emeritus Scientistship, under scheme
number 21(084)/10/EMR-II.
90
M AMORU N UNOKAWA , S.P. G OYAL AND R AKESH K UMAR
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(2002), 1385–1390.
[4] M. N UNOKAWA, On properties of Non-Carath é odory functions, Proc. Japan Acad. 68, 6 (1992), 152–
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[5] M. N UNOKAWA , S.O WA , S.K. L EE , M. O BRADOVIC , M.K. AOUF, H. S AITOH , H. I KADA AND N.
K OIKA, Sufficient conditions for starlikeness, Chinese Journal of Mathematics 24 (1996), 265–270.
[6] C. R AMESHA , S. K UMAR AND K.S. PADMANBHAM, A sufficient condition for starlikeness, Chinese
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Mamoru Nunokawa
University of Gunma
Hoshikuki-Cho 798-8
Chuou-Ward, Chiba
JAPAN-2600808
e-mail: Mamoru [email protected]
S.P. Goyal
Department of Mathematics
University of Rajasthan
Jaipur, INDIA-302055
e-mail: [email protected]
Rakesh Kumar
Department of Mathematics
Amity University Rajasthan
Jaipur, INDIA-302002
e-mail: [email protected]
Journal of Classical Analysis
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