General Mathematics Vol. 17, No. 3 (2009), 91–98
New sufficient conditions for univalence
1
B.A. Frasin
Abstract
The object of the present paper is to obtain new sufficient conditions on f 00 (z) which lead to some subclasses of univalent functions
defined in the open unit disk.
2000 Mathematics Subject Classification:30C45
Key words and phrases: Analytic and univalent functions, Strongly
starlike and strongly convex functions
1
Introduction
Let An denote the class of functions of the form :
f (z) = z + an+1 z n+1 + an+2 z n+2 + · · · ,
(1)
1
Received 29 July, 2008
Accepted for publication (in revised form) 15 September, 2008
91
92
B.A. Frasin
which are analytic in the open unit disk U = {z : |z| < 1}. Let A1 = A,
then a function f (z) ∈ A is said to be in the class S ∗ (α), the class of starlike
functions of order α, 0 ≤ α < 1, if and only if
µ 0 ¶
zf (z)
(2)
Re
>α
(z ∈ U).
f (z)
Then S ∗ = S ∗ (0) is the class of starlike functions in U. Further, if f (z) ∈
A satisfies
¯
¯
0
¯
¯
¯arg zf (z) ¯ < απ ,
¯
f (z) ¯
2
(3)
(z ∈ U)
for some 0 < α ≤ 1, then f (z) said to be strongly starlike function of order
∗
∗
α in U, and this class denoted by S (α). Note that S (1) = S ∗ .
Furthermore, let K(α), 0 ≤ α < 1, which consists of functions f (z) ∈ A
such that
µ
zf 00 (z)
Re 1 + 0
f (z)
(4)
¶
>α
(z ∈ U),
and K = K(0) is the class of convex functions in U. Also, if f (z) ∈ A
satisfies
¯
¶¯
µ
00
¯
¯
¯arg 1 + zf (z) ¯ < απ ,
¯
f 0 (z) ¯
2
(5)
(z ∈ U)
for some 0 < α ≤ 1, then f (z) said to be strongly convex function of
order α in U, and this class denoted by K(α). Note that K(1) = K and if
∗
zf 0 (z) ∈ S (α) then f (z) ∈ K(α).
All the above mentioned classes are subclasses of univalent fictions in U.
There are many results for sufficient conditions of functions f (z) which
are analytic in U to be starlike, convex, strongly starlike and strongly convex
New sufficient conditions for univalence
93
functions have been given by several researchers .(see [9],[1],[2],[7],[6],[5]).
In this paper we will study λ such that the conditions|f 00 (z)| ≤ λ , z ∈ U ,
implies that f (z) belongs to one of the classes defined above.
In order to prove our main results, we shall need the following lemmas.
Lemma 1 ([9]) If f (z) ∈ A satisfies
s
0
(6)
|f (z) − 1| < 2a
√
5 − 4 1 − a2
16a2 + 9
(z ∈ U )
∗
where a = sin(απ/2), 0 < α ≤ 1, then f (z) ∈ S (α).
Lemma 2 ([4])If f (z) ∈ An , satisfies
|f 0 (z) − 1| <
(7)
(1 − α)(n + 1)
p
,
α + (n + 1)2 + 1
(z ∈ U)
where 0 ≤ α < 1, then f (z) ∈ S ∗ (α).
Lemma 3 ([8])If f (z) ∈ An , satisfies
|f 0 (z) + αzf 00 (z) − 1| <
(8)
(α − 2)(nα + 1)
α(n + 1)
(z ∈ U)
where α > 2, then f (z) ∈ K.
2
Main Results
Employing the same method used by Nunokawa et al.[6], we prove the
following
94
B.A. Frasin
Theorem 1 If f (z) ∈ A satisfies
s
00
(9)
|f (z)| ≤ 2a
√
5 − 4 1 − a2
16a2 + 9
(z ∈ U; 0 < α ≤ 1)
∗
where a = sin(απ/2), then f (z) ∈ S (α).
Proof. Noting that
¯
¯ z
s
√
¯ Z|z|
¯Z
Z|z|
2
¯
¯
¯
¯
5
−
4
1
−
a
|f 0 (z) − 1| = ¯¯ f 00 (σ)dσ ¯¯ ≤ ¯f 00 (teiθ )¯ dt ≤ 2a
dt
16a2 + 9
¯
¯
0
0
0
s
s
√
√
5 − 4 1 − a2
5 − 4 1 − a2
= 2a
|z|
<
2a
16a2 + 9
16a2 + 9
∗
Hence, by Lemma 1, we conclude that f (z) ∈ S (α).
Corollary 1 Let f (z) ∈ A, z ∈ U then
√
(i) |f 00 (z)| ≤ 2 5/5 = 0.8944 . . . implies f (z) ∈ S ∗ ;
q
√
∗
00
(ii) |f (z)| ≤ (5 − 2 3)/13 = 0.3437 . . .implies f (z) ∈ S (1/3);.
q
√
∗
(iii) |f 00 (z)| ≤ (10 − 4 2)/17 = 0.5054 . . . implies f (z) ∈ S (1/2);
and
(iii) |f 00 (z)| ≤
√
∗
21/7 = 0.6546 . . . implies f (z) ∈ S (2/3).
Remark 1 The result from Corollary 1 (i) was obtained by Nunokawa et
al.[6].
Now, we derive
New sufficient conditions for univalence
Theorem 2 If f (z) ∈ A satisfies
s
√
5 − 4 1 − a2
00
(10)
|f (z)| ≤ a
16a2 + 9
95
(z ∈ U; 0 < α ≤ 1)
where a = sin(απ/2), then f (z) ∈ K(α).
Proof. It follows that
|(zf 0 (z))0 − 1| = |f 0 (z) + zf 00 (z) − 1| ≤ |f 0 (z) − 1| + |zf 00 (z)|
¯ z
¯
s
√
¯Z
¯
Z|z|
¯
¯
5−4
1−a2
|z|
≤ ¯¯ f 00 (t)dt¯¯ + |zf 00 (z)| ≤ |f 00 (t)dt| + a
16a2 +9
¯
¯
0
0
s
s
√
√
2
5−4 1−a
5 − 4 1 − a2
≤ 2a
|z| < 2a
16a2 + 9
16a2 + 9
∗
Therefore, using Lemma 1, we see that zf 0 (z) ∈ S (α), or f (z) ∈ K(α).
Corollary 2 Let f (z) ∈ A, z ∈ U then
(i) |f 00 (z)| ≤
√
5/5 = 0.4472 . . . implies f (z) ∈ K;
q
√
1
00
(ii) |f (z)| ≤ 2 (5 − 2 3)/13 = 0.1718 . . .implies f (z) ∈ K(1/3);.
q
√
1
00
(iii) |f (z)| ≤ 2 (10 − 4 2)/17 = 0.2527 . . . implies f (z) ∈ K(1/2);
and
(iii) |f 00 (z)| ≤
1
2
√
21/7 = 0.3273 . . . implies f (z) ∈ K(2/3).
Remark 2 The result from Corollary 2 (i) was obtained by Nunokawa et
al.[6].
Applying the same method as in the proof of Theorem 1 and using
Lemmas 2 instead of Lemma 1, we obtain the following theorem
96
B.A. Frasin
Theorem 3 If f (z) ∈ An satisfies
|f 00 (z)| ≤
(11)
(1 − α)(n + 1)
p
α + (n + 1)2 + 1
(z ∈ U ; 0 ≤ α < 1)
then f (z) ∈ S ∗ (α).
Letting α = 0 in Theorem 3, we obtain
Corollary 3 If f (z) ∈ An satisfies
n+1
|f 00 (z)| ≤ p
(n + 1)2 + 1
(12)
(z ∈ U)
then f (z) ∈ S ∗ .
Remark 3 Letting n = 1 in Corollary 3 , we obtain the result (i) from
Corollary 2.5 which was obtained by Nunokawa et al.[6].
Finally, we prove
Theorem 4 If f (z) ∈ An , satisfies
|f 00 (z)| <
(13)
(α − 2)(nα + 1)
α(α + 1)(n + 1)
(z ∈ U)
where α > 2, then f (z) ∈ K.
Proof. It follows that
|f 0 (z) + αzf 00 (z) − 1| ≤ |f 0 (z) − 1| + α |zf 00 (z)|
¯ z
¯
¯Z
¯
Z|z|
¯
¯
(α − 2)(nα + 1)
00
00
≤ ¯¯ f (t)dt¯¯ + α |zf (z)| ≤ |f 00 (t)dt| +
|z|
(α + 1)(n + 1)
¯
¯
0
≤
0
(α − 2)(nα + 1)
(α − 2)(nα + 1)
|z| <
,
α(n + 1)
α(n + 1)
using Lemma 3, we have f (z) ∈ K.
Letting α → ∞ in Theorem 4, we have the following result obtained by
Mocanu[3].
New sufficient conditions for univalence
97
Corollary 4 If f (z) ∈ An , satisfies
(14)
|f 00 (z)| <
n
n+1
(z ∈ U)
then f (z) ∈ K.
Letting n = 1 in Corollary 4, we have the following result obtained by
Obradovič[7].
Corollary 5 If f (z) ∈ A, satisfies
(15)
|f 00 (z)| <
1
2
(z ∈ U)
then f (z) ∈ K.
References
[1] P.T. Mocanu, Some starlikeness conditions for analytic functions, Rev.
Roumaine Math. Pures. Appl. 33 (1988), 117-124.
[2] P.T. Mocanu, Two simple conditions for starlikeness, Mathematica
(Cluj). 34 (57) (1992), 175-181.
[3] P.T. Mocanu, Some simple criteria for for starlikeness and convexity,
Libertas Mathematica, 13 (1993), 27-40.
[4] P.T. Mocanu and G. Oros, A sufficient condition for starlikeness of order
α, IJMMS 28: 9 (2001), 557-560.
[5] M. Nunokawa, On the order of strongly starlikeness of strongly convex
functions, Proc. Japan. Acad., Sec.A 68 (1993), 234-237.
98
B.A. Frasin
[6] M. Nunokawa, S. Owa, Y. Polatoğlu, M. Cağlar and E. Yavuz, Some
sufficient conditions for starlikeness and convexity, Submited.
[7] M. Obradovič, Simple sufficients conditions for starlikeness, Matematicki Vesnik. 49 (1997), 241-244.
[8] G. Oros, A condition for convexity, Gen. Math.Vol. 8, No. 3-4 (2000),1521.
[9] S. N. Tuneski, On simple sufficient conditions for univalence, Mathematica Bohemica, Vol. 126, No. 1, (2001), 229-236.
B.A. Frasin
Department of Mathematics
Al al-Bayt University
P.O. Box: 130095 Mafraq, Jordan
e-mail: bafrasin@yahoo.com