Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 86493, 5 pages
doi:10.1155/2007/86493
Research Article
Sufficient Univalence Conditions for Analytic Functions
Daniel Breaz and Nicoleta Breaz
Received 30 October 2007; Accepted 4 December 2007
Recommended by Narendra Kumar K. Govil
We consider a general integral operator and the class of analytic functions. We extend
some univalent conditions of Becker’s type for analytic functions using a general integral
transform.
Copyright © 2007 D. Breaz and N. Breaz. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let ᐁ = {z ∈ C, |z| < 1} be the unit disk, let Ꮽ denote the class of the functions f of the
form
f (z) = z + a2 z2 + a3 z3 + · · · , z ∈ ᐁ ,
(1.1)
which are analytic in the open disk, and let ᐁ satisfy the condition f (0) = f (0) − 1 = 0.
Consider ᏿ = { f ∈ Ꮽ : f is univalent functions in ᐁ}.
In [1], Pescar needs the following theorem.
Theorem 1.1 [1]. Let c and β be complex numbers with Re β > 0, |c|≤1, and c= − 1, and
let h(z) = z + a2 z2 + · · · be a regular function in ᐁ.If
c|z|2β + 1 − |z|2β zh (z) ≤1
βh (z) (1.2)
for all the z ∈ ᐁ, then the function
z
Fβ (z) = β
is regular and univalent in ᐁ.
0
t β−1 h (t)dt
1/β
= z + ···
(1.3)
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Journal of Inequalities and Applications
In [2], Ozaki and Nunokawa give the next result.
Theorem 1.2 [2]. Let f ∈ Ꮽ satisfy the following condition:
2 z f (z)
f 2 (z) − 1≤1
(1.4)
for all z ∈ ᐁ, then f is univalent in ᐁ.
Lemma 1.3 (The Schwarz lemma) [3, 4]. Let the analytic function f be regular in the unit
disk and let f (0) = 0 . If | f (z)|≤1, then
f (z)≤|z|
(1.5)
for all z ∈ ᐁ, where the equality can hold only if | f (z)| = Kz and K = 1.
In [5], Seenivasagan and Breaz consider, for fi ∈ Ꮽ2 (i = 1,2,...,n) and α1 ,α2 ,...,αn ,
β ∈ C, the integral operator
z
Fα1 ,α2 ,...,αn ,β (z) = β
t
0
β −1
n fi (t) 1/αi
i=1
t
1/β
dt
.
(1.6)
When αi = α for all i = 1,2,...,n, Fα1 ,α2 ,...,αn ,β (z) becomes the integral operator Fα,β (z)
considered in [6].
2. Main results
fi ∈ Ꮽ, for i ∈ {1,...,n}, satisfy the condition
Theorem 2.1. Let M ≥ 1 and the functions
(1.4), and let β be a real number, β ≥ ni=1 (2M + 1)/ |αi | and c is a complex number.
If
1 2M + 1
,
β i=1 αi n
|c|≤1 −
(2.1)
fi (z)≤M
(2.2)
for all z ∈ ᐁ, then the function Fα1 ,α2 ,...,αn ,β defined in (1.6) is in the class ᏿.
Proof. Define a function
h(z) =
z
n 0 i=1
fi (t)
t
1/αi
dt,
(2.3)
then we have h(0) = h (0) − 1 = 0. Also, a simple computation yields
h (z) =
n fi (z) 1/αi
i =1
n
z
,
(2.4)
zh (z)
1 z fi (z)
=
−1 .
h (z)
α
fi (z)
i =1 i
(2.5)
D. Breaz and N. Breaz 3
From (2.5), we have
n
2 zh (z) n 1
fi (z) 1 z fi (z)
≤ z fi (z) + 1 =
+1 .
h (z) αi fi (z) |α | fi (z) 2 z i=1
i =1 i
(2.6)
From the hypothesis, we have | fi (z)|≤M (z ∈ ᐁ, i = 1,2,...,n), then by Lemma 1.3,
we obtain that
(z ∈ ᐁ, i = 1,2,...,n).
| fi (z)|≤M |z|
(2.7)
We apply this result in inequality (2.6), and we obtain
2 zh (z) n 1
≤ z fi (z)
M
+
1
2 h (z) α i=1
n
≤
i
fi (z)
i
fi (z)
1 z2 fi (z)
−
1
M
+
M
+
1
2
α i=1
n
=
i =1
1
(M + M + 1) =
αi (2.8)
n
2M + 1
.
αi i =1
We have
n
1 z fi (z)
c|z|2β + 1 − |z|2β zh (z) = c|z|2β + 1 − |z|2β 1
−
1
βh (z)
β α f (z)
i =1
n
1
≤|c| + ·
β i=1
i
i
1 z2 fi (z) · fi (z) + 1 .
αi f 2 (z) |z|
(2.9)
i
We obtain
n
2M + 1
c|z|2β + 1 − |z|2β zh (z) ≤|c| + 1
.
α βh (z) β
i=1
(2.10)
i
So from (2.1), we have
c|z|2β + 1 − |z|2β zh (z) ≤1.
βh (z) Applying Theorem 1.1, we obtain that Fα1 ,α2 ,...,αn ,β is univalent.
(2.11)
Theorem 2.2. Let M ≥ 1 and the functions fi ∈ Ꮽ, for i ∈ {1,...,n} satisfy the condition
(1.4), and let β be a real number, β ≥ n(2M + 1)/ |α| and c is a complex number.
If
|c|≤1 −
1 n(2M + 1)
,
β
|α|
fi (z)≤M
(2.12)
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Journal of Inequalities and Applications
for all z ∈ ᐁ, then the function
z
Fα,β (z) = β
n fi (t) 1/α
1/β
dt
(2.13)
Proof. In Theorem 2.1, we consider α1 = α2 = · · · = αn = α.
0
t
β −1
t
i=1
is in the class ᏿.
Corollary 2.3. Let the functions
fi ∈ Ꮽ, for i ∈ {1,...,n}, satisfy the condition (1.4), and
let β be a real number, β ≥ ni=1 (3/ |αi |) and c is a complex number.
If
1 3
,
β i=1 αi n
|c|≤1 −
(2.14)
fi (z)≤1
for all z ∈ ᐁ, then the function Fα1 ,α2 ,...,αn ,β defined in (1.6) is in the class ᏿.
Proof. In Theorem 2.1, we consider M = 1.
Corollary 2.4. Let M ≥ 1 and the function f ∈ Ꮽ, satisfy the condition (1.4), and let β be
a real number, β ≥ (2M + 1)/ |α| and c is a complex number.
If
|c|≤1 −
1 2M + 1
,
β |α|
(2.15)
f (z)≤M
for all z ∈ ᐁ, then the function
z
Gα,β (z) = β
0
t
β −1
f (t)
t
1/β
1/α
dt
(2.16)
is in the class ᏿.
Proof. In Theorem 2.1, we consider n = 1.
Corollary 2.5. Let the function f ∈ Ꮽ satisfy the condition (1.4), and let β be a real
number, β ≥ 3/ |α| and c is a complex number.
If
|c|≤1 −
1 3
,
β |α|
f (z)≤1
(2.17)
D. Breaz and N. Breaz 5
for all z ∈ ᐁ, then the function
z
Gα,β (z) = β
0
t
β −1
f (t)
t
1/β
1/α
dt
(2.18)
is in the class ᏿.
Proof. In Corollary 2.4, we consider M = 1.
Acknowledgment
This resaerch was supported by the Grant of the Romanian Academy no. 20/2007.
References
[1] V. Pescar, “A new generalization of Ahlfors’s and Becker’s criterion of univalence,” Bulletin of the
Malaysian Mathematical Society, vol. 19, no. 2, pp. 53–54, 1996.
[2] S. Ozaki and M. Nunokawa, “The Schwarzian derivative and univalent functions,” Proceedings
of the American Mathematical Society, vol. 33, no. 2, pp. 392–394, 1972.
[3] Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, 1952.
[4] Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.
[5] N. Seenivasagan and D. Breaz, “Certain sufficient conditions for univalence,” to appear in General Mathematics.
[6] D. Breaz and N. Breaz, “The univalent conditions for an integral operator on the calsses S p and
T2 ,” Journal of Approximation Theory and Applications, vol. 1, no. 2, pp. 93–98, 2005.
Daniel Breaz: Department of Mathematics, “1 Decembrie 1918” University, Alba Iulia, Romania
Email address: dbreaz@uab.ro
Nicoleta Breaz: Department of Mathematics, “1 Decembrie 1918” University, Alba Iulia, Romania
Email address: nbreaz@uab.ro