AN EXTENSION OF OZAKI AND NUNOKAWA’S
UNIVALENCE CRITERION
Ozaki and Nunokawa’s
Univalence Criterion
HORIANA TUDOR
Faculty of Mathematics and Informatics
"Transilvania" University
2200 Braşov, ROMANIA
EMail: htudor@unitbv.ro
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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Received:
11 May, 2008
Accepted:
14 October, 2008
Communicated by:
N.E. Cho
2000 AMS Sub. Class.:
30C55.
Key words:
Univalent functions, Univalence criteria.
Abstract:
In this paper we obtain a sufficient condition for the analyticity and the univalence
of the functions defined by an integral operator. In a particular case we find the
well known condition for univalency established by S. Ozaki and M. Nunokawa.
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Contents
1
Introduction
3
2
Preliminaries
4
3
Main Results
5
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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1.
Introduction
We denote by Ur = {z ∈ C : |z| < r} a disk of the z-plane, where r ∈ (0, 1], U1 =
U and I = [0, ∞). Let A be the class of functions f analytic in U such that f (0) =
0, f 0 (0) = 1.
Theorem 1.1 ([1]). Let f ∈ A. If for all z ∈ U
2 0
z f (z)
(1.1)
f 2 (z) − 1 < 1,
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
then the function f is univalent in U .
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2.
Preliminaries
In order to prove our main result we need the theory of Löewner chains; we recall
the basic result of this theory, from Pommerenke.
Theorem 2.1 ([2]). Let L(z, t) = a1 (t)z + a2 (t)z 2 + · · · , a1 (t) 6= 0 be analytic in
Ur , for all t ∈ I, locally absolutely continuous in I and locally uniform with respect
to Ur .For almost all t ∈ I, suppose that
∂L(z, t)
∂L(z, t)
z
= p(z, t)
,
∂z
∂t
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
∀z ∈ Ur ,
where p(z, t) is analytic in U and satisfies the condition Re p(z, t) > 0, for all z ∈ U ,
t ∈ I. If |a1 (t)| → ∞ for t → ∞ and {L(z, t)/a1 (t)} forms a normal family in Ur ,
then for each t ∈ I, the function L(z, t) has an analytic and univalent extension to
the whole disk U .
vol. 9, iss. 4, art. 117, 2008
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3.
Main Results
Theorem 3.1. Let f ∈ A and α be a complex number, Re α > 0. If the following
inequalities
2 0
z f (z)
<1
(3.1)
−
1
f 2 (z)
and
(3.2)
2 0
z f (z)
1 − |z|2α z 2 f 0 (z)
2α
−1
f 2 (z) − 1 |z| + 2
α
f 2 (z)
2 0
(1 − |z|2α )2
z f (z)
f (z)
≤1
+
−
1
+
(1
−
α)
−
1
α2 |z|2α
f 2 (z)
z
are true for all z ∈ U \ {0}, then the function Fα ,
Z z
α1
α−1 0
(3.3)
Fα (z) = α
u f (u)du
0
is analytic and univalent in U , where the principal branch is intended.
Proof. Let us consider the function g1 (z, t) given by
e2αt − 1 f (e−t z)
g1 (z, t) = 1 −
−1 .
α
e−t z
For all t ∈ I and z ∈ U we have e−t z ∈ U and because f ∈ A, the function g1 (z, t)
is analytic in U and g1 (0, t) = 1. Then there is a disk Ur1 , 0 < r1 < 1 in which
g1 (z, t) 6= 0, for all t ∈ I. For the function
Z e−t z
g2 (z, t) = α
uα−1 f 0 (u)du ,
0
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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g2 (z, t) = z α · g3 (z, t), it can be easily shown that g3 (z, t) is analytic in Ur1 and
g3 (0, t) = e−αt . It follows that the function
−t 2
z)
(eαt − e−αt ) f (e
e−t z
g4 (z, t) = g3 (z, t) +
g1 (z, t)
is also analytic in a disk Ur2 , 0 < r2 ≤ r1 and g4 (0, t) = eαt . Therefore, there is
a disk Ur3 , 0 < r3 ≤ r2 in which g4 (z, t) 6= 0, for all t ∈ I and we can choose an
analytic branch of [g4 (z, t)]1/α , denoted by g(z, t). We choose the branch which is
equal to et at the origin.
From these considerations it follows that the function
L(z, t) = z · g(z, t) = et z + a2 (t)z 2 + · · ·
is analytic in Ur3 , for all t ∈ I and can be written as follows

 α1
Z e−t z
2αt
(2−α)t α−2 2 −t
(e − 1)e
z f (e z) 
−t
(3.4) L(z, t) = α
uα−1 f 0 (u)du +
.
2αt
z)
0
1 − e α−1 f (e
−
1
e−t z
From the analyticity of L(z, t) in Ur3 , it follows that there is a number r4 , 0 < r4 <
r3 , and a constant K = K(r4 ) such that
|L(z, t)/et | < K,
∀z ∈ Ur4 ,
t ∈ I,
and then {L(z, t)/et } is a normal family in Ur4 . From the analyticity of ∂L(z, t)/∂t,
for all fixed numbers T > 0 and r5 , 0 < r5 < r4 , there exists a constant K1 > 0
(that depends on T and r5 ) such that
∂L(z, t) ∀z ∈ Ur5 , t ∈ [0, T ].
∂t < K1 ,
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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It follows that the function L(z, t) is locally absolutely continuous in I, locally uniform with respect to Ur5 . We also have that the function
∂L(z, t)
∂L(z, t)
p(z, t) = z
∂z
∂t
is analytic in Ur , 0 < r < r5 , for all t ∈ I.
In order to prove that the function p(z, t) has an analytic extension with positive
real part in U for all t ∈ I, it is sufficient to show that the function w(z, t) defined in
Ur by
p(z, t) − 1
w(z, t) =
p(z, t) + 1
can be continued analytically in U and that |w(z, t)| < 1 for all z ∈ U and t ∈ I.
By simple calculations, we obtain
(3.5) w(z, t)
−2t 2 0 −t
e z f (e z)
1 − e−2αt e−2t z 2 f 0 (e−t z)
−2αt
=
−1 e
+2
−1
f 2 (e−t z)
α
f 2 (e−t z)
−2t 2 0 −t
e z f (e z)
f (e−t z)
(1 − e−2αt )2
− 1 + (1 − α)
−1 .
+
α2 e−2αt
f 2 (e−t z)
e−t z
From (3.1) and (3.2) we deduce that the function w(z, t) is analytic in the unit disk
and
2 0
z f (z)
(3.6)
|w(z, 0)| = 2
− 1 < 1.
f (z)
We observe that w(0, t) = 0. Let t be a fixed number, t > 0, z ∈ U, z 6= 0. Since
|e−t z| ≤ e−t < 1 for all z ∈ U = {z ∈ C : |z| ≤ 1} we conclude that the function
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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w(z, t) is analytic in U . Using the maximum modulus principle it follows that for
each arbitrary fixed t > 0, there exists θ = θ(t) ∈ R such that
(3.7)
|w(z, t)| < max |w(ξ, t)| = |w(eiθ , t)|,
|ξ|=1
We denote u = e−t · eiθ . Then |u| = e−t < 1 and from (3.5) we get
2 0
u f (u)
1 − |u|2α u2 f 0 (u)
2α
iθ
w(e , t) =
− 1 |u| + 2
−1
f 2 (u)
α
f 2 (u)
2 0
u f (u)
f (u)
(1 − |u|2α )2
− 1 + (1 − α)
−1 .
+
α2 |u|2α
f 2 (u)
u
Since u ∈ U , the inequality (3.2) implies that |w(eiθ , t)| ≤ 1 and from (3.6) and
(3.7) we conclude that |w(z, t)| < 1 for all z ∈ U and t ≥ 0.
From Theorem 2.1 it results that the function L(z, t) has an analytic and univalent
extension to the whole disk U for each t ∈ I, in particular L(z, 0). But L(z, 0) =
Fα (z). Therefore the function Fα (z) defined by (3.3) is analytic and univalent in
U.
If in Theorem 3.1 we take α = 1 we obtain the following corollary which is just
Theorem 1.1, namely Ozaki-Nunokawa’s univalence criterion.
Corollary 3.2. Let f ∈ A. If for all z ∈ U , the inequality (3.1) holds true, then the
function f is univalent in U .
Proof. For α = 1 we have F1 (z) = f (z) and the inequality (3.2) becomes
2 0
2 2 z f (z)
(1
−
|z|
)
2
2
(3.8) − 1 |z| + 2(1 − |z| ) +
f 2 (z)
|z|2
2 0
z f (z)
1
≤ 1.
= −
1
·
f 2 (z)
|z|2 Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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It is easy to check that if the inequality (3.1) is true, then the inequality (3.8) is also
true. Indeed, the function g,
g(z) =
z 2 f 0 (z)
−1
f 2 (z)
is analytic in U , g(z) = b2 z 2 + b3 z 3 + · · · , which shows that g(0) = g 0 (0) = 0.
In view of (1.1) we have |g(z)| < 1 and using Schwarz’s lemma we get |g(z)| <
|z|2 .
Example 3.1. Let n be a natural number, n ≥ 2, and the function
f (z) =
(3.9)
z
n+1 .
1− zn
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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Then f is univalent in U and F n+1 is analytic and univalent in U , where
2
(3.10)
F n+1 (z) =
2
n+1
2
Z
z
u
n−1
2
f 0 (u)du
2
n+1
.
0
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Proof. We have
(3.11)
z 2 f 0 (z)
− 1 = z n+1
2
f (z)
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and
(3.12)
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z n+1
f (z)
−1=
.
z
n − z n+1
It is clear that condition (3.1) of Theorem 3.1 is satisfied, and the function f is
univalent in U .
Taking into account (3.11) and (3.12), condition (3.2) of Theorem 3.1 becomes
2(n+1)
4
4
|z|
|z|n+1 (1 − |z|n+1 ) +
(1 − |z|n+1 )2
+
n+1
(n + 1)2
2(1 − n)
1
n+1 2
+
(1
−
|z|
)
(n + 1)2
n − |z|n+1 1
2
2(n+1)
n+1
n+1 2
≤
(n
+
1)
|z|
+
4(n
+
1)(1
−
|z|
)
+
6(1
−
|z|
)
(n + 1)2
2
1
2(n+1)
n+1
=
(n
−
2n
+
3)|z|
+
(4n
−
8)|z|
+
6
≤ 1,
(n + 1)2
because the greatest value of the function
2
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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g(x) = (n − 2n + 3)x + (4n − 8)x + 6,
for x ∈ [0, 1], n ≥ 2 is taken for x = 1 and is g(1) = (n + 1)2 . Therefore the
function F n+1 is analytic and univalent in U .
2
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References
[1] S. OZAKI AND M. NUNOKAWA, The Schwartzian derivative and univalent
functions, Proc. Amer. Math. Soc., 33(2) (1972), 392–394.
[2] Ch. POMMERENKE, Univalent Functions, Vandenhoech Ruprecht, Göttingen,
1975.
Ozaki and Nunokawa’s
Univalence Criterion
Horiana Tudor
vol. 9, iss. 4, art. 117, 2008
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