137
Acta Math. Univ. Comenianae
Vol. LXXVI, 2(2007), pp. 137–142
AN INTEGRAL UNIVALENT OPERATOR
D. BREAZ and N. BREAZ
Abstract. In˛ this paper ˛ we consider the class of univalent functions defined by
˛ z2 f 0 (z)
˛
the condition ˛ f 2 (z) − 1˛ < 1, z ∈ U, where f (z) = z + a2 z 2 + . . . , is an analytic
function in the unit disc U = {z ∈ C : |z| < 1} . We present univalence conditions
for the operator
1
0
1
n(α−1)+1
Zz
α−1
α−1
@
A
.
Gα,n (z) = (n (α − 1) + 1) g1
(t) . . . gn (t) dt
0
1. Introduction
Let U be the unit disc, U = {z ∈ C : |z| < 1}. Denote by H (U ) the class of
holomorphic functions on U and consider the set of analytic functions
An = f ∈ H (U ) : f (z) = z + an+1 z n+1 + . . .
(1)
If n = 1 then A1 = A. Let Hu (U ) be the class of univalent functions on the
unit disc U and S the class of regular and univalent functions f (z) = z +a2 z 2 +. . .
in U, which satisfy the condition f (0) = f 0 (0) − 1 = 0.
In their paper [3], Ozaki and Nunokawa proved the following results:
Theorem 1. If we assumethat g ∈ A satisfies the condition
2 0
z f (z)
(2)
f 2 (z) − 1 < 1, z ∈ U
then f is univalent in U .
Lemma 1 (The Schwartz Lemma). Let the analytic function g be regular in
the unit disc U and g (0) = 0. If |g (z)| ≤ 1, ∀z ∈ U, then
(3)
|g (z)| ≤ |z| ,
∀z ∈ U
and equality holds only if g (z) = εz, where |ε| = 1.
Received October 18, 2004.
2000 Mathematics Subject Classification. Primary 30C45.
Key words and phrases. univalent operator, unit disc, univalent functions.
138
D. BREAZ and N. BREAZ
Theorem 2. Let α be a complex number with Re α > 0, and let f = z +
a2 z 2 + . . . be a regular function on U . If
2Re α 1 − |z|
zf 00 (z) (4)
∀z ∈ U
f 0 (z) ≤ 1,
Re α
then for any complex number β with Re β ≥ Re α the function
 z
 β1
Z
(5)
Fβ (z) = β tβ−1 f 0 (t) dt = z + . . .
0
is regular and univalent in U .
Theorem 3. Assume that g ∈ A satisfies condition (2), and let α be a complex
number with
|α − 1| ≤
(6)
Re α
.
3
If
|g (z)| ≤ 1,
(7)
∀z ∈ U
then the function

(8)
Zz
Gα (z) = α
 α1
g (α−1) (t) dt
0
is of class S.
2. Main results
Theorem 4. Let gi ∈ A, ∀i = 1, . . . , n, n ∈ N∗ , satisfy the properties
z 2 g 0 (z)
i
(9)
− 1 < 1, ∀z ∈ U, ∀i = 1, . . . , n
2
gi (z)
and α ∈ C, with
|α − 1| ≤
(10)
Re α
.
3n
If |gi (z)| ≤ 1, ∀z ∈ U , ∀i = 1, . . . , n, then the function

(11)
Zz
Gα,n (z) = (n (α − 1) + 1)
0
is univalent.
1
 n(α−1)+1
g1α−1 (t) . . . gnα−1 (t) dt
139
AN INTEGRAL UNIVALENT OPERATOR
Proof. From (11) we have:
Zz
Gα,n (z) = ( (n(α − 1) + 1)
t
n(α−1)
g1 (t)
t
α−1
· ...
0
(12)
·
gn (t)
t
1
! n(α−1)+1
α−1
dt
We consider the function
Zz (13)
g1 (t)
t
f (z) =
α−1
...
gn (t)
t
α−1
dt.
0
The function f is regular in U , and from (13) we obtain
f 0 (z) =
(14)
g1 (z)
z
α−1
...
gn (z)
z
α−1
and
00
f (z) = (α − 1)
(15)
+
g1 (z)
z
g1 (z)
z
α−1
α−2
α−1
α−1
zg10 (z) − g1 (z) g2 (z)
gn (z)
...
+ ...
z2
z
z
α−1
α−2 0
gn−1 (z)
gn (z)
zgn (z) − gn (z)
...
(α − 1)
.
z
z
z2
Next we calculate the expresion
zf 00 (z)
=
f 0 (z)
z (α − 1)
g1 (z)
z
α−2
z
(16)
g1 (z)
z
+
= (α − 1)
α−1
zf 00
f0 .
...
zg10 (z)−g1 (z)
z2
g1 (z)
z
α−1
...
α−1
g2 (z)
z
gn (z)
z
α−1
...
gn (z)
z
α−1
α−2
(α − 1) gnz(z)
α−1
α−1
g1 (z)
gn (z)
.
.
.
z
z
gn−1 (z)
z
zg10 (z) − 1
zg 0 (z) − 1
+ · · · + (α − 1) n
.
g1 (z)
gn (z)
The modulus
(17)
00 zf f0 α−1 + ...
0
zgn
(z)−gn (z)
z2
140
D. BREAZ and N. BREAZ
can then be evaluated as
00
0
0
zf (z) = (α − 1) zg1 (z) − 1 + · · · + (α − 1) zgn (z) − 1 f 0 (z) g1 (z)
gn (z) zg10 (z) − 1 zgn0 (z) − 1 ≤ (α − 1)
(18)
+ · · · + (α − 1)
g1 (z) gn (z) 0
0
zg1 (z) − 1 zgn (z) − 1 .
= |α − 1| + · · · + |α − 1| g1 (z) gn (z) 2Re α
By multiplying the first and the last terms of (18) with 1−|z|
> 0, we obtain
Re α
2Re α
2Re α 00
zg10 (z) 1 − |z|
zf (z) ≤ 1 − |z|
|α
−
1|
f 0 (z) g1 (z) + 1 + . . .
Re α
Re α
0
2Re α
zg (z) 1 − |z|
+1
+
|α − 1| n
Re α
gn (z) (19)
2Re α
z 2 g10 (z) |g1 (z)|
1 − |z|
≤ |α − 1|
g 2 (z) |z| + 1 + . . .
Re α
1
2Re α 2 0
1 − |z|
z gn (z) |gn (z)| + 1 .
+ |α − 1|
g 2 (z) |z|
Re α
n
By applying the Schwartz Lemma and using (19), we obtain
2Re α 2Re α 2 0
00
1 − |z|
zf (z) ≤ |α − 1| 1 − |z|
z g1 (z) − 1 + 2 + . . .
f 0 (z) g 2 (z)
Re α
Re α
1
(20)
2Re α
z 2 gn0 (z)
1 − |z|
+ |α − 1|
g 2 (z) − 1 + 2 .
Reα
n
Since gi satisfies the condition (2) ∀i = 1, . . . , n, then from (20) we obtain:
2Re α
2Re α
2Re α 00
1 − |z|
1 − |z|
zf (z) ≤ 3 |α − 1| 1 − |z|
+
·
·
·
+
3
|α
−
1|
f 0 (z) Re α
Re α
(21) Re α
3 |α − 1|
3 |α − 1|
3n |α − 1|
≤
+ ··· +
=
.
Re α
Re α
Reα
But |α − 1| ≤
(22)
Re α
3n
so from (7) we obtain that
2Re α 00
1 − |z|
zf (z) ≤ 1,
0
Re α
f (z) for all z ∈ U and the Theorem 2 implies that the function Gα,n is in the class
S.
Corollary 1. Let g ∈ A satisfy (2) and α a complex number such that
(23)
|α − 1| ≤
Re α
,
3k
k ∈ N∗
AN INTEGRAL UNIVALENT OPERATOR
If |g (z)| ≤ 1, ∀z ∈ U, then the function

Gkα (z) = (k (α − 1) + 1)
(24)
Zz
141
1
 k(α−1)+1
g k(α−1) (t) dt
0
is univalent.
Proof. We consider the functions
1
 k(α−1)+1

k(α−1)
Zz
g (t)
dt
(25) Gkα (z) = (k (α − 1) + 1) tk(α−1)
t
0
and
Zz f (z) =
(26)
g (t)
t
k(α−1)
dt.
0
The function f is regular in U. From (26) we obtain
k(α−1)
g (z)
f 0 (z) =
z
and
f 00 (z) = k (α − 1)
g (z)
z
k(α−1)−1
zg 0 (z) − g (z)
.
z2
Next we have
2Re α
(27)
1 − |z|
Re α
00
0
zg (z) zf (z) 1 − |z|2Re α
≤
k |α − 1| +1 ,
f 0 (z) Re α
g (z) ∀z ∈ U.
Applying the Schwartz Lemma and using (27) we obtain
2Re α 2Re α 2 0
00
1 − |z|
zf (z) ≤ |α − 1| 1 − |z|
z g (z) − 1 + 2 .
(28)
g 2 (z)
f 0 (z) Re α
Re α
Since g satisfies conditions (2) then from (28) and (23) we obtain
2Re α 00
1 − |z|
zf (z) ≤ 3k |α − 1| 1 − |z|2Re α ≤ 3k |α − 1| ≤ 1.
(29)
0
Re α
f (z)
Re α
Re α
Now Theorem 2 and (29) imply that Gkα ∈ S.
Corollary 2. Let f, g ∈ A, satisfy (1) and α the complex number with the
property
(30)
|α − 1| ≤
Re α
.
6
142
D. BREAZ and N. BREAZ
If |f (z)| ≤ 1, ∀z ∈ U and |g (z)| ≤ 1, ∀z ∈ U , then the function
1

 2α−1
Zz
(31)
Gα (z) = (2α − 1) f (α−1) (t) g (α−1) (t) dt
0
is univalent.
Proof. In Theorem 1 we set n = 2, g1 = f , g2 = g.
Remark 1. Theorem 1 is a generalization of Theorem 3.
Remark 2. From Corollary 1, for k = 1, we obtain Theorem 3.
References
1. Nehari Z., Conformal Mapping, Mc Graw-Hill Book Comp., New York, 1952 (Dover. Publ.
Inc. 1975).
2. Nunokava M., On the theory of multivalent functions, Tsukuba J. Math. 11(2) (1987), 273–
286.
3. Ozaki S. and Nunokawa M., The Schwartzian derivative and univalent functions, Proc. Amer.
Math. Soc. 33(2) (1972), 392–394.
4. Pascu N. N., On univalent criterion II, Itinerant seminar on functional equations approximation and convexity, Cluj-Napoca, Preprint nr. 6, (1985), 153–154.
5. Pascu N. N., An improvement of Beker’s univelence criterion, Proceedings of the Commemorative Session Cimion Stoilow, Brasov (1987), 43–48.
6. Pescar V., New criteria for univalence of certain integral operators, Demonstratio Mathematica, XXXIII (2000), 51–54.
D. Breaz, Department of Mathematics, “1 Decembrie 1918” University, Alba Iulia, Romania,
e-mail: dbreaz@uab.ro
N. Breaz, Department of Mathematics, “1 Decembrie 1918” University, Alba Iulia, Romania,
e-mail: nbreaz@uab.ro