Paper27-Acta26-2011. 112KB Jun 04 2011 12:03:17 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 26/2011
pp. 251-256

ON THE UNIVALENCE OF A CERTAIN INTEGRAL OPERATOR

Daniel Breaz, Nicoleta Breaz, Virgil Pescar
Abstract. In view of an integral operator Hγ1 ,γ2 ,...,γ[Reη] ,β,η for analytic functions f in the open unit disk U, sufficient conditions for univalence of this integral
operator are discussed.
2000 Mathematics Subject Classification: 30C45.
Key Words and Phrases: Integral operator, univalence, starlike, integer part.

1. Introduction
Let A be the class of functions f of the form
f (z) = z +

∞
X

an z n

n=2

which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. Let S denote the
subclass of A consisting of all univalent functions f in U.
For f ∈ A, the integral operator Gα is defined by
Gα (z) =

Z

z

0



f (u)
u

1

α

du

for some complex numbers α(α 6= 0).
In [1] Kim-Merkes prove that the integral operator Gα is in the class S for
and f ∈ S.
Also, the integral operator Jγ for f ∈ A is given by
 Z z
γ
1
1
−1
γ
u (f (u)) du
Mγ (z) =
γ 0
γ be a complex number, γ 6= 0.

251

(1.1)

1
|α|

≤

1
4

(1.2)

D. Breaz, N. Breaz, V. Pescar - On the univalence of a certain integral operator

Miller and Mocanu in [3] have studied that the integral operator Mγ is in the
class S for f ∈ S ∗ , γ > 0, S ∗ is the subclass of S consisting of all starlike functions
f in U.
We introduce the general integral operator

Hγ1 ,γ2 ,...,γ[Reη] ,β,η (z) =

(

ηβ

Z

0

z

uηβ−1



f1 (u)
u



1
γ1

...



f[Reη] (u)
u

1
[Reη]

γ

du

1

) ηβ

(1.3)

for fj ∈ A, γj , η, β complex numbers, [Reη] ≥ 1, γj 6= 0, j = 1, [Reη], β 6= 0, and
[Reη] is the integer part of η.
If in (1.3) we take η = 1, γ1 = γ, β = γ1 and f1 = f we obtain the integral
operator Mγ .
From (1.3) we take η · β = 1, [Reη] = 1, γ1 = α, f1 = f , we obtain the integral
operator Gγ , given by (1.1).
2.Preliminary results
We need the following lemmas.
Lemma 2.1.[5]Let α be a complex number, Re α > 0 and f ∈ A. If


1 − |z|2Re α  zf ′′ (z) 
 f ′ (z)  ≤ 1
Re α

(2.1)

for all z ∈ U, then for any complex number β, Re β ≥ Re α the function
 Z
Fβ (z) = β

z

u

β−1 ′

f (u)du

0

1

β

(2.2)

is in the class S.
Lemma 2.2.(Schwarz [2])Let f the function regular in the disk
UR = {z ∈ C : |z| < R} with |f (z)| < M , M fixed. If f (z) has in z = 0 one zero
with multiply ≥ m, then
M
(2.3)
|f (z)| ≤ m |z|m , z ∈ UR
R
the equality (in the inequality (2.3) for z 6= 0) can hold only if
f (z) = eiθ

M m
z ,
Rm

where θ is constant.
252

D. Breaz, N. Breaz, V. Pescar - On the univalence of a certain integral operator

3.Main results
Theorem 3.1.Let γj , η complex numbers, Reη ≥ 1 j = 1, [Reη], a =

[Reη]
P

Re

j=1

0 and fj ∈ A, fj (z) = z + b2j z 2 + b3j z 3 + . . . ,
If

j = 1, [Reη].

 ′

2a+1
 zfj (z)

 (2a + 1) 2a


|γj |, j = 1, [Reη],
 fj (z) − 1 ≤
2 [Reη]

1
γj

>

(3.1)

for all z ∈ U, then for any complex number β, Re ηβ ≥ a, the function

Hγ1 ,γ2 ,...,γ[Reη] ,β,η (z) =

(

ηβ

Z

z

uηβ−1

0



f1 (u)
u



1
γ1

...



f[Reη] (u)
u

1
[Reη]

γ

du

1
) ηβ

(3.2)

is in the class S.
Proof. We consider the function
g(z) =

Z

0

z



f1 (u)
u



1
γ1

...



f[Reη] (u)
u

γ

1
[Reη]

(3.3)

du

The function g is regular in U. We define the function p(z) =
z ∈ U and we obtain

[Reη] 
X 1  zfj′ (z)
zg ′′ (z)
=
−1 , z ∈U
p(z) = ′
g (z)
γj fj (z)

zg ′′ (z)
g ′ (z) ,

(3.4)

j=1

From (3.1) and (3.4) we have
(2a + 1)
|p(z)| ≤
2

2a+1
2a

(3.5)

for all z ∈ U and applying Lemma 2.2 we get
2a+1

(2a + 1) 2a
|z|, z ∈ U
|p(z)| ≤
2
From (3.4) and (3.6) we have


2a+1
1 − |z|2a  zg ′′ (z)  (1 − |z|2a )|z| (2a + 1) 2a
·
 g ′ (z)  ≤
a
a
2
253

(3.6)

(3.7)

D. Breaz, N. Breaz, V. Pescar - On the univalence of a certain integral operator

for all z ∈ U.
2a
Because max (1−|z|a )|z| =
|z|≤1

2
(2a+1)

2a+1
2a

1 − |z|2a
a

, from (3.7) we have
 ′′ 
 zg (z) 


 g ′ (z)  ≤ 1

(3.8)

for all z ∈ U. So, by the Lemma 2.1, the integral operator Hγ1 ,γ2 ,...,γ[Reη] ,β,η belongs
to class S.
Corollary 3.2.Let γ be a complex number, a = Re γ1 > 0 and f ∈ A, f (z) =
z + b21 z 2 + b31 z 3 + . . ..
If
 ′

2a+1
 zf (z)
 (2a + 1) 2a


|γ|
(3.9)
 f (z) − 1 ≤
2
for all z ∈ U, then the integral operator Mγ define by (3.9) belongs to the class S.
Proof. We take η = 1, β = γ1 , γ1 = γ, f1 = f in Theorem 3.1.
Corollary 3.3.Let α, η complex numbers a = Re α1 ∈ (0, 1], Reη ≥ 1 and
fj ∈ A, fj (z) = z + b2j z 2 + . . . , j = 1, [Reη].
If
 ′

2a+1
 zfj (z)
 (2a + 1) 2a


(3.10)
|α|, j = 1, [Reη]
 fj (z) − 1 ≤
2 [Reη]

for all z ∈ U, then the function
Lα,η (z) =

Z

z

0



f1 (u)
u

1

α

...



f[Reη] (u)
u

 α1

du

(3.11)

is in the class S.
Proof. For γ1 = γ2 = ... = γ[Reη] = α and ηβ = 1 in Theorem 3.1. we have the
Corollary 3.3.
Remark. For [Reη] = 1, f1 = f , f ∈ A, a = Re α1 ∈ (0, 1] from Corollary 3.3,
we obtain that the function Gα (z) is in the class S.
Theorem 3.5.Let γj , η complex numbers, [Reη] ≥ 1. j = 1, [Reη] , a =
[Reη]
∞
P
P
Re γ1j ≥ 21 and fj ∈ S, fj (z) = z +
bkj z k , j = 1, [Reη].
j=1

k=2

If

[Reη]

X 1
1
1
≤ , for a ≥
|γj |
4
2
j=1

254

(3.12)

D. Breaz, N. Breaz, V. Pescar - On the univalence of a certain integral operator

then for any complex number β, Re ηβ ≥ a, the integral operator Hγ1 ,γ2 ,...,γ[Reη] ,β,η
given by (1.3) is in the class S.
Proof. We consider the function
g(z) =

Z

0

z



f1 (u)
u



1
γ1

...



f[Reη] (u)
u

γ

1
[Reη]

du

(3.13)

The function g is regular in U. We have
1 − |z|2a
a


 ′′ 

X  1  zfj′ (z)

 zg (z)  1 − |z|2a [Reη]



− 1 .
 g ′ (z)  ≤

a
|γj | fj (z)

(3.14)

j=1

Because fj ∈ S, j = 1, [Reη] we have

 ′ 
 zfj (z)  1 + |z|


 fj (z)  ≤ 1 − |z| , z ∈ U, j = 1, [Reη]

(3.15)

From (3.14) and (3.15) we obtain



[Reη]
X 1
1 − |z|2a  zg ′′ (z)  1 − |z|2a 2
≤
 g ′ (z) 
a
a
1 − |z|
|γj |

(3.16)

j=1

for all z ∈ U.
1−|z| 2a
For a ≥ 21 we have max 1−|z|
= 2a and from (3.12), (3.16) we obtain
|z|≤1



1 − |z|2a  zg ′′ (z) 
 g ′ (z)  ≤ 1, z ∈ U
a

(3.17)

From (3.17) and Lemma 2.1 it results that the integral operator
Hγ1 ,γ2 ,...,γ[Reη] ,β,η belongs to class S.


Corollary 3.6.Let α, η complex numbers, [Reη] ≥ 1, a = Re α1 ∈ 12 , 1 and
∞
P
bkj z k , j = 1, [Reη].
fj ∈ S, fj (z) = z +
If

k=2

1
1
≤ ,
|α|
4
then the integral operator Lα,η given by (3.11) is in the class S.
Proof. We take ηβ = 1, γ1 = γ2 = ... = γ[Reη] = α in Theorem 3.5.
255

(3.18)

D. Breaz, N. Breaz, V. Pescar - On the univalence of a certain integral operator
Corollary 3.7.Let γ be a complex number, a = Re γ1 ≥ 12 and f ∈ S f (z) =
z + b21 z 2 + b31 z 3 + ....
If
1
1
≤ ,
(3.19)
|γ|
4
then the integral operator Mγ define by (1.2) belongs to class S.
Proof. For η = 1, β = γ1 , γ1 = γ, f1 = f in Theorem 3.5 we have the Corollary
3.7.
References
[1] Y. J. Kim, E. P. Merkes, On an Integral of Powers of a Spirallike Function,
Kyungpook Math. J., 12 (1972), 249-253.
[2] O. Mayer, The Functions Theory of One Variable Complex, Bucureşti, 1981.
[3] S. S. Miller, P. T. Mocanu, Differential Subordinations, Theory and Applications, Monographs and Text Books in Pure and Applied Mathematics, 225, Marcel
Dekker, New York, 2000.
[4] Z. Nehari, Conformal Mapping, Mc Graw-Hill Book Comp., New York, 1952
(Dover. Publ. Inc., 1975).
[5] N. N. Pascu, An Improvement of Becker’s Univalence Criterion, Proceedings
of the Commemorative Session Simion Stoilow, University of Braşov, 1987, 43-48.
[6] V. Pescar, D. V. Breaz, The Univalence of Integral Operators, Academic
Publishing House, Sofia, 2008.
Daniel Breaz
Department of Mathematics and Computer Science
”1 Decembrie 1918” University of Alba Iulia
Str. N. Iorga, No. 11-13, Alba Iulia, 510009, Alba, Romania
email:dbreaz@uab.ro
Nicoleta Breaz
Department of Mathematics and Computer Science
”1 Decembrie 1918” University of Alba Iulia
Str. N. Iorga, No. 11-13, Alba Iulia, 510009, Alba, Romania
email:nbreaz@uab.ro
Virgil Pescar
Department of Mathematics
Transilvania University of Braşov
500091, Braşov, Romania
email:virgilpescar@unitbv.ro
256