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ACTA UNIVERSITATIS APULENSIS

No 19/2009

UNIVALENCE CRITERION FOR AN INTEGRAL OPERATOR

Virgil Pescar, Daniel Breaz
Abstract.We consider the integral operator denoted by Tα,β and for the function f ∈ A we proved a sufficient condition for univalence from this integral operator.
2000 Mathematics Subject Classification: 30C45.
Key Words and Phrases: Integral operator, univalence, starlike.
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
γ
 Z z
1
1
−1
γ
Mγ (z) =
u (f (u)) du
γ 0
γ be a complex number, γ 6= 0.

203

(1.1)

1
|α|

≤

1
4

(1.2)

V. Pescar, D. Breaz - Univalence criterion for an integral operator

Miller and Mocanu [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 consider the integral operator Tα,β defined by

Tα,β

" Z
= β

z



uβ−1

0

f (u)
u

1

α

du

# β1

(1.3)

for f ∈ A and α, β be complex numbers, α 6= 0, β 6= 0.
We need the following lemmas.
Lemma 1.1.[6].Let α be a complex number, Re α > 0 and f ∈ A. If


1 − |z|2Re α  zf ′′ (z) 
 f ′ (z)  ≤ 1
Re α

(1.4)

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

(1.5)

is in the class S.
Lemma 1.2.(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

|f (z)| ≤ m |z|m , z ∈ UR
(1.6)
R
the equality (in the inequality (1.6) for z 6= 0) can hold only if
f (z) = eiθ

M m
z ,
Rm

where θ is constant.
2.Main results
Theorem 2.1.Let α be a complex number, a = Re
f (z) = z + a2 z 2 + . . ..

204

1
α

> 0 and f ∈ A,

V. Pescar, D. Breaz - Univalence criterion for an integral operator

If
 ′

2a+1
 zf (z)
 (2a + 1) 2a


|α|
 f (z) − 1 ≤
2

(2.1)

for all z ∈ U, then for any complex number β, Re β ≥ Re α1 , the function
" Z

Tα,β (z) = β

z

uβ−1

0



f (u)
u

1

α

du

# β1

(2.2)

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

Z

z



0

f (u)
u

1

α

du

(2.3)

The function f is regular in U. From (2.3) we have
′

g (z) =
g ′′ (z) =

1
f (z) α
,
z

1
1 f (z) α −1 zf ′ (z) − f (z)
α
z
z2



We define the function h(z) =

zg ′′ (z)
g ′ (z) ,

1
zg ′′ (z)
=
h(z) = ′
g (z)
α



z ∈ U and we obtain

zf ′ (z)
−1 , z ∈U
f (z)

(2.4)

The function h satisfies the condition h(0) = 0. From (2.1) and (2.4) we have
(2a + 1)
|h(z)| ≤
2

2a+1
2a

(2.5)

for all z ∈ U. Applying Lemma 1.2 we get
(2a + 1)
|h(z)| ≤
2

2a+1
2a

|z|

(2.6)

for all z ∈ U.
From (2.4) and (2.6) we obtain


2a+1
1 − |z|2a  zg ′′ (z)  (2a + 1) 2a 1 − |z|2a
|z|
 g ′ (z)  ≤
a
2
a
205

(2.7)

V. Pescar, D. Breaz - Univalence criterion for an integral operator

for all z ∈ U.
Because
max
|z|≤1




1 − |z|2a
2
|z| =
2a+1
a
(2a + 1) 2a

from (2.7) we have
1 − |z|2a
a
for all z ∈ U.

 ′′ 
 zg (z) 


 g ′ (z)  ≤ 1

(2.8)

1

α
, and by Lemma 1.1 we obtain that the
From (2.3) we have g ′ (z) = f (z)
z
integral operator Tα,β define by (2.2) is in the class S.
Corollary. 2.2.Let α be a complex number, a = Re

1
α

> 0 and f ∈ A,

f (z) = z + a2 z 2 + a3 z 3 + . . .
If

 ′
2a+1
 (2a + 1) 2a
 zf (z)
≤

−
1
|α|

 f (z)
2

(2.9)

for all z ∈ U, then the integral operator Mα given by (1.2) belongs to class S.
Proof. For β = α1 , from Theorem 2.1 we obtain Corollary 2.2.
Corollary. 2.3.Let α be a complex number, a = Re
A, f (z) = z + a2 z 2 + . . ..
If

 ′
2a+1
 (2a + 1) 2a
 zf (z)


|α|
 f (z) − 1 ≤
2

1
α

∈ (0, 1] and f ∈

(2.10)

for all z ∈ U, then the integral operator Gα , is in the class S.
Proof. We take β = 1 in Theorem 2.1.
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, Bucure¸sti, 1981.

206

V. Pescar, D. Breaz - Univalence criterion for an integral operator

[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] P. T. Mocanu, T. Bulboac˘a, G. St. S˘al˘
agean, The Geometric Theory of
Univalent Functions, Cluj, 1999.
[5] Z. Nehari, Conformal Mapping, Mc Graw-Hill Book Comp., New York, 1952
(Dover. Publ. Inc., 1975).
[6] N. N. Pascu, An Improvement of Becker’s Univalence Criterion, Proceedings
of the Commemorative Session Simion Stoilow, University of Bra¸sov, 1987, 43-48.
[7] N.N. Pascu and V. Pescar, On the Integral Operators of Kim-Merkes and
Pfaltzgraff, Mathematica, Univ. Babe¸s-Bolyai, Cluj-Napoca, 32 (55), 2 (1990), 185192.
Virgil Pescar
Department of Mathematics
”Transilvania” University of Bra¸sov
500091 Bra¸sov, Romania
e-mail: virgilpescar@unitbv.ro
Daniel Breaz
Department of Mathematics
”1 Decembrie 1918” University of Alba Iulia
510009 Alba Iulia, Romania
e-mail: dbreaz@uab.ro

207