paper 24-18-2009. 121KB Jun 04 2011 12:03:08 AM

ACTA UNIVERSITATIS APULENSIS

No 18/2009

ON THE UNIVALENCE OF SOME INTEGRAL OPERATORS

Virgil Pescar

Abstract. In view of two integral operators Hγ1 ,γ2 ,...,γn and Jγ1 ,γ2 ,...,γn
for analytic functions fj , j = 1, n in the open unit disk U, sufficient conditions
for univalence of these integral operators are discussed.
2000 Mathematics Subject Classification: 30C45.
Key Words and Phrases: Integral operator, univalence, starlike functions.
1. Introduction
We consider the unit open disk U = {z ∈ C : |z| < 1} and A the class of
the functions f which are analytic in U and f (0) = f ′ (0) − 1 = 0.
We denote by S the subclass of the functions f ∈ A which are univalent
in U and S ∗ denote the subclass of S consisting of all starlike functions f in
U.
We consider the integral operators
 Z z

γ
1
1
−1
Jγ (z) =
u (f (u)) γ du
(1)
γ 0
for f ∈ A, γ be a complex number, γ 6= 0 and
Hγ1 ,γ2 ,...,γn (z) =

Z

0

z



f1 (u)

u

 γ1

1

...



fn (u)
u

 γ1

n

du

(2)

for fj ∈ A and γj complex numbers, γj 6= 0, j = 1, n.
Miller and Mocanu [5] have studied that the integral operator Jγ is in the
class S for f ∈ S ∗ .
255

V. Pescar - On the univalence of some integral operators
From (2) for n = 1, f1 = f and γ11 = α we obtain the integral operator
Kim-Merkes, Hα , given by
α
Z z
f (u)
du
(3)
Hα (z) =
u
0
In [3] Kim-Merkes prove that the integral operator Hα is in the class S
for |α| ≤ 41 and f ∈ S.
Pescar in [8], [9], has obtained univalence sufficient conditions for the

integral operator Hα .
Pescar, Owa [12] have studied univalence problems for integral operator
Hα .
In [2], D. Breaz and N. Breaz have studied the univalence of integral
operator Hγ1 ,γ2 ,...,γn .
In this paper we introduce a general integral operator
Jγ1 ,γ2 ,...,γn (z) =

(

n
X
1
γ
j=1 j

!Z

z

1
γ1

−1

1
γn

) Pn 1

u f1 (u) ...fn (u) du

0

1
j=1 γj

(4)

for fj ∈ A, γj complex numbers, γj 6= 0, j = 1, n, which is a generalization

of integral operator Jγ , given by (1).
For n = 1, f1 = f and γ1 = γ, from (4) we obtain the integral operator
Jγ .
In the present paper, we obtain some sufficient conditions for the integral
operators Hγ1 ,γ2 ,...,γn and Jγ1 ,γ2 ,...,γn to be in the class S.
2. Preliminary results
We need the following lemmas.
Lemma 2.1. [7] Let α be a complex number, Re α > 0 and f ∈ A. If


1 − |z|2Re α  zf ′′ (z) 
(5)
 f ′ (z)  ≤ 1
Re α

for all z ∈ U, then the integral operator Fα defined by
 Z
Fα (z) = α

z

u

 α1
f (u)du

α−1 ′

0

256

(6)

V. Pescar - On the univalence of some integral operators
is in the class S.
Lemma 2.2. [1] If f (z) = z + a2 z 2 + .. is analytic in U and
 ′′ 



2  zf (z) 
(1 − |z| )  ′
≤1
f (z) 

(7)

for all z ∈ U, then the function f (z) is univalent in U.

Lemma 2.3. (Schwarz [4]). 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
|f (z)| ≤

M m
|z| , z ∈ UR
Rm

(8)

the equality (in the inequality (8) for z 6= 0) can hold only if
f (z) = eiθ

M m
z ,
Rm

where θ is constant.
3. Main results
Theorem 3.1
positive numPn Let γj1 be complex numbers, γj 6= 0, Mj real
2
3
=
1
and
f
∈
A,
f

(z)
=
z
+
a
z
+
a
Re
bers, j = 1, n,
j
j
2j
3j z + ..., j =
j=1
γj
1, n. If

 ′


 zfj (z)


(9)
 fj (z) − 1 ≤ Mj , j = 1, n
and

√
n
X
3 3
Mj
≤
|γj |
2
j=1

(10)

then the integral operators Jγ1 ,γ2 ,...,γn given by (4) and Hγ1 ,γ2 ,...,γn given by (2)
are in the class S.
Proof. We observe that

Jγ1 ,γ2 ,...,γn (z) =
257

V. Pescar - On the univalence of some integral operators

=

(

n
X
1
γ
j=1 j

!Z

z

u

Pn

1
j=1 γj

−1



z



0

Let us consider the function
Z
Hγ1 ,γ2 ,...,γn (z) =

0

f1 (u)
u

f1 (u)
u

 γ1

...



...



1

 γ1

1

fn (u)
u

) Pn 1

 γ1

du

 γ1

du

fn (u)
u

n

n

1
j=1 γj

(11)

(12)

for fj ∈ A, j = 1, n. The function Hγ1 ,γ2 ,...,γn (z) is regular in U.
zH ′′

(z)

We define the function p by p(z) = H ′γ1 ,γ2 ,...,γn(z) . The function p satisfies
γ1 ,γ2 ,...,γn
p(0) = 0 and


n 
X

1  zfj′ (z)
 ,z ∈ U
(13)
−
1
|p(z)| ≤

 fj (z)
|γ
|
j
j=1
From (9) and (13) we have

n
X
Mj
|p(z)| ≤
|γj |
j=1

(14)

for all z ∈ U.
Applying Lemma 2.3 we obtain
n
X
Mj
|p(z)| ≤
|z| , z ∈ U
|γ
|
j
j=1

(15)

and hence, we get


n
X

 zHγ′′1 ,γ2 ,...,γn (z) 
Mj
2


≤ (1 − |z| )|z|
1 − |z|  ′

Hγ1 ,γ2 ,...,γn (z)
|γj |
j=1
2

(16)

for all z ∈ U.
We have



2
max (1 − |z|2 )|z| = √
|z|≤1
3 3
and from (10) and (16) we obtain
 ′′





zH
(z)
γ
,γ
,...,γ
n
≤1
1 − |z|2  ′ 1 2
Hγ1 ,γ2 ,...,γn (z) 
258

(17)

V. Pescar - On the univalence of some integral operators
for all z ∈ U.
From (17) and by Lemma 2.2, we obtain that the integral operator
Hγ1 ,γ2 ,...,γn is in the class S.


1
1
f1 (z) γ1
fn (z) γn
′
Because Hγ1 ,γ2 ,...,γn (z) =
and
...
z
z
Re α =

n
X

Re

j=1

1
= 1,
γj

from (17) and by Lemma 2.1 it results that the integral operator Jγ1 ,γ2 ,...,γn
belongs to class S.

Corollary 3.2 Let γ be a complex number, Re γ1 = 1 and f ∈ A,
f (z) = z + a21 z 2 + a31 z 3 + . . .
If

√

 ′
 3 3
 zf (z)


 f (z) − 1 ≤ 2 |γ|

(18)

for all z ∈ U, then the integral operators Jγ and Hα , α = γ1 , are in the class
S.

Proof. From Theorem 3.1, for n = 1, γ1 = γ, α = γ1 , f1 = f it results that
Jγ ∈ S and Hα ∈ S.

Corollary 3.3. If f ∈ A and
√
 ′

 zf (z)
 3 3


 f (z) − 1 ≤ 2 , z ∈ U

then the integral operator of Alexander H given by
Z z
f (u)
du
H(z) =
u
0

(19)

(20)

is in the class S.

Proof. From Theorem Theorem 3.1, for n = 1, γ1 = 1, f1 = f, H = H1 we
obtain H ∈ S.


Theorem 3.4. Let γj be complex numbers and fj ∈ S, fj (z) = z +
a2j z 2 + a3j z 3 + . . . , j = 1.n.
259

V. Pescar - On the univalence of some integral operators
If
n
X
1
1
≤
|γj |
4
j=1

(21)

then the integral operator Hγ1 ,γ2 ,...,γn given by (2) is in the class S.
zH ′′

(z)

Proof. We consider the function p(z) = H ′γ1 ,γ2 ,...,γn(z) , where Hγ1 ,γ2 ,...,γn is
γ1 ,γ2 ,...,γn
defined by (2). We obtain


n
X

1  zfj′ (z)
, z ∈ U
(22)
−
1
|p(z)| ≤

 fj (z)
|γ
|
j
j=1

 ′ 
 zf (z) 
Because fj ∈ S we have  fjj(z)  ≤

1+|z|
,
1−|z|

z ∈ U, j = 1, n and

 ′
  ′ 
 zfj (z)
  zfj (z) 
2
 


 fj (z) − 1 ≤  fj (z)  + 1 ≤ 1 − |z| , j = 1, n, z ∈ U

(23)

From (21), (23) and (22) we obtain
|p(z)| ≤

1
, z∈U
2(1 − |z|)

(24)

and hence, we have
 ′′

 zHγ1 ,γ2 ,...,γn (z) 

≤1
(1 − |z| )  ′
Hγ1 ,γ2 ,...,γn (z) 
2

for all z ∈ U.
By Lemma 2.2 we have Hγ1 ,γ2 ,...,γn is in the class S.

(25)



Corollary 3.5. Let γ be a complex number and f ∈ S, f = z +a21 z 2 +. . .
If
|γ| ≤

1
4

then the integral operator Hγ ∈ S.

(26)

Proof. In Theorem Theorem 3.4 for n = 1, γ11 = γ, f1 = f it results that
Hγ is in the class S.

260

V. Pescar - On the univalence of some integral operators
Theorem 3.6. Let γj be complex numbers,
γj 6= 0, fj ∈ A, fj (z) =
P
z + a2j z 2 + a3j z 3 + . . . , j = 1, n and a = nj=1 Re γ1j > 0. If

 ′
 (2a + 1) 2a+1
 zfj (z)
2a


|γj |, j = 1, n
 fj (z) − 1 ≤
2n

(27)

then the integral operator Jγ1 ,γ2 ,...,γn is in the class S.

Proof. The integral operator Jγ1 ,γ2 ,...,γn has the form (11).
We consider the function
zHγ′′1 ,γ2 ,...,γn (z)
, z∈U
p(z) =
Hγ′ 1 ,γ2 ,...,γn (z)

where Hγ1 ,γ2 ,...,γn (z) is define by (2).
The function p satisfies p(0) = 0 and from (28) we obtain


n 
X

1  zfj′ (z)
 , z∈U
−
1
|p(z)| ≤

 fj (z)
|γ
|
j
j=1

(28)

(29)

From (27) and (29) we have

(2a + 1)
|p(z)| ≤
2

2a+1
2a

(30)

for all z ∈ U.
Applying Lemma 2.3 we obtain
(2a + 1)
|p(z)| ≤
2

2a+1
2a

|z|, z ∈ U

(31)

From (28) and (31) we have
1 − |z|2a
a
for all z ∈ U.
Because

 ′′

2a
 zHγ1 ,γ2 ,...,γn (z)  (2a + 1) 2a+1
2a 1 − |z|

≤
|z|
 H′

2
a
γ1 ,γ2 ,...,γn (z)
max
|z|≤1




2
1 − |z|2a
|z| =
2a+1 ,
a
(2a + 1) 2a
261

(32)

V. Pescar - On the univalence of some integral operators
from (32) we obtain
1 − |z|2a
a
for all z ∈ U.
From (2) we have
Hγ′ 1 ,γ2 ,...,γn (z)

=




 ′′
 zHγ1 ,γ2 ,...,γn (z) 
≤1


 H′
γ1 ,γ2 ,...,γn (z)

f1 (z)
z

 γ1

1

...



fn (z)
z

(33)

 γ1

n

, z∈U

(34)

and from (33) by Lemma 2.1 it results that the integral operator Jγ1 ,γ2 ,...,γn
belongs to class S.

Remark 3.7. From Theorem 3.6 for n = 1, γ1 = γ, f1 = f, a = Re γ1 = 1
we obtain the condition
√

 ′
 3 3
 zf (z)


(35)
 f (z) − 1 ≤ 2 |γ|

and, hence, we obtain Corollary 3.2.

References
[1] J. Becker, L¨ownersche Differentialgleichung Und Quasikonform
Fortsetzbare Schlichte Functionen, J. Reine Angew. Math. , 255 (1972),
23-43.
[2] D. Breaz, N. Breaz, Two Integral Operators, Studia Universitatis
”Babe¸s-Bolyai”, Ser. Math., Cluj-Napoca, 3 (2002), 13-21.
[3] Y. J. Kim, E. P. Merkes, On an Integral of Powers of a Spirallike
Function, Kyungpook Math. J., 12 (1972), 249-253.
[4] O. Mayer, The Functions Theory of One Variable Complex, Bucure¸sti,
1981.
[5] 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.
[6] Z. Nehari, Conformal Mapping, Mc Graw-Hill Book Comp., New York,
1952 (Dover. Publ. Inc., 1975).
262

V. Pescar - On the univalence of some integral operators
[7] N. N. Pascu, On a Univalence Criterion II, Itinerant Seminar Functional Equations, Approximation and Convexity, University ”Babe¸s-Bolyai”,
Cluj-Napoca, 85 (1985), 153-154.
[8] V. Pescar, On Some Integral Operations which Preserves the Univalence, Journal of Mathematics, The Punjab University, XXX (1997), 1-10.
[9] V. Pescar, On the Univalence of an Integral Operator, Studia Universitatis ”Babe¸s-Bolyai”, Ser. Math., Cluj-Napoca, XLIII, Number 4 (1998),
95-97.
[10] V. Pescar, New Univalence Criteria, ”Transilvania” University of
Bra¸sov, 2002.
[11] V. Pescar, D. V. Breaz, The Univalence of Integral Operators, Academic Publishing House, Sofia, 2008.
[12] V. Pescar, S. Owa, Univalence Problems for Integral Operators by
Kim-Merkes and Pfaltzgraff, Journal of Approximation Theory and Applications, New Delhi, vol. 3, 1-2 (2007), 17-21.
Author:
Virgil Pescar
Department of Mathematics
”Transilvania” University of Bra¸sov
Faculty of Science
Bra¸sov, Romania
e-mail:virgilpescar@unitbv.ro

263