General Mathematics Vol. 17, No. 4 (2009), 97–109
Certain Sufficient Conditions For Univalence
Virgil Pescar, Daniel Breaz, Nicoleta Breaz
Abstract
We introduce the integral operator Hγ1 ,γ2 ,...,γn ,β,δ for analytic
functions f in the open unit disk U, sufficient conditions for univalence of this integral operator are discussed.
2000 Mathematics Subject Classification: Primary 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.
97
98
Virgil Pescar, Daniel Breaz, Nicoleta Breaz
For f ∈ A, the integral operator Gα is defined by
¶1
Z zµ
f (u) α
(1)
Gα (z) =
du
u
0
for some complex numbers α(α 6= 0).
In [2] Kim-Merkes prove that the integral operator Gα is in the class S
for
1
|α|
≤
1
4
and f ∈ S.
Also, the integral operator Mγ for f ∈ A is given by
½ Z z
¾γ
1
1
−1
γ
Mγ (z) =
(2)
u (f (u)) du
γ 0
γ be a complex number, γ 6= 0.
Miller and Mocanu [4] 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.
Pescar in [9] define a general integral operator
Jγ1 ,γ2 ...,γn (z) =
(3)
½µ
=
1
1
1
+
+ ... +
γ1 γ2
γn
¶Z
z
u−1 (f1 (u))
1
γ1
¾
1
γn
. . . (fn (u)) du
1
1
1
1
γ1 + γ2 +...+ γn
0
for fj ∈ A and complex numbers γj , (γj 6= 0), j = 1, n, which is a generalization of integral operator Mγ .
For n = 1, f1 = f and γ1 = γ, from (3) we obtain the integral operator
Mγ .
We introduce the general integral operator
(
(4)Hγ1 ,γ2 ...,γn ,β,δ (z) =
Z
z
δβ
0
µ
uδβ−1
f1 (u)
u
¶ γ1
1
µ
...
fn (u)
u
¶ γ1
n
) δβ1
du
Certain Sufficient Conditions For Univalence
99
for fj ∈ A, γj complex numbers, γj 6= 0, j = 1, n, δ, β complex numbers,
δ 6= 0, β 6= 0, n ∈ N − {0}.
P
From (4), for β = 1δ nj=1
1
γj
we obtain the integral operator Jγ1 ,γ2 ...,γn
defined by (3), for n = 1, δ = 1, β = 1, γ1 = α and f1 = f we have the
integral operator Gα given by (1).
If in (4) we take n = 1, γ1 = γ, β = γ1 , δ = 1 and f1 = f we obtain the
integral operator Mγ .
For n = 1, γ1 = α, δ = 1 and f1 = f from (4) we obtain the integral
operator Tα,β defined in [8] by
" Z
µ
¶ α1 # β1
z
f (u)
Tα,β = β
uβ−1
du
(5)
u
0
for f ∈ A and α, β be complex numbers, α 6= 0, β 6= 0.
For δβ = 1 we have the integral operator given in [1] and for δ = 1, γ1 =
γ2 = . . . = γn = α we obtain the integral operator defined in [1].
2
Preliminary results
We need the following lemmas.
Lemma 1 [7]. Let α be a complex number, Re α > 0 and f ∈ A. If
¯
¯
1 − |z|2Re α ¯¯ zf 00 (z) ¯¯
(6)
¯ f 0 (z) ¯ ≤ 1
Re α
for all z ∈ U, then for any complex number β, Re β ≥ Re α the function
· Z z
¸ β1
(7)
Fβ (z) = β
uβ−1 f 0 (u)du
0
is in the class S.
100
Virgil Pescar, Daniel Breaz, Nicoleta Breaz
Lemma 2 (Schwarz [3]). 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
(8)
|f (z)| ≤
M m
|z| , z ∈ UR
Rm
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 1 Let γj , α complex numbers, j = 1, n,
fj ∈ A, fj (z) = z + b2j z 2 + b3j z 3 + . . . ,
a = Reα > 0 and
j = 1, n, n ∈ N − {0}.
If
(9)
¯ 0
¯
¯ zfj (z)
¯ (2a + 1) 2a+1
2a
¯
¯≤
−
1
|γj |, j = 1, n,
¯ fj (z)
¯
2n
for all z ∈ U , then for any complex numbers β and δ, Re δβ ≥ a, the
function
(
(10)
Hγ1 ,γ2 ,...,γn ,β,δ (z) =
is in the class S.
Z
z
δβ
0
µ
uδβ−1
f1 (u)
u
¶ γ1
1
µ
...
fn (u)
u
¶ γ1
n
) δβ1
du
Certain Sufficient Conditions For Univalence
101
Proof. We consider the function
¶1
µ
¶1
Z zµ
f1 (u) γ1
fn (u) γn
(11)
g(z) =
...
du
u
u
0
The function g is regular in U. We define the function p(z) =
zg 00 (z)
,
g 0 (z)
z ∈ U and we obtain
(12)
µ
¶¸
n ·
zg 00 (z) X 1 zfj0 (z)
p(z) = 0
=
−1 , z ∈U
g (z)
γj fj (z)
j=1
From (9) and (12) we have
(2a + 1)
|p(z)| ≤
2
(13)
2a+1
2a
for all z ∈ U and applying Lemma 2 we get
(2a + 1)
|p(z)| ≤
2
(14)
2a+1
2a
|z|, z ∈ U
From (12) and (14) we have
¯
¯
2a+1
1 − |z|2a ¯¯ zg 00 (z) ¯¯ (1 − |z|2a )|z| (2a + 1) 2a
(15)
·
¯ g 0 (z) ¯ ≤
a
a
2
for all z ∈ U .
Because
max
|z|≤1
(1 − |z|2a )|z|
2
=
2a+1 ,
a
(2a + 1) 2a
from (15) we have
(16)
¯
¯
1 − |z|2a ¯¯ zg 00 (z) ¯¯
¯ g 0 (z) ¯ ≤ 1
a
for all z ∈ U . So, by the Lemma 1, the integral operator Hγ1 ,γ2 ,...,γn ,β,δ
belongs to class S.
102
Virgil Pescar, Daniel Breaz, Nicoleta Breaz
Corollary 1 Let γj , α complex numbers, j = 1, n, a = Reα > 0,
n
P
j=1
Re γ1j ≥
2
Reα and fj ∈ A, fj (z) = z + b2j z + . . . , j = 1, n, n ∈ N − {0, 1}.
If
(17)
¯ 0
¯
¯ zfj (z)
¯ (2a + 1) 2a+1
2a
¯
¯
|γj |, j = 1, n
¯ fj − 1¯ ≤
2n
for all z ∈ U, then the function
(Ã n
!
) Pn 1 1
X 1 Z z
j=1 γj
1
1
u−1 (f1 (u)) γ1 ... (fn (u)) γn du
(18) Jγ1 ,γ2 ,...γn (z) =
γ
0
j=1 j
is in the class S.
Proof. For β =
1
δ
Pn
1
j=1 γj ,
from Theorem 1 we obtain Corollary 1.
Corollary 2 Let γ, α complex numbers, a = Reα > 0, Re γ1 ≥ Reα and
f ∈ A, f (z) = z + b21 z 2 + b31 z 3 + . . ..
If
(19)
¯ 0
¯
¯ zf (z)
¯ (2a + 1) 2a+1
2a
¯
¯
|γ|
¯ f (z) − 1¯ ≤
2
for all z ∈ U, then the integral operator Mγ define by (2) belongs to the
class S.
Proof. We take n = 1, δ = 1, β = γ1 , γ1 = γ, f1 = f in Theorem 1.
Corollary 3 Let γj , α complex numbers, j = 1, n, a = Reα, a ∈ (0, 1] and
fj ∈ A, fj (z) = z + b2j z 2 + . . . , j = 1, n, n ∈ N − {0}.
Certain Sufficient Conditions For Univalence
103
If
¯ 0
¯
¯ zfj (z)
¯ (2a + 1) 2a+1
2a
¯
¯
|γj |, j = 1, n
¯ fj (z) − 1¯ ≤
2n
(20)
for all z ∈ U, then the function
Z
(21)
Kγ1 ,γ2 ,...γn (z) =
z
0
µ
f1 (u)
u
µ
¶ γ1
1
...
fn (u)
u
¶ γ1
n
du
belongs to class S.
Proof. We take β =
1
δ
in Theorem 1.
Corollary 4 Let α, γ complex numbers a = Reγ > 0, n ∈ N − {0} and
fj ∈ A, fj (z) = z + b2j z 2 + . . . , j = 1, n.
If
¯ 0
¯
¯ zfj (z)
¯ (2a + 1) 2a+1
2a
¯
¯≤
−
1
|α|, j = 1, n
¯ fj (z)
¯
2n
(22)
for all z ∈ U , then for any complex number β with Re β ≥ Reγ the function
(23)
( Z
Lα,β (z) = β
z
0
µ
uβ−1
f1 (u)
u
¶ α1
µ
...
fn (u)
u
¶ α1
) β1
du
is in the class S.
Proof. For δ = 1 and γ1 = γ2 = ... = γn = α in Theorem 3.1. we have the
Corollary 4.
104
Virgil Pescar, Daniel Breaz, Nicoleta Breaz
Theorem 2 Let γj , α complex numbers, j = 1, n , a = Reα > 0 and
P
k
fj ∈ S, fj (z) = z + ∞
k=2 bkj z , j = 1, n.
If
(24)
n
X
1
a
1
≤ , for 0 < a <
|γj |
2
2
j=1
or
n
X
1
1
1
≤ , for a ≥
|γj |
4
2
j=1
(25)
then for any complex numbers β, δ, Re δβ ≥ a, the integral operator
Hγ1 ,γ2 ,...,γn ,β,δ given by (4) is in the class S.
Proof. We consider the function
¶1
¶1
µ
Z zµ
f1 (u) γ1
fn (u) γn
g(z) =
(26)
...
du
u
u
0
The function g is regular in U. We have
¯
¯
¯
¯¸
n ·
¯
1 − |z|2a ¯¯ zg 00 (z) ¯¯ 1 − |z|2a X 1 ¯¯ zfj0 (z)
¯
≤
−
1
(27)
¯ g 0 (z) ¯
¯ fj (z)
¯
a
a
|γ
|
j
j=1
Because fj ∈ S, j = 1, n we have
¯ 0 ¯
¯ zfj (z) ¯ 1 + |z|
¯
¯
(28)
¯ fj (z) ¯ ≤ 1 − |z| , z ∈ U, j = 1, n
From (27) and (28) we obtain
(29)
for all z ∈ U.
1 − |z|2a
a
¯ 00 ¯
n
¯ zg (z) ¯ 1 − |z|2a 2 X
1
¯
¯≤
¯ g 0 (z) ¯
a
1 − |z| j=1 |γj |
Certain Sufficient Conditions For Univalence
For 0 < a <
1
2
105
we have
1 − |z| 2a
=1
max
|z|≤1 1 − |z|
and from (24), (29) we get
¯
¯
1 − |z|2a ¯¯ zg 00 (z) ¯¯
¯ g 0 (z) ¯ ≤ 1, z ∈ U
a
(30)
For a ≥
1
2
we have
1 − |z| 2a
max
= 2a
|z|≤1 1 − |z|
and from (25), (29) we obtain
¯
¯
1 − |z|2a ¯¯ zg 00 (z) ¯¯
(31)
¯ g 0 (z) ¯ ≤ 1, z ∈ U
a
From (30), (31) and Lemma 1 it results that the integral operator
Hγ1 ,γ2 ,...,γn ,β,δ belongs to class S.
Corollary 5 Let γj , α complex numbers, j = 1, n, a = Reα > 0,
n
P
j=1
Reα and fj ∈ S, fj (z) = z + b2j z 2 + . . . , j = 1, n.
If
(32)
n
X
1
a
1
≤ , for 0 < a <
|γj |
2
2
j=1
or
(33)
δ
X
1
1
1
≤ , for a ≥
|γj |
4
2
j=1
then the integral operator Jγ1 ,γ2 ,...γn given by (3) is in the class S.
Re γ1j ≥
106
Proof. For β =
Virgil Pescar, Daniel Breaz, Nicoleta Breaz
1
δ
Pn
1
j=1 γj ,
from Theorem 2 we obtain that the integral
operator Jγ1 ,γ2 ,...γn is in the class S.
Corollary 6 Let γj , α complex numbers, j = 1, n, a = Reα, a ∈ (0, 1] and
fj ∈ S, fj (z) = z + b2j z 2 + . . . , j = 1, n.
If
(34)
n
X
1
a
1
≤ , for 0 < a <
|γj |
2
2
j=1
or
(35)
n
X
1
1
1
≤ , for
≤a≤1
|γj |
4
2
j=1
then the integral operator Kγ1 ,γ2 ,...γn given by (21) is in the class S.
Proof. We take β =
1
δ
and from Theorem 2 it results that Kγ1 ,γ2 ,...γn given
by (21) belongs to class S.
Corollary 7 Let α, γ complex numbers, a = Reγ > 0 and fj ∈ S, fj (z) =
P
k
z+ ∞
k=2 bkj z , j = 1, n.
If
(36)
1
a
1
≤ , for 0 < a <
|α|
2
2
or
(37)
1
1
1
≤ , for a ≥
|α|
4
2
then for any complex number β, Re β ≥ Re γ the integral operator Lα,β
given by (23) is in the class S.
Certain Sufficient Conditions For Univalence
107
Proof. We take δ = 1, γ1 = γ2 = ... = γn = α in Theorem 2.
Corollary 8 Let α, γ complex numbers, a = Re α > 0 and f ∈ S,
f (z) = z + b21 z 2 + b31 z 3 + ....
If
(38)
1
a
1
≤ , for 0 < a <
|γ|
2
2
or
(39)
1
1
1
≤ , for a ≥
|γ|
4
2
then the integral operator Mγ define by (2) belongs to class S.
Proof. For n = 1, δ = 1, β = γ1 , γ1 = γ, f1 = f in Theorem 2 we have
the Corollary 8.
Corollary 9 Let α, γ complex numbers, a = Reγ > 0 and f ∈ S, f (z) =
z + b21 z 2 + b31 z 3 + ....
If
(40)
1
a
1
≤ , for 0 < a <
|α|
2
2
or
(41)
1
1
1
≤ , for a ≥
|α|
4
2
then for any complex number β, Re β ≥ Re γ, the integral operator Tα,β
given by (5) is in the class S.
Proof. For n = 1, δ = 1, γ1 = α and f1 = f from Theorem 2 we obtain
Corollary 9.
108
Virgil Pescar, Daniel Breaz, Nicoleta Breaz
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”Babeş-Bolyai”, Mathematica, Cluj-Napoca, No. 3, 2002, 13-21.
[2] Y. J. Kim, E. P. Merkes, On an Integral of Powers of a Spirallike
Function, Kyungpook Math. J., 12 (1972), 249-253.
[3] O. Mayer, The Functions Theory of One Variable Complex, Bucureşti,
1981.
[4] 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.
[5] P. T. Mocanu, T. Bulboacă, G. St. Sălăgean, The Geometric Theory
of Univalent Functions, Cluj, 1999.
[6] Z. Nehari, Conformal Mapping, Mc Graw-Hill Book Comp., New York,
1952 (Dover. Publ. Inc., 1975).
[7] N. N. Pascu, An Improvement of Becker’s Univalence Criterion, Proceedings of the Commemorative Session Simion Stoilow, University of
Braşov, 1987, 43-48.
[8] N. N. Pascu, V. Pescar On the Integral Operators of Kim-Merkes and
Pfaltzgraff, Mathematica, Univ. Babeş-Bolyai Cluj-Napoca, 32(55),
2(1990), 185-192.
Certain Sufficient Conditions For Univalence
109
[9] V. Pescar, On the Univalence of Some Integral Operators, Acta Universitatis Apulensis, No. 18/2009, 255-264.
Virgil Pescar
”Transilvania” University of Braşov
Department of Mathematics
500091 Braşov, Romania
E-mail: virgilpescar@unitbv.ro
Daniel Breaz, Nicoleta Breaz
”1 Decembrie 1918” University of Alba Iulia
Department of Mathematics
510009 Alba, Romania
E-mails: dbreaz@uab.ro, nbreaz@uab.ro