Univalence condition and properties for
two integral operators
N. Ularu and D. Breaz
Abstract. In this paper we consider the class N(β) and two integral
operators I(f, g)(z) and I1 (f, g)(z), for analytic functions f, g in the open
unit disk U, where g is a functions that belongs to the family B(µ, β).
For this two operators we obtain some properties in this class and the
univalence of I1 (f, g)(z).
M.S.C. 2010: 30C45.
Key words: Analytic functions; univalent functions; unit disk; regular functions;
functions class.
1
Introduction and preliminaries
Let U = {z : |z| < 1} be the unit disk and A the class of all functions of the form:
(1.1)
f (z) = z +
∞
X
an z n ,
n=2
which are analytic in U and satisfy the condition f (0) = f 0 (0) − 1 = 0. We denote
by S the class of univalent and regular functions. In this paper, we introduce two
integral operators, defined by:
Zz
(1.2)
f 0 (t)eg(t) dt
I(f, g)(z) =
0
and
Zz ³
(1.3)
I1 (f, g)(z) =
f 0 (t)eg(t)
´α
dt,
0
where f, g ∈ A and α ∈ C. The operator I1 (f, g) was introduced and studied by
Ularu and Breaz in [7].
Applied Sciences, Vol.15, 2013, pp.
112-117.
c Balkan Society of Geometers, Geometry Balkan Press 2013.
°
Univalence condition and properties for two integral operators
113
We further consider some classes of analytical functions. Many authors studied
this classes, proving some interesting properties. We say that a function f (z) ∈ S∗β is
in the class of starlike functions of order β if it satisfies
µ 0 ¶
zf (z)
Re
>β
(z ∈ U)
f (z)
for some β(0 ≤ β < 1). The class of starlike functions was introduced by Robertson
in [6]. As well, a function f ∈ A is said to be in the class Rβ if and only if
Re (f 0 (z)) > β,
(z ∈ U)
for some β(0 ≤ β < 1).
Let N(β) be the subclass of A that contains all the functions f (z), which satisfy
the inequality
½ 00
¾
zf (z)
Re
+
1
<β
(β > 1, z ∈ U).
f 0 (z)
Uralegaddi, Ganigi and Sarangi in [8] and Owa and Srivastava in [5] introduced and
studied the class N(β) .
The family B(µ, β), µ ≥ 0, 0 ≤ β < 1, consisting of the functions f (z), which
satisfy the condition
¯
¯
µ
¶µ
¯
¯
z
¯ 0
¯
(1.4)
− 1¯ < 1 − β
(z ∈ U).
¯f (z)
¯
¯
f (z)
was studied by Frasin and Jahangiri in [3]. The family B(µ, β) is a comprehensive class
of analytic functions that includes various new classes of analytic univalent functions,
for example, B(1, β)⊂S∗β , and B(0, β)≡Rβ . The subclass B(2, β) ≡ B(β) has been
introduced by Frasin and Darus [2].
To prove the univalence condition for the operator I1 (f, g)(z) we will use the
Becker univalence criterion given by the following result:
Theorem 1.1. [1] If the function f is regular in the unit disk U, f (z) = z+a2 z 2 +. . .
and
¯ 00 ¯
2 ¯ zf (z) ¯¯
(1 − |z| )¯¯ 0
≤1
f (z) ¯
for all z ∈ U, then the function f is univalent in U.
Lemma 1.1. (General Schwarz-Lemma, [4]) Let f the function regular in the
disk UR = {z ∈ C : |z| < R}, with |f (z)| < M , M fixed. If f has in z = 0 one zero
with multiply ≥ m, then
(1.5)
|f (z)| ≤
M
m
|z| , z ∈ UR
Rm
the equality (in the inequality (1.5) for z 6= 0) can hold only if f (z) = eiθ RMm z m , where
θ is constant.
The purpose of this paper is to prove the univalence condition for the integral
operator I1 (f, g)(z) and to show that the operators I(f, g)(z) and I1 (f, g)(z) are in
the class N(δ) by using functions from the class B(µ, β).
114
2
N. Ularu and D. Breaz
Main results
Theorem 2.1. Let α be a complex number with Reα ≥ 1, M be a positive real
number (M ≥ 1) and the functions f, g ∈ A of the form (1.1). If
¯ 00 ¯
¯ f (z) ¯
0
¯
¯
¯ f 0 (z) ¯ ≤ 1, z ∈ U, |g (z)| ≤ M, z ∈ U
and
√
3 3
,
|α| ≤
2(M + 1)
then the function I1 (f, g)(z) defined by (1.3) is in the class S.
Proof. From (1.3), we have that
³
´α
I10 (f, g)(z) = f 0 (z)eg(z)
(2.1)
and
(2.2)
³
´α−1 h
i
I100 (f, g)(z) = α f 0 (z)eg(z)
· eg(z) (f 00 (z) + f 0 (z)g 0 (z)) .
From (2.1) and (2.2) we get
I100 (f, g)(z)
=α
I10 (f, g)(z)
and
µ
¶
f 00 (z)
0
+
g
(z)
f 0 (z)
¯ 00
¯
¯ µ 00
¶¯
¯ zI1 (f, g)(z) ¯
¯
f (z)
2 ¯¯
0
¯
¯
(1 − |z| )¯ 0
= (1 − |z| )¯zα
+ g (z) ¯¯
¯
0
I1 (f, g)(z)
f (z)
2
(2.3)
2
≤ (1 − |z| )|z||α|(1 + M ).
Let us consider the function T : [0, 1] → R, T (x) = x(1 − x2 ), x = |z|. Then we have
(2.4)
2
T (x) ≤ √
3 3
for all x ∈ [0, 1]. From (2.3), (2.4) we obtain
¯ 00
¯
2 ¯¯ zI1 (f, g)(z) ¯¯
(2.5)
(1 − |z| )¯ 0
≤ 1.
I1 (f, g)(z) ¯
From Theorem 1.1 and (2.5) we yield that the function I1 (f, g)(z) is in the class S. ¤
Corollary 2.1. Let α be a complex number with Reα ≥ 1 and let be the functions
f, g ∈ A of the form (1.1). If
¯ 00 ¯
¯ f (z) ¯
0
¯
¯
¯ f 0 (z) ¯ ≤ 1, z ∈ U, |g (z)| < 1, z ∈ U
and
√
3 3
,
|α| ≤
4
then the function I1 (f, g)(z) defined by (1.3) is in the class S.
Univalence condition and properties for two integral operators
Proof. We consider M = 1 in Theorem 2.1.
115
¤
Example 2.1. We consider the analytical functions f (z) = z+ 21 z 2 and g(z) = z·ez .
Because the functions f (z) and g(z) satisfy the conditions from Theorem 2.1, we
obtain that
Zz h
i
t α
I(z) =
(1 + t) · et·e
dt
0
is in the univalent functions class S.
Theorem 2.2. Let the functions f, g ∈ A, with g in the class B(µ, β),¯ µ ≥ 1,
¯ 0≤
¯ f 00 (z) ¯
β < 1. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ¯ f 0 (z) ¯ < 1,
then the integral operator I(f, g)(z) defined by (1.2) is in N(ρ), where
ρ = 2 + (2 − β)M µ .
Proof. From (1.2) we obtain that
zI 00 (f, g)(z)
=z
I 0 (f, g)(z)
µ
¶
f 00 (z)
0
+ g (z) .
f 0 (z)
Thus we have
µ 00
¶
· µ 00
¶
¸
zI (f, g)(z)
f (z)
0
Re
+
1
=
Re
z
+
g
(z)
+
1
I 0 (f, g)(z)
f 0 (z)
µ¯ 00 ¯
¶
¯ f (z) ¯
¯ + |g 0 (z)| + 1
< |z| ¯¯ 0
(2.6)
f (z) ¯
¯
¯ ¶
µ
¶µ ¯¯
µ
¯
¯¯ g(z) ¯µ
z
¯¯
¯ + 1.
< |z| 1 + ¯¯g 0 (z)
g(z) ¯¯ z ¯
Because g ∈ B(µ, β), |g(z)| < M , applying the General Schwarz Lemma and from
(2.6), we obtain
¯
µ 00
¶ ·
µ¯
µ
¶µ
¶
¸
¯ 0
¯
zI (f, g)(z)
z
¯g (z)
¯ + 1 Mµ + 1
Re
+
1
<
1
+
−
1
¯
¯
I 0 (f, g)(z)
g(z)
< 2 + (2 − β)M µ = ρ,
which implies that the integral operator I(f, g)(z) is in the class N(ρ)
¤
Corollary 2.2. Consider the functions f, g ∈ A, with g in the class
¯ 00Rβ ,¯0 ≤ β < 1.
¯ (z) ¯
If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ¯ ff 0 (z)
¯ < 1, then
the integral operator I(f, g)(z) defined by (1.2) is in N(ρ), where
ρ = 4 − β.
Proof. We put µ = 0 in Theorem 2.2.
For µ = 1 in Theorem 2.2 we obtain:
¤
116
N. Ularu and D. Breaz
Corollary 2.3. Let the functions f, g ∈ A, with g in the class S∗β , 0 ≤ β < 1. If
¯ 00 ¯
¯ (z) ¯
|g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ¯ ff 0 (z)
¯ < 1, then the
integral operator I(f, g)(z) defined by (1.2) is in N(ρ), where
ρ = 2 + (2 − β)M.
Theorem 2.3. Let the functions f, g ∈ A, with g in the class B(µ, β), µ ≥ 1, 0 ≤
β < 1 and α a complex number
¯ Reα ≥ 1. If |g(z)| < M , for M a positive real
¯ 00 with
¯ f (z) ¯
number (M ≥ 1), z ∈ U and ¯ f 0 (z) ¯ < 1, then the integral operator I1 (f, g)(z) defined
by (1.3) is in N(ρ), where
ρ = |α|(1 + (2 − β)M µ ) + 1.
Proof. From (1.3) we obtain
zI100 (f, g)(z)
= zα
I10 (f, g)(z)
So,
µ
Re
(2.7)
zI100 (f, g)(z)
+1
I10 (f, g)(z)
¶
µ
¶
f 00 (z)
0
+
g
(z)
f 0 (z)
· µ 00
¶
¸
f (z)
0
= Re αz
+
g
(z)
+
1
f 0 (z)
µ¯ 00 ¯
¶
¯ f (z) ¯
¯ + |g 0 (z)| + 1
< |α||z| ¯¯ 0
f (z) ¯
¯ ¶
¯
¶µ ¯¯
µ
µ
¯¯ g(z) ¯µ
¯
z
¯¯
¯ + 1.
< |α||z| 1 + ¯¯g 0 (z)
g(z) ¯¯ z ¯
Since g ∈ B(µ, β), |g(z)| < M from General Schwarz Lemma and from (2.7), we infer
¯
µ 00
¶
µ
µ¯
µ
¶µ
¶
¶
¯ 0
¯
zI1 (f, g)(z)
z
µ
¯
¯
Re
+ 1 < |α| 1 + ¯g (z)
− 1¯ + 1 M
+1
I10 (f, g)(z)
g(z)
< |α|(1 + (2 − β)M µ ) + 1 = ρ.
So the integral operator I1 (f, g)(z) ∈ N(ρ).
¤
Corollary 2.4. Let the functions f, g ∈ A, with g in the class B(1, β) ⊂ S∗β , 0 ≤
β < 1 and α a complex number
¯ 00 with
¯ Reα ≥ 1. If |g(z)| < M , for M a positive real
¯ (z) ¯
number (M ≥ 1), z ∈ U and ¯ ff 0 (z)
¯ < 1, then the integral operator I1 (f, g)(z) defined
by (1.3) is in N(ρ), where
ρ = |α|(1 + (2 − β)M ) + 1.
Proof. In Theorem 2.3 we put µ = 1.
¤
For µ = 0 in Theorem 2.3 we obtain:
Corollary 2.5. Let the functions f, g ∈ A, with g in the class Rβ , 0 ≤ β < 1 and
α a complex number with
¯ ≥ 1. If |g(z)| < M , for M a positive real number
¯ 00 Reα
¯ f (z) ¯
(M ≥ 1), z ∈ U and ¯ f 0 (z) ¯ < 1, then the integral operator I1 (f, g)(z) defined by
(1.3) is in N(ρ), where
ρ = |α|(3 − β) + 1.
Univalence condition and properties for two integral operators
3
117
Conclusions
The future research will study similar properties using other classes of analytical functions for the integral operators which were defined in this paper, further generalizing
the integral operators and getting results related to convexity, starlikeness, univalence
conditions, etc.
Acknowledgements. This work was partially supported by the strategic project
POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 co-financed by the
European Social Fund-Investing in People.
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Author’s address:
Nicoleta Ularu,
University of Piteşti, 1 Târgul din Vale Str., RO-110040, Piteşti, Romania.
E-mail: nicoletaularu@yahoo.com
Daniel Breaz,
”1 Decembrie 1918” University of Alba Iulia,
11-13 N. Iorga Str., RO-510009, Alba Iulia, Romania.
E-mail: dbreaz@uab.ro