Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 702715, 8 pages
doi:10.1155/2008/702715
Research Article
On the Univalence Criterion of a General
Integral Operator
Daniel Breaz1 and H. Özlem Güney2
1
2
Department of Mathematics, “1 Decembrie 1918” University, Alba Iulia 510009, Romania
Department of Mathematics, Faculty of Science and Arts, University of Dicle, Diyarbakir 21280, Turkey
Correspondence should be addressed to Daniel Breaz, dbreaz@uab.ro
Received 12 March 2008; Accepted 16 May 2008
Recommended by Paolo Ricci
In this paper we considered an general integral operator and three classes of univalent functions for
which the second order derivative is equal to zero. By imposing supplimentary conditions for these
functions we proved some univalent conditions for the considered general operator. Also some
interesting particullar results are presented.
Copyright q 2008 D. Breaz and H. Özlem Güney. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let unit disk and let A denote the class of functions f of the form
fz z a2 z2 a3 z3 · · ·
z ∈ U,
1.1
which are analytic in the open disk U and satisfy the conditions f0 f 0 − 1 0. Consider
S {f ∈ A : f are univalent functions in U}.
Let A2 be the subclass of A consisting of functions of the form
fz z ∞
ak zk .
1.2
k3
Let T be the univalent subclass of A which satisfies
z2 f z
−
1
< 1 z ∈ U.
fz2
1.3
2
Journal of Inequalities and Applications
Let T2 be the subclass of T for which f 0 0. Let T2,μ be the subclass of T2 consisting
of functions of the form 1.2 which satisfy
z2 f z
1.4
−
1
≤ μ z ∈ U
fz 2
for some μ 0 < μ ≤ 1, and let us denote T2,1 ≡ T2 . Furthermore, for some real p with 0 < p ≤ 2
we define a subclass Sp of A consisting of all functions fz which satisfy
z
1.5
fz ≤ p z ∈ U.
In 1, Singh has shown that if fz ∈ Sp, then fz satisfies
z2 f z
−
1
≤ p|z|2 z ∈ U.
fz 2
1.6
Ahlfors 2 and Becker 3 had obtained the following univalence criterion.
Theorem 1.1. Let c be a complex number, |c| ≤ 1, c /
− 1. If fz z a2 z2 · · · is a regular
function in U and
2
2 zf z 1.7
c|z| 1 − |z|
≤1
f z for all z ∈ U, then the function f is regular and univalent in U.
In 4, Pescar had obtained the following theorem.
Theorem 1.2 see 4. Let β be a complex number, Re β > 0, c a complex number, |c| ≤ 1, c / − 1,
2
and hz z a2 z · · · a regular function in U. If
2β
2β zh z 1.8
≤1
c|z| 1 − |z|
βh z for all z ∈ U, then the function
z
1/β
β−1 Fβ z β t h tdt
z ···
1.9
0
is regular and univalent in U.
Lemma 1.3 the general Schwarz lemma 5. Let the function fz be regular in the disk UR {z ∈ C; |z| < R}, with |fz| < M for fixed M. If fz has one zero with multiplicity order bigger than
m for z 0, then
fz ≤ M |z|m ,
Rm
z ∈ UR .
The equality can hold only if fz eiθ M/Rm zm , where θ is constant.
1.10
D. Breaz and H. Özlem Güney
3
In 6, Seenivasagan and Breaz consider for fi ∈ A2 i 1, 2, . . . , n and α1 , α2 , . . . , αn , β ∈
C, the integral operator
z
1/β
n fi t 1/αi
β−1
Fα1 ,α2 ,...,αn ,β z β t
dt
.
t
0
i1
1.11
When αi α for all i 1, 2, . . . , n, Fα1 ,α2 ,...,αn ,β z becomes the integral operator Fα,β z
considered in 7.
2. Main results
Theorem 2.1. Let M ≥ 1, let the functions fi ∈ Sp, for i ∈ {1, . . . , n}, satisfy the condition 1.6,
let αi , β be complex numbers, let Re β ≥ ni1 1 pM 1/|αi |, and let c be a complex number.
If
|c| ≤ 1 −
n
1 pM 1
1 ,
Re β i1
|αi |
fi z ≤ M,
2.1
for all z ∈ U, then the functions Fα1 ,α2 ,...,αn ,β defined in 1.11 are in the class S.
Proof. Define a function
hz z
n 0 i1
fi t
t
1/αi
dt,
2.2
then we have h0 h 0 − 1 0. Also, a simple computation yields
h z n fi z 1/αi
i1
n
zh z 1
h z
α
i1 i
z
,
zfi z
−1 .
fi z
2.3
From 2.3, we have
n
n
zh z 1 zfi z 1 z2 fi z fi z ≤
1 1 .
h z fi z |αi | fi2 z z i1 αi
i1
2.4
From the hypothesis, we have |fi z| ≤ M z ∈ U, i 1, 2, . . . , n, then by the general
Schwarz lemma, we obtain that
fi z ≤ M|z|
z ∈ U, i 1, 2, . . . , n.
2.5
4
Journal of Inequalities and Applications
We apply this result in inequality 2.4, then we obtain
2 n
z f z zh z 1
i
≤
M1
h z f z2 i
i1 αi
2 n
z f z
1
i
− 1 M M 1
≤
f z2
i
i1 αi
n
1 pM|z|2 M 1
αi i1
<
2.6
n
1 pM 1
.
αi i1
We have
n
1 zfi z
2β
2β zh z 2β
2β 1
−1 c|z| 1 − |z|
c|z| 1 − |z|
βh z β i1 αi fi z
n
1
1 ·
≤ |c| |β| i1 |αi |
We obtain
2 z f z |f z|
i
i
1 .
·
2
f z |z|
i
n
1 pM 1
c|z|2β 1 − |z|2β zh z < |c| 1
αi βh z
|β| i1
n
1 pM 1
1 ≤ |c| .
αi Re β i1
So, from 2.1 we have
c|z|2β 1 − |z|2β zh z ≤ 1.
βh z 2.7
2.8
2.9
Applying Theorem 1.2, we obtain that Fα1 ,α2 ,...,αn ,β is univalent.
Theorem 2.2. Let M ≥ 1, let the functions fi ∈ Sp, for i ∈ {1, . . . , n}, satisfy the condition 1.6,
let α, β be complex numbers, let Re β ≥ n1 pM 1/|α|, and let c be a complex number.
If
1 n p 1M 1
|c| ≤ 1 −
,
Re β
|α|
2.10
fi z ≤ M,
for all z ∈ U, then the function
z
1/β
n fi t 1/α
β−1
Fα,β z β t
dt
t
0
i1
is in the class S.
2.11
D. Breaz and H. Özlem Güney
5
Proof. In Theorem 2.1, we consider α1 α2 · · · αn α.
Corollary 2.3. Let the functions fi ∈ Sp, for i ∈ {1, . . . , n}, satisfy the condition 1.6, let αi , β be
complex numbers, let Re β ≥ ni1 p 2/|αi |, and let c be a complex number.
If
n
p2
1 ,
Re β i1 αi fi z ≤ 1,
|c| ≤ 1 −
2.12
for all z ∈ U, then the function Fα1 ,α2 ,...,αn ,β defined in 1.11 is in the class S.
Proof. In Theorem 2.1, we consider M 1.
Corollary 2.4. Let M ≥ 1, let the function f ∈ Sp satisfy the condition 1.6, let α, β be complex
numbers, let Re β ≥ 1 pM 1/|α|, and let c be a complex number.
If
1 1 pM 1
|c| ≤ 1 −
,
Re β
|α|
fz ≤ M
2.13
z
1/α 1/β
β−1 ft
dt
Gα,β z β t
t
0
2.14
for all z ∈ U, then the function
is in the class S.
Proof. In Theorem 2.1, we consider n 1.
Corollary 2.5. Let the function f ∈ Sp satisfy the condition 1.6, let α, β be complex numbers, let
Re β ≥ p 2/|α|, and let c be a complex number.
If
|c| ≤ 1 −
1 p2
,
Re β |α|
2.15
|fz| ≤ 1
for all z ∈ U, then the function
z
1/α 1/β
β−1 ft
dt
Gα,β z β t
t
0
is in the class S.
Proof. In Corollary 2.4, we consider M 1.
2.16
6
Journal of Inequalities and Applications
Theorem 2.6. Let M ≥ 1, let the functions fi ∈ T2,μi , for i ∈ {1, . . . , n}, satisfy the condition 1.4, let
αi , β be complex numbers, let Re β ≥ ni1 1 μi M 1/|αi |, and let c be a complex number.
If
|c| ≤ 1 −
n 1μ M1
1 i
,
αi Re β i1
fi z ≤ M
2.17
for all z ∈ U, then the function Fα1 ,α2 ,...,αn ,β defined in 1.11 is in the class S.
Proof. Define a function
hz z
n 0 i1
fi t
t
1/αi
dt,
2.18
then we have h0 h 0 − 1 0. Also a simple computation yields
h z n fi z 1/αi
i1
n
zh z 1
h z
α
i1 i
z
,
zfi z
−1 .
fi z
2.19
From 2.19, we have
n
n
zh z 1 zfi z 1 z2 fi z fi z ≤
1
1
.
h z fi z f 2 z z i1 αi
i1 αi
i
2.20
From the hypothesis, we have |fi z| ≤ M z ∈ U, i 1, 2, . . . , n, then by the general
Schwarz lemma, we obtain that
fi z ≤ M|z|
z ∈ U, i 1, 2, . . . , n.
2.21
From 2.20, we obtain
2 n
z f z zh z 1
i
≤
M1
h z f z2 i1 αi
i
2 n
z f z
1
i
≤
− 1M M 1
f z2
i1 αi
i
n
1 μi M M 1
≤
αi i1
n 1μ M1
i
.
αi i1
2.22
D. Breaz and H. Özlem Güney
7
We have
n
1 zfi z
c|z|2β 1 − |z|2β zh z c|z|2β 1 − |z|2β 1
−1 βh z β i1 αi fi z
n
1 z2 f z |fi z|
1 2 i
·
·
1
.
≤ |c| |β| i1 αi fi z |z|
We obtain
n 1μ M1
i
c|z|2β 1 − |z|2β zh z ≤ |c| 1
αi βh z |β| i1
n 1μ M1
1 i
≤ |c| .
αi Re β i1
2.23
2.24
So, from 2.17 we have
c|z|2β 1 − |z|2β zh z ≤ 1.
βh z 2.25
Applying Theorem 1.2, we obtain that Fα1 ,α2 ,...,αn ,β is univalent.
Corollary 2.7. Let M ≥ 1, let the functions fi ∈ T2,μi , for i ∈ {1, . . . , n}, satisfy the condition 1.4,
let α, β be complex numbers, let Re β ≥ n1 μi M 1/|α|, and let c be a complex number.
If
1 n 1 μi M 1
,
|c| ≤ 1 −
Re β
|α|
2.26
fi z ≤ M
for all z ∈ U, then the function
z
1/β
n fi t 1/α
β−1
Fα,β z β t
dt
t
0
i1
2.27
is in the class S.
Proof. In Theorem 2.6, we consider α1 α2 · · · αn α.
Corollary 2.8. Let the functions fi ∈ T2,μi , for i ∈ {1, . . . , n}, satisfy the condition 1.4, let αi , β be
complex numbers, let Re β ≥ ni1 μi 2/|αi |, and let c be a complex number.
If
n
μi 2
1 ,
Re β i1 αi fi z ≤ 1
|c| ≤ 1 −
for all z ∈ U, then the function Fα1 ,α2 ,...,αn ,β defined in 1.11 is in the class S.
2.28
8
Journal of Inequalities and Applications
Proof. In Theorem 2.6, we consider M 1.
Corollary 2.9. Let M ≥ 1, let the function f ∈ T2,μ satisfy the condition 1.4, let α, β be complex
numbers, let Re β ≥ 1 μM 1/|α|, and let c be a complex number.
If
1 1 μM 1
|c| ≤ 1 −
,
Re β
|α|
2.29
fz ≤ M
for all z ∈ U, then the function
z
1/α 1/β
β−1 ft
dt
Gα,β z β t
t
0
2.30
is in the class S.
Proof. In Theorem 2.6, we consider n 1.
Corollary 2.10. Let the function f ∈ T2,μ satisfy the condition 1.4, let α, β be complex numbers, let
Re β ≥ μ 2/|α|, and let c be a complex number.
If
1 μ2
|c| ≤ 1 −
,
Re β |α|
2.31
fz ≤ 1
for all z ∈ U, then the function
z
1/α 1/β
β−1 ft
dt
Gα,β z β t
t
0
2.32
is in the class S.
Proof. In Corollary 2.9, we consider M 1.
Acknowledgment
D. Breaz was supported by the GAR 20/2007.
References
1 V. Singh, “On a class of univalent functions,” International Journal of Mathematics and Mathematical
Sciences, vol. 23, no. 12, pp. 855–857, 2000.
2 L. V. Ahlfors, “Sufficient conditions for quasiconformal extension,” Annals of Mathematics Studies, vol.
79, pp. 23–29, 1973.
3 J. Becker, “Löwnersche Differentialgleichung und Schlichtheitskriterien,” Mathematische Annalen, vol.
202, no. 4, pp. 321–335, 1973.
4 V. Pescar, “A new generalization of Ahlfors’s and Becker’s criterion of univalence,” Bulletin of the
Malaysian Mathematical Society. Second Series, vol. 19, no. 2, pp. 53–54, 1996.
5 Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.
6 N. Seenivasagan and D. Breaz, “Certain sufficient conditions for univalence,” General Mathematics, vol.
15, no. 4, pp. 7–15, 2007.
7 D. Breaz and N. Breaz, “The univalent conditions for an integral operator on the classes Sp and T2 ,”
Journal of Approximation Theory and Applications, vol. 1, no. 2, pp. 93–98, 2005.