Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 719354, 4 pages
doi:10.1155/2008/719354
Research Article
Certain Integral Operators on the Classes
Mβi and Nβi
Daniel Breaz
Department of Mathematics, 1st December 1918, University of Alba Iulia, 510009 Alba, Romania
Correspondence should be addressed to Daniel Breaz, dbreaz@uab.ro
Received 13 September 2007; Revised 21 October 2007; Accepted 2 January 2008
Recommended by Vijay Gupta
We consider the classes Mβi and Nβi of the analytic functions and two general integral operators. We prove some properties for these operators on these classes.
Copyright q 2008 Daniel Breaz. 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 U {z ∈ C, |z| < 1} be the open unit disk and let A denote the class of the functions fz
of the form
fz z a2 z2 a3 z3 · · · ,
z ∈ U,
1.1
which are analytic in the open disk U.
Let Mβ be the subclass of A, consisting of the functions fz, which satisfy the inequality
zf z
< β, z ∈ U, β > 1,
1.2
Re
fz
and let Nβ be the subclass of A, consisting of functions fz, which satisfy the inequality
zf z
Re
1
< β, z ∈ U.
1.3
f z
These classes are studied by Uralegaddi et al. in 1, and Owa and Srivastava in 2.
Consider the integral operator Fn introduced by D. Breaz and N. Breaz in 3, having the
form
Fn z z 0
f1 t
t
α1
where fi z ∈ A and αi > 0, for all i ∈ {1, . . . , n}.
···
fn t
t
αn
dt,
1.4
2
Journal of Inequalities and Applications
Remark
1.1. This operator extends the integral operator of Alexander given by Fz z
ft/tdt.
0
Also, we consider the next integral operator denoted by Fα1 ,...,αn that was introduced by
Breaz et al. in 4, having the form
z
Fα1 ,...,αn z 0
α
α1
f1 t · · · fn t n dt,
1.5
where fi z ∈ A and αi > 0 for all i ∈ {1, . . . , n}.
It is easy to see that these integral operators are analytic operators.
2. Main results
Theorem 2.1. Let fi ∈ Mβi , for each i 1, 2, 3, . . . , n with βi > 1. Then Fn z ∈ Nμ with
μ 1 ni1 αi βi − 1 and αi > 0, (i 1, 2, 3, . . . , n).
Proof. After some calculi, we obtain that
n
n
zf z zFn z −
αi i
αi .
fi z
Fn z
i1
i1
2.1
The relation 2.1 is equivalent to
zFn z
Re
1
Fn z
n
zfi z
αi Re
fi z
i1
−
n
αi 1.
2.2
αi βi − 1 1.
2.3
i1
Since fi ∈ Mβi , we have
zFn z
1
Re
Fn z
Because
n
i1
<
n
i1
αi β i −
n
αi 1 i1
n
i1
αi βi − 1 > 0, we obtain that Fn ∈ Nμ, where μ 1 n
i1
αi βi − 1.
Corollary 2.2. Let fi ∈ Mβ for each i 1, 2, 3, . . . , n with β > 1. Then Fn z ∈ Nγ with
γ 1 β − 1 ni1 αi and αi > 0, i 1, 2, 3, . . . , n.
Proof. In Theorem 2.1, we consider β1 β2 · · · βn β.
Corollary 2.3. Let f ∈ Mβ with β > 1. Then the integral operator Fz ∈ Nδ with δ αβ − 1 1 and α > 0.
z
0
ft/tα dt
Proof. In Corollary 2.2, we consider n 1 and α1 α.
Corollary
2.4. Let f ∈ Mβ with β > 1. Then the integral operator of Alexander Fz z
ft/tdt
∈ Nβ.
0
Daniel Breaz
3
Proof. We have
zF z zf z
− 1.
F z
fz
2.4
From 2.4, we have
zF z
Re
1
F z
Re
zf z
< β.
fz
2.5
So relation 2.5 implies that Alexander operator is in Nβ.
Theorem 2.5. Let fi ∈ Nβi for each i 1, 2, 3, . . . , n, with βi > 1. Then Fα1 ,...,αn z ∈ Nρ with
ρ 1 ni1 αi βi − 1 and αi > 0, i 1, 2, 3, . . . , n.
Proof. After some calculi, we have
zFα1 ,...,αn z
α1
Fα 1 ,...,αn z
zf1 z
zfn z
·
·
·
α
n
f1 z
fn z
2.6
that is equivalent to
zFα1 ,...,αn z
Fα 1 ,...,αn z
1 α1
n
zf1 z
zfn z
1 · · · αn
1 −
αi 1.
f1 z
fn z
i1
2.7
Since fi ∈ Nβi , for all i ∈ {1, . . . , n}, we have
Re
zfn z
1
fn z
2.8
< βi .
So we obtain
Re
zFα1 ,...,αn z
Fα 1 ,...,αn z
1
<
n
i1
αi β i −
which implies that Fα1 ,...,αn ∈ Nρ, where ρ 1 n
αi 1 i1
n
i1 αi βi
n
i1
αi βi − 1 1
2.9
− 1.
Corollary 2.6. Let fi ∈ Nβ for each i 1, 2, 3, . . . , n with β > 1. Then Fα1 ,...,αn z ∈ Nη with
η 1 ni1 αi β − 1 and αi > 0, i 1, 2, 3, . . . , n.
Proof. In Thorem 2.5, we consider β1 β2 · · · βn β.
Corollary 2.7. Let f ∈ Nβ with β > 1. Then the integral operator
Fα z z
0
is in the class Nαβ − 1 1 and α > 0.
α
f t dt
2.10
4
Journal of Inequalities and Applications
Proof. We have
zf z
zFα z
α .
f z
Fα z
2.11
From 2.11 we have
zFα z
1
Re
Fα z
zf z
α Re
1 1 − α < αβ 1 − α αβ − 1 1.
f z
2.12
So the relation 2.12 implies that the operator Fα is in Nαβ − 1 1.
Example 2.8. Let fz 1/2β − 1{1 − 1 − z2β−1 } ∈ Nβ. After some calculi, we obtain that
Fα z z
0
α
f t dt 1
1 − z2αβ−11 ∈ N αβ − 1 1 .
2α1 − β − 1
2.13
Acknowledgment
The paper is supported by Grant no. 2-CEx 06-11-10/25.07.2006.
References
1 B. A. Uralegaddi, M. D. Ganigi, and S. M. Sarangi, “Univalent functions with positive coefficients,”
Tamkang Journal of Mathematics, vol. 25, no. 3, pp. 225–230, 1994.
2 S. Owa and H. M. Srivastava, “Some generalized convolution properties associated with certain subclasses of analytic functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 3, Article
ID 42, 13 pages, 2002.
3 D. Breaz and N. Breaz, “Two integral operators,” Studia Universitatis Babeş-Bolyai, Mathematica, vol. 47,
no. 3, pp. 13–19, 2002.
4 D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,” in press.