Hindawi Publishing CorporationJournal of Inequalities and Applications Volume 2008, Article ID 719354, 4 pages doi:10.1155/2008/719354 Research Article Certain Integral Operators on the
Trang 1Hindawi 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
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 classesMβ i and Nβ i of the analytic functions and two general integral opera-tors 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
LetU {z ∈ C, |z| < 1} be the open unit disk and let A denote the class of the functions fz
of the form
fz z a2z2 a3z3 · · · , z ∈ U, 1.1 which are analytic in the open diskU.
LetMβ be the subclass of A, consisting of the functions fz, which satisfy the
inequal-ity
Re
zfz
fz
and letNβ be the subclass of A, consisting of functions fz, which satisfy the inequality
Re
zfz
fz 1
These classes are studied by Uralegaddi et al in1, and Owa and Srivastava in 2
Consider the integral operator F nintroduced by D Breaz and N Breaz in3, having the form
F n z
z
0
f1t
t
α1
· · ·
f n t
t
α n
where f i z ∈ A and α i > 0, for all i ∈ {1, , n}.
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Remark 1.1 This operator extends the integral operator of Alexander given by Fz
z
0ft/tdt.
Also, we consider the next integral operator denoted by F α1, ,α n that was introduced by Breaz et al in4, having the form
F α1, ,α n z
z
0
f1t α1· · ·f nt α n
where f i z ∈ A and α i > 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 f i ∈ Mβ i , for each i 1, 2, 3, , n with β i > 1 Then F n z ∈ Nμ with
μ 1 n i1 α i β i − 1 and α i > 0, (i 1, 2, 3, , n).
Proof After some calculi, we obtain that
zF nz
F nz
n
i1
α i
zf iz
f i z −
n
i1
α i 2.1 The relation2.1 is equivalent to
Re
zF nz
F nz 1
n
i1
α iRe
zf iz
f i z
− n
i1
α i 1. 2.2
Since f i ∈ Mβ i, we have
Re
zF nz
F nz 1
<
n
i1
α i β i− n
i1
α i 1 n
i1
α i
β i− 1 1. 2.3
Because n i1 α i β i − 1 > 0, we obtain that F n ∈ Nμ, where μ 1 n
i1 α i β i− 1
Corollary 2.2 Let f i ∈ Mβ for each i 1, 2, 3, , n with β > 1 Then F n z ∈ Nγ with
γ 1 β − 1 n i1 α i and α i > 0, i 1, 2, 3, , n.
Proof InTheorem 2.1, we consider β1 β2 · · · β n β.
Corollary 2.3 Let f ∈ Mβ with β > 1 Then the integral operator Fz z
0ft/t α dt
∈ Nδ with δ αβ − 1 1 and α > 0.
Proof InCorollary 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
0ft/tdt ∈ Nβ.
Trang 3Daniel Breaz 3
Proof We have
zFz
Fz
zfz
fz − 1. 2.4 From2.4, we have
Re
zFz
Fz 1
Rezfz
So relation2.5 implies that Alexander operator is in Nβ.
Theorem 2.5 Let f i ∈ Nβ i for each i 1, 2, 3, , n, with β i > 1 Then F α1, ,α n z ∈ Nρ with
ρ 1 n i1 α i β i − 1 and α i > 0, i 1, 2, 3, , n.
Proof After some calculi, we have
zF α1, ,α n z
F α1, ,α n z α1
zf1z
f1z · · · α n zf nz
f nz 2.6
that is equivalent to
zF α1, ,α n z
F α1, ,α n z 1 α1
zf
1z
f1z 1
· · · α n
zf nz
f nz 1
− n
i1
α i 1. 2.7
Since f i ∈ Nβ i , for all i ∈ {1, , n}, we have
Re
zf nz
f nz 1
So we obtain
Re
zF
α1, ,α n z
F α1, ,α n z 1
<
n
i1
α i β i− n
i1
α i 1 n
i1
α i
β i− 1 1 2.9
which implies that F α1, ,α n ∈ Nρ, where ρ 1 n
i1 α i β i− 1
Corollary 2.6 Let f i ∈ Nβ for each i 1, 2, 3, , n with β > 1 Then F α1, ,α n z ∈ Nη with
η 1 n i1 α i β − 1 and α i > 0, i 1, 2, 3, , n.
Proof In Thorem2.5, we consider β1 β2 · · · β n β.
Corollary 2.7 Let f ∈ Nβ with β > 1 Then the integral operator
F α z
z
0
ft α
is in the class Nαβ − 1 1 and α > 0.
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Proof We have
zF αz
F αz α
zfz
fz . 2.11
From2.11 we have
Re
zF αz
F αz 1
α Re
zfz
fz 1
1 − α < αβ 1 − α αβ − 1 1. 2.12
So the relation2.12 implies that the operator F αis inNαβ − 1 1.
Example 2.8 Let fz 1/2β − 1{1 − 1 − z 2β−1 } ∈ Nβ After some calculi, we obtain that
F α z
z
0
ft α
2α1 − β − 1 1 − z 2αβ−11 ∈ N αβ − 1 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
sub-classes 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 Babes¸-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.
...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
sub -classes of analytic functions,” Journal... Journal of Inequalities in Pure and Applied Mathematics, vol 3, no 3, Article< /small>
ID 42, 13 pages, 2002.
3 D Breaz and N Breaz, “Two integral operators, ”