Paper-7-Acta23.2010. 126KB Jun 04 2011 12:03:14 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 23/2010
pp. 79-89
SOME PROPERTIES FOR CERTAIN INTEGRAL OPERATORS
Aabed Mohammed, Maslina Darus and Daniel Breaz
Abstract. Recently Breaz and Breaz [4] and Breaz et.al[5] introduced two
general integral operators Fn and Fα1 ,...,αn . Considering the classes N(γ), MT(µ, β)
and KD(µ, β) we derived some properties for Fn and Fα1 ,...,αn . Two new subclasses
KDFn (µ, β, α1 , ..., αn ) and KDFα1 ,...,αn (µ, β, α1 , ..., αn ) are defined. Necessary and
sufficient conditions for a family of functions fj to be in the KDFn (µ, β, α1 , ..., αn )
and KDFα1 ,...,αn (µ, β, α1 , ..., αn ) are determined.
AMS Subject Classification: 30C45
Keywords and phrases: Analytic and univalent functions, convex functions, integral operators.
1. Introduction.
Let A denote the class of functions of the form
∞
X
f (z) = z +
an z n
n=2
which are analytic in the open unite disc U = {z ∈ C : |z| < 1}. We also denote by S
the subclass of A consisting of functions which are also univalent in U. Furthermore,
we denote by T the subclass of S consisting of functions whose nonzero coefficients,
from the second one, are negative and has the form:
f (z) = z −
∞
X
an z n , an ≥ 0.
n=2
A function f ∈ A is the convex function of order α, 0 ≤ α < 1, if f satisfies the
following inequality
′′
zf (z)
Re
+ 1 > α, z ∈ U
f ′ (z)
and we denote this class by K(α).
Similarly, if f ∈ A satisfies the following inequality:
79
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Re
zf ′ (z)
f (z)
> α, z ∈ U
for some α, 0 ≤ α < 1, then f is said to be starlike of order α and we denote this
class by S∗ (α).
Let N(γ) be the subclass of A consisting of the functions f which satisfy the inequality
′′
zf (z)
Re
+
1
< γ, z ∈ U, γ > 1.
f ′ (z)
This class was studied by Owa and Srivastava [8].
Let MT(µ, β) be the subclass of A consisting of the functions f which satisfy the
analytic characterization
′
zf (z)
zf ′ (z)
−
1
+
1
<
β
µ
f (z)
f (z)
for some 0 < β ≤ 1, and 0 ≤ µ < 1,
Definition 1.([9])A function f is said to be in the class KD(µ, β), if satisfies
the following inequality:
Re
′′
zf (z)
zf ′′ (z)
+β
+ 1 ≥ µ ′
′
f (z)
f (z)
for some µ ≥ 0 and 0 ≤ β < 1.
For fj (z) ∈ A and αj > 0 for all j ∈ {1, 2, 3, ..., n}, D. Breaz and N. Breaz [4]
introduced the following integral operator
Fn (z) =
Zz Y
n
fj (t) αj
0 j=1
t
dt,
(1)
Recently Breaz et.al [5] introduced the following integral operator
Zz Y
n
′ αj
fj (t) dt,
Fα1 ,...,αn (z) =
0 j=1
where fj ∈ A and αj > 0, for all j ∈ {1, 2, 3, ..., n}.
80
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A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
For univalence, starlike and convexity of these integral operators see ([4]-[7]), see
also ([1]-[3]) for several properties.
Now by using the equations (1) and (2) and the Definition 1, we introduce the
following two new subclasses of KD(µ, β).
Definition 2. A family of functions fj , j ∈ {1, ..., n} is said to be in the class
KDFn (µ, β, α1 , ..., αn ), if satisfies the inequality:
′′
′′
zF (z)
zFn (z)
+ β,
+ 1 ≥ µ ′n
Re
′
Fn (z)
Fn (z)
(3)
for some µ ≥ 0 and 0 ≤ β < 1, where Fn is defined in (1).
Definition 3. A family of functions fj , j ∈ {1, ..., n} is said to be in the class
KDFα1 ,...,αn (µ, β, α1 , ..., αn ) if satisfies the inequality:
′′
′′
zFα1 ,...,αn (z)
zFα1 ,...,αn (z)
+ β,
+ 1 ≥ µ ′
Re
Fα′ 1 ,...,αn (z)
Fα1 ,...,αn (z)
(4)
for some µ ≥ 0 and 0 ≤ β < 1, where Fα1 ,...,αn is defined as in (2).
2. Main results
Our first result is the following:
Theorem 1. Let αj ∈ R, αj > 0 for j ∈ {1, ..., n} and fj ∈ A and suppose that
f ′ (z)
n
P
j
αj βj (µj Mj + 1) + 1.
fj (z) < Mj .If fj ∈ M T (µj , βj ) then Fn ∈ N (σ), where σ =
j=1
Proof. From (1), we observe that Fn ∈ A. On the other hand, it is easy to see that
Fn′ (z)
=
n
Y
fj (z) αj
j=1
z
.
Differentiating (5) logarithmically and multiply by z , we obtain
" ′
#
′′
n
zfj (z)
zFn (z) X
=
−1 .
αj
fj (z)
Fn′ (z)
j=1
81
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A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Thus we have
′
n
X
zfj (z)
zFn′′ (z)
− 1 + 1.
+1=
αj
Fn′ (z)
fj (z)
j=1
We calculate the real part from both terms of the above expression and obtain
Re
X
′
n
zfj (z)
zFn′′ (z)
+1 =
− 1 + 1.
αj Re
Fn′ (z)
fj (z)
(6)
j=1
Since ℜw ≤ | w| , then
Re
′
X
n
zfj (z)
zFn′′ (z)
− 1 + 1.
+1 ≤
αj
′
Fn (z)
fj (z)
j=1
Since fj ∈ MT (µj , βj ) for j ∈ {1, ..., n}, we have
Re
′ X
X
n
n
n
X
fj (z)
zfj′ (z)
zFn′′ (z)
+
µ
+
1
≤
+
1
+
1
≤
α
β
µ
αj βj + 1
α
β
j
j
j
j
j
j
fj (z)
fj (z)
Fn′ (z)
j=1
j=1
<
n
X
αj βj µj Mj +
j=1
Hence Fn ∈ N (σ), σ =
n
X
αj βj + 1 =
j=1
n
P
n
X
j=1
αj βj (µj Mj + 1) + 1.
j=1
αj βj (µj Mj + 1) + 1.
j=1
Letting n = 1, α1 = α, α2 = ... = αn = 0, M1 = M and f1 = f, in the Theorem 1,
we have
′
(z)
Corollary 1. Let α ∈ R , α > 0, f ∈ A and suppose that ff (z)
< M, M fixed.
α
Rz
dt ∈ N(σ) , σ = αβ(µM + 1) + 1.
If f ∈ M T (µ, β) then F1 (z) = 0 f (t)
t
Letting α = 1 in Corollary 1, we have
′
(z)
Corollary 2. Let f ∈ A and suppose that ff (z)
< M, M fixed.
Rz
dt ∈ N(σ) , σ = β(µM + 1) + 1.
then F1 (z) = 0 f (t)
t
If f ∈ M T (µ, β)
Theorem 2. Let αj > 0 for j ∈ {1, ..., n}, let βj > 0 be real number with the
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A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
property 0 ≤ βj < 1 and let fj ∈ KD(µj , βj ) for j ∈ {1, ..., n}, µj ≥ 0. If
n
P
αj (1 − βj ) ≤ 1 then the functions Fα1 ,...,αn given by (2) is convex of or0 <
j=1
der ρ = 1 −
n
P
αj (1 − βj ).
j=1
Proof. From (2), we observe that Fα1 ,...,αn ∈ A and
n
X
zFα′′1 ,...,αn (z)
+
1
=
αj
Fα′ 1 ,...,αn (z)
j=1
!
n
X
fj′′ (z)
+1 −
+1.
z ′
fj (z)
j=1
We calculate the real part from both terms of the above expression and obtain
!
X
′′
n
n
X
fj′′ (z)
zFα1 ,...,αn (z)
+1 −
+1 =
αj + 1.
αj Re z ′
Re
Fα′ 1 ,...,αn (z)
fj (z)
j=1
j=1
Since fj ∈ KD(µj , βj ) for j = {1, ..., n}, we have
!
X
′′
n
n
f ′′ (z)
X
zFα1 ,...,αn (z)
j
+1 >
Re
αj µj z ′
αj + 1.
+ βj −
fj (z)
Fα′ 1 ,...,αn (z)
j=1
j=1
This relation is equivalent to
Re
X
n
n
f ′′ (z) X
zFα′′1 ,...,αn (z)
j
+
+
1
>
αj (βj − 1) + 1.
α
µ
z
j j
fj′ (z)
Fα′ 1 ,...,αn (z)
j=1
j=1
f ′′ (z)
Since αj µj z fj′ (z) > 0 we obtain
j
Re
!
′′
n
X
zFα1 ,...αn (z)
+
1
≥
1
−
αj (1 − βj ),
Fα′ 1 ,...αn (z)
j=1
which implies that Fα1 ,...,αn is convex of order ρ = 1 −
n
P
αj (1 − βj ).
j=1
Letting n = 1, α1 = α, α2 = ... = αn = 0 and f1 = f, in the Theorem 2, we have
Corollary 3. Let α be a real number, α > 0. Suppose that the function fj ∈
83
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
KD(µ, β) and o ≤ α(1 − β) < 1 . In these conditions the function Fα (z) =
Rz ′
(f (t))α dt is convex of order 1 − (1 − β)α.
0
Letting α = 1 in Corollary 3, we have
Corollary 4. Let f ∈ KD(µ, β) and consider the integral operator F1 (z) =
Rz
f ′ (t)dt.
0
In this condition F1 is convex of order β.
A necessary and sufficient condition for a family of analytic functions
fj ∈ KDFn (µ, β, α1 , ..., αn )
In this section, we give a necessary and sufficient condition for a family of functions
fj ∈ KDFn (µ, β, α1 , ..., αn ). Before embarking on the proof of our result, let us
′′ (z)
calculate the expression zF
F ′ (z) , required for proving our result.
n
Recall that, from (1), we have
Fj′ (z)
=
n
Y
fj (z) αj
z
j=1
, z ∈ U.
After some calculation, we obtain that
n
Fn′′ (z) X
=
αj
Fn′ (z)
j=1
that is equivalent to
n
zFn′′ (z) X
=
αj
Fn′ (z)
j=1
Let fj (z) = z −
∞
P
fj′ (z) 1
−
fj (z) z
,
zfj′ (z)
−1 .
fj (z)
∞
P
an,j z n . Then fj′ (z) = 1 −
nan,j z n−1 and we get
n=2
n=2
∞
P
n
z
−
na
z
n
n,j
zFn′′ (z) X
n=2
=
α
− 1
j
∞
=
′
P
Fn (z)
j=1
z−
an,j z n
n=2
=
n
X
j=1
αj
1−
∞
P
nan,j z n−1 − 1 +
n=2
∞
P
an,j z n−1
n=2
1−
∞
P
an,j z n−1
n=2
P
∞
n−1
(n
−
1)
a
z
n
n,j
X
n=2
=−
.
α
j
∞
P
j=1
1−
an,j z n−1
n=2
(7)
84
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Theorem 3. Let the function fj ∈ T for j ∈ {1, ..., n}. Then the functions fj ∈
KDFn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n} if and only if
P
∞
α
(n
−
1)
(µ
+
1)
a
n
n,j
X
n=2 j
≤ 1 − β.
(8)
∞
P
j=1
1−
an,j
n=2
Proof. First consider
′′
′′
′′ (z)
zF (z)
zFn (z)
zF
n
− Re 1 +
.
≤ (µ + 1) ′n
µ ′
′
Fn (z)
Fn (z)
Fn (z)
From (7) we obtain
P
∞
n−1
′′
∞
(n
−
1)
a
z
n,j
zF (z)
X n=2
≤
= (µ + 1)
(µ + 1) ′n
α
j
∞
P
Fn (z)
n=2
n−1
1−
an,j z
n=2
P
P
∞
∞
n−1
α (n − 1) an,j
α (n − 1) |an,j | |z|
n
n
X
X
n=2 j
n=2 j
.
≤ (µ + 1)
≤ (µ + 1)
∞
∞
P
P
n−1
j=1
j=1
1−
|an,j | |z|
1−
an,j
n=2
n=2
If (8) holds then the above expression is bounded by 1 − β, and consequently
′′
zFn (z)
zFn′′ (z)
µ ′
< −β,
− Re 1 + ′
Fn (z)
Fn (z)
which equivalent to
zF ′′ (z)
Re 1 + ′n
Fn (z)
′′
zF (z)
+ β.
≥ µ ′n
Fn (z)
Hence fj ∈ KDFn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n}.
Conversely. Let fj ∈ KDFn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n} and prove that (8)
holds. If fj ∈ KDFn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n} and z is real, we get from (3)
and (7)
P
P
∞
∞
n−1
n−1
∞
(n − 1) an,j z
(n − 1) an,j z
n
X
n=2
X n=2
1−
αj
αj
∞
∞
≥ µ
+ β ≥
P
P
n−1
n−1
j=1
1−
1−
an,j z
an,j z
j=1
n=2
n=2
85
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
P
∞
n−1
(n − 1) an,j z
n
X
n=2
+ β.
µ
αj
∞
P
n−1
j=1
1−
an,j z
n=2
That is equivalent to
P
P
∞
∞
n−1
n−1
α (n − 1) an,j z
α µ (n − 1) an,j z
n
n
X
X
n=2 j
n=2 j
+
≤ 1 − β.
∞
∞
P
P
n−1
n−1
j=1
j=1
1−
an,j z
1−
an,j z
n=2
n=2
The above inequality reduce to
P
∞
n−1
α
(µ
+
1)
(n
−
1)
a
z
n
n,j
X
n=2 j
≤ 1 − β.
∞
P
n−1
j=1
1−
an,j z
n=2
Let z → 1− along the real axis, then we get
P
∞
α (µ + 1) (n − 1) an,j
n
X
n=2 j
≤ 1 − β.
∞
P
j=1
1−
an,j
n=2
Which give the required result.
A necessary and sufficient condition for a family of analytic functions
fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn )
In this section, we give a necessary and sufficient condition for a family of analytic
functions fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn ). Let us first calculate the expression,
zFα′′1 ,...,αn (z)
Fα′ 1 ,...,αn (z) ,
required for proving our result. From (2) we obtain
Fα′ 1 ,...,αn (z) =
n
Y
′ αj
fj (z) .
j=1
After some calculus, we obtain
n
zfj′ (z)
zFα′′1 ,...,αn (z) X
.
=
α
j
Fα′ 1 ,...,αn (z)
fj′ (z)
j=1
86
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Let fj (z) = z −
1)an,j z n−2 we get
∞
P
n=2
an,j z n , fj′ (z) = 1 −
∞
P
n=2
nan,j z n−1 and fj′′ (z) = −
∞
P
n(n −
n=2
P
∞
n−1
n(n
−
1)a
z
n,j
n=2
zFα′′1 ,...,αn (z)
.
=
−
α
j
∞
′
P
Fα1 ,...,αn (z)
n−1
j=1
nan,j z
1−
n
X
(9)
n=2
Theorem 4. Let the functions fj ∈ T for j ∈ {1, ..., n}. Then the functions
fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n} if and only if
P
∞
α n(n − 1)(µ + 1)an,j
n
X
n=2 j
≤ 1 − β.
(10)
∞
P
j=1
1−
nan,j
n=2
Proof. First consider
′′
′′
zFα1 ,...,αn (z)
zFα1 ,...,αn (z)
zFα′′1 ,...,αn (z)
.
− Re 1 + ′
≤ (µ + 1) ′
µ ′
Fα1 ,...,αn (z)
Fα1 ,...,αn (z)
Fα1 ,...,αn (z)
From (9) we obtain
P
∞
n−1
′′
∞
n
(n
−
1)
a
z
n,j
zFα ,...,αn (z)
X n=2
≤
= (µ + 1)
α
(µ + 1) ′ 1
j
∞
P
Fα1 ,...,αn (z)
n=2
nan,j z n−1
1−
n=2
P
P
∞
∞
n−1
α
n
(n
−
1)
|a
|
|z|
α
n
(n
−
1)
a
n
n
n,j
n,j
X
X
n=2 j
n=2 j
.
≤ (µ + 1)
≤ (µ + 1)
∞
∞
P
P
n−1
j=1
j=1
1−
n |an,j | |z|
1−
nan,j
n=2
n=2
If (10) holds then the above expression is bounded by 1 − β and consequently
′′
zFα1 ,...,αn (z)
zFα′′1 ,...,αn (z)
− Re 1 +
µ ′
< −β,
Fα1 ,...,αn (z)
Fα′ 1 ,...,αn (z)
which equivalent to
zFα′′ ,...,αn (z)
Re 1 + ′ 1
Fα1 ,...,αn (z)
′′
zFα1 ,...,αn (z)
+ β.
≥ µ ′
Fα1 ,...,αn (z)
87
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Hence fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n}.
Conversely,Let fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn ) and prove that (10) holds. If fj ∈
KDFα1 ,...,αn (µ, β, α1 , ..., αn ) and z is real we get from (4) and (9)
P
P
∞
∞
n−1
n−1
X
n (n − 1) an,j z
n (n − 1) an,j z
n
X
n
n=2
n=2
αj
1−
αj
∞
∞
≥ µ
+ β ≥
P
P
n−1
n−1
j=1
1−
nan,j z
1−
nan,j z
j=1
n=2
n=2
P
∞
n−1
n (n − 1) an,j z
∞
X
n=2
+ β,
≥µ
αj
∞
P
n−1
j=1
nan,j z
1−
n=2
which is equivalent to
P
P
∞
∞
n−1
n−1
α µn (n − 1) an,j z
α n (n − 1) an,j z
n
n
X
n=2 j
n=2 j
X
+
≤ 1 − β.
∞
∞
P
P
n−1
n−1
j=1
j=1
1−
nan,j z
1−
nan,j z
n=2
n=2
The above inequality reduce to
P
∞
n−1
α n(µ + 1) (n − 1) an,j z
n
X
n=2 j
≤ 1 − β.
∞
P
n−1
j=1
nan,j z
1−
n=2
Let z → 1− along the real axis, then we get
P
∞
α
n(µ
+
1)
(n
−
1)
a
n
j
n,j
X n=2
≤ 1 − β,
∞
P
j=1
nan,j
1−
n=2
which give the required result.
Acknowledgement: The work here is supported by UKM-ST-06-FRGS0107-2009.
88
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
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operators, Acta Universitatis Apulensis, No 19/2009, pp. 209-116.
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Ineq., 2:4 (2008), 527-335.
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[5] D. Breaz, S. Owa and N. Breaz, A new integral univalent operator, Acta
Universitatis Apulensis, No 16/2008, pp. 11-16.
[6] D. Breaz, A convexity property for an integral operator on the class Sp (β),
Journal of Inequalities and Applications, vol. 2008, Article ID 143869.
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[8]S. Owa and H.M. Srivastava, Some generalized convolution properties associated with certain subclasses of analytic functions, JIPAM, 3(3)(2003), 42: 1-13.
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and convex functions, Internat. J. Math. Math. Sci., 55(2004), 2959-2961.
Aabed Mohammed and Maslina Darus
School of Mathematical Sciences,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor D. Ehsan,
Malaysia
e-mails: [email protected], [email protected]
Daniel Breaz
Department of Mathematics
” 1 Decembrie 1918 ” University
Alba Iulia, str. N. Iorga, 510009, No. 11-13
Alba, Romania
e-mail: [email protected]
89
ISSN: 1582-5329
No. 23/2010
pp. 79-89
SOME PROPERTIES FOR CERTAIN INTEGRAL OPERATORS
Aabed Mohammed, Maslina Darus and Daniel Breaz
Abstract. Recently Breaz and Breaz [4] and Breaz et.al[5] introduced two
general integral operators Fn and Fα1 ,...,αn . Considering the classes N(γ), MT(µ, β)
and KD(µ, β) we derived some properties for Fn and Fα1 ,...,αn . Two new subclasses
KDFn (µ, β, α1 , ..., αn ) and KDFα1 ,...,αn (µ, β, α1 , ..., αn ) are defined. Necessary and
sufficient conditions for a family of functions fj to be in the KDFn (µ, β, α1 , ..., αn )
and KDFα1 ,...,αn (µ, β, α1 , ..., αn ) are determined.
AMS Subject Classification: 30C45
Keywords and phrases: Analytic and univalent functions, convex functions, integral operators.
1. Introduction.
Let A denote the class of functions of the form
∞
X
f (z) = z +
an z n
n=2
which are analytic in the open unite disc U = {z ∈ C : |z| < 1}. We also denote by S
the subclass of A consisting of functions which are also univalent in U. Furthermore,
we denote by T the subclass of S consisting of functions whose nonzero coefficients,
from the second one, are negative and has the form:
f (z) = z −
∞
X
an z n , an ≥ 0.
n=2
A function f ∈ A is the convex function of order α, 0 ≤ α < 1, if f satisfies the
following inequality
′′
zf (z)
Re
+ 1 > α, z ∈ U
f ′ (z)
and we denote this class by K(α).
Similarly, if f ∈ A satisfies the following inequality:
79
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Re
zf ′ (z)
f (z)
> α, z ∈ U
for some α, 0 ≤ α < 1, then f is said to be starlike of order α and we denote this
class by S∗ (α).
Let N(γ) be the subclass of A consisting of the functions f which satisfy the inequality
′′
zf (z)
Re
+
1
< γ, z ∈ U, γ > 1.
f ′ (z)
This class was studied by Owa and Srivastava [8].
Let MT(µ, β) be the subclass of A consisting of the functions f which satisfy the
analytic characterization
′
zf (z)
zf ′ (z)
−
1
+
1
<
β
µ
f (z)
f (z)
for some 0 < β ≤ 1, and 0 ≤ µ < 1,
Definition 1.([9])A function f is said to be in the class KD(µ, β), if satisfies
the following inequality:
Re
′′
zf (z)
zf ′′ (z)
+β
+ 1 ≥ µ ′
′
f (z)
f (z)
for some µ ≥ 0 and 0 ≤ β < 1.
For fj (z) ∈ A and αj > 0 for all j ∈ {1, 2, 3, ..., n}, D. Breaz and N. Breaz [4]
introduced the following integral operator
Fn (z) =
Zz Y
n
fj (t) αj
0 j=1
t
dt,
(1)
Recently Breaz et.al [5] introduced the following integral operator
Zz Y
n
′ αj
fj (t) dt,
Fα1 ,...,αn (z) =
0 j=1
where fj ∈ A and αj > 0, for all j ∈ {1, 2, 3, ..., n}.
80
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A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
For univalence, starlike and convexity of these integral operators see ([4]-[7]), see
also ([1]-[3]) for several properties.
Now by using the equations (1) and (2) and the Definition 1, we introduce the
following two new subclasses of KD(µ, β).
Definition 2. A family of functions fj , j ∈ {1, ..., n} is said to be in the class
KDFn (µ, β, α1 , ..., αn ), if satisfies the inequality:
′′
′′
zF (z)
zFn (z)
+ β,
+ 1 ≥ µ ′n
Re
′
Fn (z)
Fn (z)
(3)
for some µ ≥ 0 and 0 ≤ β < 1, where Fn is defined in (1).
Definition 3. A family of functions fj , j ∈ {1, ..., n} is said to be in the class
KDFα1 ,...,αn (µ, β, α1 , ..., αn ) if satisfies the inequality:
′′
′′
zFα1 ,...,αn (z)
zFα1 ,...,αn (z)
+ β,
+ 1 ≥ µ ′
Re
Fα′ 1 ,...,αn (z)
Fα1 ,...,αn (z)
(4)
for some µ ≥ 0 and 0 ≤ β < 1, where Fα1 ,...,αn is defined as in (2).
2. Main results
Our first result is the following:
Theorem 1. Let αj ∈ R, αj > 0 for j ∈ {1, ..., n} and fj ∈ A and suppose that
f ′ (z)
n
P
j
αj βj (µj Mj + 1) + 1.
fj (z) < Mj .If fj ∈ M T (µj , βj ) then Fn ∈ N (σ), where σ =
j=1
Proof. From (1), we observe that Fn ∈ A. On the other hand, it is easy to see that
Fn′ (z)
=
n
Y
fj (z) αj
j=1
z
.
Differentiating (5) logarithmically and multiply by z , we obtain
" ′
#
′′
n
zfj (z)
zFn (z) X
=
−1 .
αj
fj (z)
Fn′ (z)
j=1
81
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A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Thus we have
′
n
X
zfj (z)
zFn′′ (z)
− 1 + 1.
+1=
αj
Fn′ (z)
fj (z)
j=1
We calculate the real part from both terms of the above expression and obtain
Re
X
′
n
zfj (z)
zFn′′ (z)
+1 =
− 1 + 1.
αj Re
Fn′ (z)
fj (z)
(6)
j=1
Since ℜw ≤ | w| , then
Re
′
X
n
zfj (z)
zFn′′ (z)
− 1 + 1.
+1 ≤
αj
′
Fn (z)
fj (z)
j=1
Since fj ∈ MT (µj , βj ) for j ∈ {1, ..., n}, we have
Re
′ X
X
n
n
n
X
fj (z)
zfj′ (z)
zFn′′ (z)
+
µ
+
1
≤
+
1
+
1
≤
α
β
µ
αj βj + 1
α
β
j
j
j
j
j
j
fj (z)
fj (z)
Fn′ (z)
j=1
j=1
<
n
X
αj βj µj Mj +
j=1
Hence Fn ∈ N (σ), σ =
n
X
αj βj + 1 =
j=1
n
P
n
X
j=1
αj βj (µj Mj + 1) + 1.
j=1
αj βj (µj Mj + 1) + 1.
j=1
Letting n = 1, α1 = α, α2 = ... = αn = 0, M1 = M and f1 = f, in the Theorem 1,
we have
′
(z)
Corollary 1. Let α ∈ R , α > 0, f ∈ A and suppose that ff (z)
< M, M fixed.
α
Rz
dt ∈ N(σ) , σ = αβ(µM + 1) + 1.
If f ∈ M T (µ, β) then F1 (z) = 0 f (t)
t
Letting α = 1 in Corollary 1, we have
′
(z)
Corollary 2. Let f ∈ A and suppose that ff (z)
< M, M fixed.
Rz
dt ∈ N(σ) , σ = β(µM + 1) + 1.
then F1 (z) = 0 f (t)
t
If f ∈ M T (µ, β)
Theorem 2. Let αj > 0 for j ∈ {1, ..., n}, let βj > 0 be real number with the
82
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
property 0 ≤ βj < 1 and let fj ∈ KD(µj , βj ) for j ∈ {1, ..., n}, µj ≥ 0. If
n
P
αj (1 − βj ) ≤ 1 then the functions Fα1 ,...,αn given by (2) is convex of or0 <
j=1
der ρ = 1 −
n
P
αj (1 − βj ).
j=1
Proof. From (2), we observe that Fα1 ,...,αn ∈ A and
n
X
zFα′′1 ,...,αn (z)
+
1
=
αj
Fα′ 1 ,...,αn (z)
j=1
!
n
X
fj′′ (z)
+1 −
+1.
z ′
fj (z)
j=1
We calculate the real part from both terms of the above expression and obtain
!
X
′′
n
n
X
fj′′ (z)
zFα1 ,...,αn (z)
+1 −
+1 =
αj + 1.
αj Re z ′
Re
Fα′ 1 ,...,αn (z)
fj (z)
j=1
j=1
Since fj ∈ KD(µj , βj ) for j = {1, ..., n}, we have
!
X
′′
n
n
f ′′ (z)
X
zFα1 ,...,αn (z)
j
+1 >
Re
αj µj z ′
αj + 1.
+ βj −
fj (z)
Fα′ 1 ,...,αn (z)
j=1
j=1
This relation is equivalent to
Re
X
n
n
f ′′ (z) X
zFα′′1 ,...,αn (z)
j
+
+
1
>
αj (βj − 1) + 1.
α
µ
z
j j
fj′ (z)
Fα′ 1 ,...,αn (z)
j=1
j=1
f ′′ (z)
Since αj µj z fj′ (z) > 0 we obtain
j
Re
!
′′
n
X
zFα1 ,...αn (z)
+
1
≥
1
−
αj (1 − βj ),
Fα′ 1 ,...αn (z)
j=1
which implies that Fα1 ,...,αn is convex of order ρ = 1 −
n
P
αj (1 − βj ).
j=1
Letting n = 1, α1 = α, α2 = ... = αn = 0 and f1 = f, in the Theorem 2, we have
Corollary 3. Let α be a real number, α > 0. Suppose that the function fj ∈
83
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
KD(µ, β) and o ≤ α(1 − β) < 1 . In these conditions the function Fα (z) =
Rz ′
(f (t))α dt is convex of order 1 − (1 − β)α.
0
Letting α = 1 in Corollary 3, we have
Corollary 4. Let f ∈ KD(µ, β) and consider the integral operator F1 (z) =
Rz
f ′ (t)dt.
0
In this condition F1 is convex of order β.
A necessary and sufficient condition for a family of analytic functions
fj ∈ KDFn (µ, β, α1 , ..., αn )
In this section, we give a necessary and sufficient condition for a family of functions
fj ∈ KDFn (µ, β, α1 , ..., αn ). Before embarking on the proof of our result, let us
′′ (z)
calculate the expression zF
F ′ (z) , required for proving our result.
n
Recall that, from (1), we have
Fj′ (z)
=
n
Y
fj (z) αj
z
j=1
, z ∈ U.
After some calculation, we obtain that
n
Fn′′ (z) X
=
αj
Fn′ (z)
j=1
that is equivalent to
n
zFn′′ (z) X
=
αj
Fn′ (z)
j=1
Let fj (z) = z −
∞
P
fj′ (z) 1
−
fj (z) z
,
zfj′ (z)
−1 .
fj (z)
∞
P
an,j z n . Then fj′ (z) = 1 −
nan,j z n−1 and we get
n=2
n=2
∞
P
n
z
−
na
z
n
n,j
zFn′′ (z) X
n=2
=
α
− 1
j
∞
=
′
P
Fn (z)
j=1
z−
an,j z n
n=2
=
n
X
j=1
αj
1−
∞
P
nan,j z n−1 − 1 +
n=2
∞
P
an,j z n−1
n=2
1−
∞
P
an,j z n−1
n=2
P
∞
n−1
(n
−
1)
a
z
n
n,j
X
n=2
=−
.
α
j
∞
P
j=1
1−
an,j z n−1
n=2
(7)
84
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Theorem 3. Let the function fj ∈ T for j ∈ {1, ..., n}. Then the functions fj ∈
KDFn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n} if and only if
P
∞
α
(n
−
1)
(µ
+
1)
a
n
n,j
X
n=2 j
≤ 1 − β.
(8)
∞
P
j=1
1−
an,j
n=2
Proof. First consider
′′
′′
′′ (z)
zF (z)
zFn (z)
zF
n
− Re 1 +
.
≤ (µ + 1) ′n
µ ′
′
Fn (z)
Fn (z)
Fn (z)
From (7) we obtain
P
∞
n−1
′′
∞
(n
−
1)
a
z
n,j
zF (z)
X n=2
≤
= (µ + 1)
(µ + 1) ′n
α
j
∞
P
Fn (z)
n=2
n−1
1−
an,j z
n=2
P
P
∞
∞
n−1
α (n − 1) an,j
α (n − 1) |an,j | |z|
n
n
X
X
n=2 j
n=2 j
.
≤ (µ + 1)
≤ (µ + 1)
∞
∞
P
P
n−1
j=1
j=1
1−
|an,j | |z|
1−
an,j
n=2
n=2
If (8) holds then the above expression is bounded by 1 − β, and consequently
′′
zFn (z)
zFn′′ (z)
µ ′
< −β,
− Re 1 + ′
Fn (z)
Fn (z)
which equivalent to
zF ′′ (z)
Re 1 + ′n
Fn (z)
′′
zF (z)
+ β.
≥ µ ′n
Fn (z)
Hence fj ∈ KDFn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n}.
Conversely. Let fj ∈ KDFn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n} and prove that (8)
holds. If fj ∈ KDFn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n} and z is real, we get from (3)
and (7)
P
P
∞
∞
n−1
n−1
∞
(n − 1) an,j z
(n − 1) an,j z
n
X
n=2
X n=2
1−
αj
αj
∞
∞
≥ µ
+ β ≥
P
P
n−1
n−1
j=1
1−
1−
an,j z
an,j z
j=1
n=2
n=2
85
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
P
∞
n−1
(n − 1) an,j z
n
X
n=2
+ β.
µ
αj
∞
P
n−1
j=1
1−
an,j z
n=2
That is equivalent to
P
P
∞
∞
n−1
n−1
α (n − 1) an,j z
α µ (n − 1) an,j z
n
n
X
X
n=2 j
n=2 j
+
≤ 1 − β.
∞
∞
P
P
n−1
n−1
j=1
j=1
1−
an,j z
1−
an,j z
n=2
n=2
The above inequality reduce to
P
∞
n−1
α
(µ
+
1)
(n
−
1)
a
z
n
n,j
X
n=2 j
≤ 1 − β.
∞
P
n−1
j=1
1−
an,j z
n=2
Let z → 1− along the real axis, then we get
P
∞
α (µ + 1) (n − 1) an,j
n
X
n=2 j
≤ 1 − β.
∞
P
j=1
1−
an,j
n=2
Which give the required result.
A necessary and sufficient condition for a family of analytic functions
fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn )
In this section, we give a necessary and sufficient condition for a family of analytic
functions fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn ). Let us first calculate the expression,
zFα′′1 ,...,αn (z)
Fα′ 1 ,...,αn (z) ,
required for proving our result. From (2) we obtain
Fα′ 1 ,...,αn (z) =
n
Y
′ αj
fj (z) .
j=1
After some calculus, we obtain
n
zfj′ (z)
zFα′′1 ,...,αn (z) X
.
=
α
j
Fα′ 1 ,...,αn (z)
fj′ (z)
j=1
86
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Let fj (z) = z −
1)an,j z n−2 we get
∞
P
n=2
an,j z n , fj′ (z) = 1 −
∞
P
n=2
nan,j z n−1 and fj′′ (z) = −
∞
P
n(n −
n=2
P
∞
n−1
n(n
−
1)a
z
n,j
n=2
zFα′′1 ,...,αn (z)
.
=
−
α
j
∞
′
P
Fα1 ,...,αn (z)
n−1
j=1
nan,j z
1−
n
X
(9)
n=2
Theorem 4. Let the functions fj ∈ T for j ∈ {1, ..., n}. Then the functions
fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n} if and only if
P
∞
α n(n − 1)(µ + 1)an,j
n
X
n=2 j
≤ 1 − β.
(10)
∞
P
j=1
1−
nan,j
n=2
Proof. First consider
′′
′′
zFα1 ,...,αn (z)
zFα1 ,...,αn (z)
zFα′′1 ,...,αn (z)
.
− Re 1 + ′
≤ (µ + 1) ′
µ ′
Fα1 ,...,αn (z)
Fα1 ,...,αn (z)
Fα1 ,...,αn (z)
From (9) we obtain
P
∞
n−1
′′
∞
n
(n
−
1)
a
z
n,j
zFα ,...,αn (z)
X n=2
≤
= (µ + 1)
α
(µ + 1) ′ 1
j
∞
P
Fα1 ,...,αn (z)
n=2
nan,j z n−1
1−
n=2
P
P
∞
∞
n−1
α
n
(n
−
1)
|a
|
|z|
α
n
(n
−
1)
a
n
n
n,j
n,j
X
X
n=2 j
n=2 j
.
≤ (µ + 1)
≤ (µ + 1)
∞
∞
P
P
n−1
j=1
j=1
1−
n |an,j | |z|
1−
nan,j
n=2
n=2
If (10) holds then the above expression is bounded by 1 − β and consequently
′′
zFα1 ,...,αn (z)
zFα′′1 ,...,αn (z)
− Re 1 +
µ ′
< −β,
Fα1 ,...,αn (z)
Fα′ 1 ,...,αn (z)
which equivalent to
zFα′′ ,...,αn (z)
Re 1 + ′ 1
Fα1 ,...,αn (z)
′′
zFα1 ,...,αn (z)
+ β.
≥ µ ′
Fα1 ,...,αn (z)
87
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
Hence fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn ) for j ∈ {1, ..., n}.
Conversely,Let fj ∈ KDFα1 ,...,αn (µ, β, α1 , ..., αn ) and prove that (10) holds. If fj ∈
KDFα1 ,...,αn (µ, β, α1 , ..., αn ) and z is real we get from (4) and (9)
P
P
∞
∞
n−1
n−1
X
n (n − 1) an,j z
n (n − 1) an,j z
n
X
n
n=2
n=2
αj
1−
αj
∞
∞
≥ µ
+ β ≥
P
P
n−1
n−1
j=1
1−
nan,j z
1−
nan,j z
j=1
n=2
n=2
P
∞
n−1
n (n − 1) an,j z
∞
X
n=2
+ β,
≥µ
αj
∞
P
n−1
j=1
nan,j z
1−
n=2
which is equivalent to
P
P
∞
∞
n−1
n−1
α µn (n − 1) an,j z
α n (n − 1) an,j z
n
n
X
n=2 j
n=2 j
X
+
≤ 1 − β.
∞
∞
P
P
n−1
n−1
j=1
j=1
1−
nan,j z
1−
nan,j z
n=2
n=2
The above inequality reduce to
P
∞
n−1
α n(µ + 1) (n − 1) an,j z
n
X
n=2 j
≤ 1 − β.
∞
P
n−1
j=1
nan,j z
1−
n=2
Let z → 1− along the real axis, then we get
P
∞
α
n(µ
+
1)
(n
−
1)
a
n
j
n,j
X n=2
≤ 1 − β,
∞
P
j=1
nan,j
1−
n=2
which give the required result.
Acknowledgement: The work here is supported by UKM-ST-06-FRGS0107-2009.
88
A. Mohammed, M. Darus, D. Breaz - Some properties for certain integral operators
References
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operators, Acta Universitatis Apulensis, No 19/2009, pp. 209-116.
[3] B.A. Frasin, Some sufficient conditions for certain integral operators, J. Math.
Ineq., 2:4 (2008), 527-335.
[4] D. Breaz and N. Breaz, Two integral operators, Studia Universitatis BabesBolyai, Mathematica, 47:3(2002), 13-19.
[5] D. Breaz, S. Owa and N. Breaz, A new integral univalent operator, Acta
Universitatis Apulensis, No 16/2008, pp. 11-16.
[6] D. Breaz, A convexity property for an integral operator on the class Sp (β),
Journal of Inequalities and Applications, vol. 2008, Article ID 143869.
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[8]S. Owa and H.M. Srivastava, Some generalized convolution properties associated with certain subclasses of analytic functions, JIPAM, 3(3)(2003), 42: 1-13.
[9]S. Shams, S. R. Kulkarni and J. M. Jahangiri, Classes of uniformly starlike
and convex functions, Internat. J. Math. Math. Sci., 55(2004), 2959-2961.
Aabed Mohammed and Maslina Darus
School of Mathematical Sciences,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor D. Ehsan,
Malaysia
e-mails: [email protected], [email protected]
Daniel Breaz
Department of Mathematics
” 1 Decembrie 1918 ” University
Alba Iulia, str. N. Iorga, 510009, No. 11-13
Alba, Romania
e-mail: [email protected]
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