Directory UMM :Journals:Journal_of_mathematics:AUA:
ACTA UNIVERSITATIS APULENSIS
No 19/2009
ON CLOSE-TO-CONVEX FOR CERTAIN INTEGRAL OPERATORS
Aabed Mohammed, Maslina Darus and Daniel Breaz
Abstract. In this paper, we consider some sufficient conditions for two integral
operators to be close-to-convex function defined in the open unit disk.
Keywords: Analytic functions, closed-to-convex, close-to-star, integral operators.
2000 Mathematics Subject Classification: 30C45
1. Introduction
Let U = {z ∈ C : |z| < 1} be the open unit disk and A denotes the class of
functions normalized by
f (z) = z +
∞
X
ak z k
(1)
k=2
which are analytic in the open unit disk U and satisfy the condition f (0) = f ′ (0)−1 =
0. We also denote by S the subclass of A consisting of functions which are also
univalent in U.
A function f ∈ A is the convex function of order ρ, 0 ≤ ρ < 1, if f satisfies the
following inequality
′′
zf (z)
Re
+ 1 > ρ, z ∈ U
f ′ (z)
and we denote this class by K(ρ).
Similarly, if f ∈ A satisfies the following inequality
′
zf (z)
> ρ, z ∈ U
Re
f (z)
for some ρ, 0 ≤ ρ < 1, then f is said to be starlike of order ρ and we denote this
class by S ∗ (ρ). We note that f ∈ K ⇔ zf ′ (z) ∈ S ∗ , z ∈ U. In particular case, the
209
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
classes K(0) = K and S ∗ = S ∗ are familiar classes of starlike and convex functions
in U.
A function f ∈ A is called close-to-convex if there exist a convex function g such
that
Re
f ′ (z)
> 0, z ∈ U.
g ′ (z)
(2)
Since f ∈ K ⇔ zf ′ (z) ∈ S ∗ , z ∈ U, we can replace (2) by the requirement that
Re
zf ′ (z)
> 0, z ∈ U,
h(z)
where h is a starlike function on U. Furthermore, f is closed-to-convex if and only
if
Zθ2
zf ′′ (z)
Re 1 + ′
f (z)
θ1
dθ > −π,
where 0 ≤ θ1 < θ2 ≤ 2π, z = reiθ and r < 1.
Let C denote the set of normalized close-to-convex functions on U it is clear that
K ⊂ S ∗ ⊂ C ⊂ S.
A function f ∈ A is called close-to-star in U if there exist a starlike g such that
Re
f (z)
> 0, z ∈ U.
g(z)
Also f is close-to-star in U if and only if
Zθ2
Re
zf ′ (z)
dθ > −π,
f (z)
θ1
where 0 ≤ θ1 < θ2 ≤ 2π, z = reiθ and r < 1. Let CS ∗ denote the class of close-tostar functions in U, it is known that the close-to-star functions are not necessarily
univalent in U.
Shukla and Kumar [5] introduced the following subclasses of C and CS ∗ .
A function f ∈ S belongs to the class C(β, ρ) of close-to-convex functions of order
β and type ρ if for some g ∈ S ∗ (ρ),
210
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
′
arg zf (z) < βπ , z ∈ U,
g(z)
2
where β ∈ [0, 1].
A function f ∈ S belongs to the class CS ∗ (β, ρ) of close-to-star functions of order
β and type ρ if for some g ∈ S ∗ (ρ),
arg f (z) < βπ , z ∈ U,
g(z)
2
where β ∈ [0, 1].
It is clear that C(0, ρ) ≡ K(ρ) and CS ∗ (0, ρ) ≡ S ∗ (ρ).
Also C(β, ρ) ⊂ C(1, 0) ≡ C and CS ∗ (β, ρ) ⊂ CS ∗ (1, 0) ≡ CS ∗ .
Now, we consider the following integral operators
Fn (z) =
Zz
f1 (t)
t
0
α 1
· ... ·
fn (t)
t
α n
dt,
(3)
and
Fα1 ,...,αn (z) =
Zz
0
α
α
f1′ (t) 1 · ... · fn′ (t) n dt,
(4)
where fj ∈ A and αj > 0, for all j ∈ {1, 2, ..., n}.
These operators are introduced by D.Breaz and N.Breaz [1] and studied by many
authors (see [2], [3], [4]).
In the present paper, we obtain some sufficient conditions for the above integral
operators to be in the class of close-to-convex function C.
Before embarking on the proof of our results, we need the following Lemmas
introduced by Shukla and Kumar [5].
Lemma 1. If f ∈ S ∗ (ρ), then
ρ(θ2 − θ1 ) ≤
Zθ2
Re
zf ′ (z)
dθ ≤ 2π(1 − ρ) + ρ(θ2 − θ1 )
f (z)
θ1
where z = reiθ and 0 ≤ θ1 ≤ θ2 ≤ 2π.
Lemma 2. If f ∈ C(β, ρ) then
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A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
−βπ + ρ(θ2 − θ1 ) ≤
Zθ2
zf ′′ (z)
Re 1 + ′
f (z)
θ1
dθ ≤ βπ + 2π(1 − ρ) + ρ(θ2 − θ1 )
where z = reiθ and 0 ≤ θ1 ≤ θ2 ≤ 2π.
Lemma 3. If f ∈ CS ∗ (β, ρ) then
−βπ + ρ(θ2 − θ1 ) ≤
Zθ2
Re
zf ′ (z)
dθ ≤ βπ + 2π(1 − ρ) + ρ(θ2 − θ1 )
f (z)
θ1
where z = reiθ and 0 ≤ θ1 ≤ θ2 ≤ 2π.
2.Main results
Theorem 1. Let fi ∈ S ∗ (ρi ), for i ∈ {1, 2, ..., n}. If
n
P
αi ≤ 1, then Fn (z) ∈ C,
i=1
z ∈ U, where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. It is clear that Fn′ (z) 6= 0 for z ∈ U. We calculate for Fn the derivatives
of the first and second order. Since
Fn (z) =
Zz
f1 (t)
t
0
α 1
· ... ·
fn (t)
t
α n
dt,
then
Fn′ (z)
=
f1 (z)
z
α 1
...
fn (z)
z
α n
.
Differentiating the above expression logarithmically, we have
′
n
fi (z) 1
Fn′′ (z) X
−
.
=
αi
Fn′ (z)
fi (z) z
i=1
By multiplying the above expression with z we obtain
′
n
zFn′′ (z) X
zfi (z)
=
−1 .
αi
Fn′ (z)
fi (z)
i=1
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A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
That is equivalent to
n
n
i=1
i=1
X
zF ′′ (z) X zfi′ (z)
1 + ′n
=
+1−
αi
αi .
Fn (z)
fi (z)
(5)
Taking real parts in (5) and integrating with respect to θ we get
Zθ2
θ1
zF ′′ (z)
Re 1 + ′n
Fn (z)
dθ =
Zθ2 X
n
αi Re
θ1 i=1
zfi′ (z)
fi (z)
1−
dθ +
n
X
αi
i=1
!
(θ2 − θ1 ) .
Since fi ∈ S ∗ (ρi ) then by applying Lemma 1, we obtain
Zθ2
θ1
Since
n
P
zF ′′ (z)
Re 1 + ′n
Fn (z)
αi ρi −
i=1
n
P
i=1
dθ ≥
n
X
αi ρi −
i=1
n
X
!
αi + 1 (θ2 − θ1 ) .
i=1
αi + 1 > 0 so, minimum is for θ2 = θ1 we obtain that
Zθ2
θ1
zF ′′ (z)
Re 1 + ′n
Fn (z)
dθ > −π,
then Fn (z) ∈ C.
Corollary 2. Let fi ∈ S ∗ (ρ), for i ∈ {1, 2, ..., n}. If
n
P
αi ≤ 1, then Fn (z) ∈ C,
i=1
z ∈ U, where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. We consider in Theorem 1, ρ1 = ρ2 = ... = ρn .
Theorem 3. Let fi ∈ CS ∗ , i = {1, 2, ..., n}. If
n
P
αi ≤ 1, then Fn (z) ∈ C,
i=1
z ∈ U, where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. Following the same steps as in Theorem 1, we obtain that
Zθ2
θ1
zF ′′ (z)
Re 1 + ′n
Fn (z)
dθ =
Zθ2 X
n
θ1
i=1
αi Re
zfi′ (z)
fi (z)
Since fi ∈ CS ∗ , then
213
dθ +
1−
n
X
i=1
αi
!
(θ2 − θ1 ) .
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
Zθ2
θ1
Since 1 −
zF ′′ (z)
Re 1 + ′n
Fn (z)
n
P
dθ > −π
n
X
αi +
1−
i=1
n
X
i=1
αi
!
(θ2 − θ1 ) .
αi > 0 so, minimum is for θ1 = θ2 we obtain that
i=1
Zθ2
θ1
zF ′′ (z)
Re 1 + ′n
Fn (z)
dθ > −π
Then Fn ∈ C.
Theorem 4. Let fi ∈ C (βi , ρi ), i = {1, 2, ..., n}. If
n
P
i=1
αi βi ≤ 1, then Fα1 ,α2 ,...,αn (z) ∈
C, z ∈ U, where Fα1 ,α2 ,...,αn is defined as in (4), and C is the class of close to convex
functions.
Proof. It is clear that Fα′ 1 ,...,αn (z) 6= 0 for z ∈ U. Since
Fα1 ,...,αn (z) =
Zz
0
α
α
f1′ (t) 1 · ... · fn′ (t) n dt.
Following the same steps as in Theorem 1, we obtain that
n
zFα′′1 ,...,αn (z) X
zfi′′ (z)
.
=
α
i
Fα′ 1 ,...,αn (z)
fi′ (z)
i=1
That is equivalent to
n
zFα′′ ,...,αn (z) X
1+ ′ 1
=
αi
Fα1 ,...,αn (z)
i=1
n
X
zfi′′ (z)
+
1
+
1
−
αi .
fi′ (z)
(6)
i=1
Taking real parts in (6) and integrating with respect to θ we get
Zθ2
θ1
zFα′′ ,...,αn (z)
Re 1 + ′ 1
Fα1 ,...,αn (z)
dθ =
Zθ2 X
n
θ1
i=1
αi Re
!
n
X
zfi′′ (z)
+ 1 dθ+ 1 −
αi (θ2 − θ1 ) .
fi′ (z)
Since fi ∈ C (βi , ρi ) then by applying Lemma 2, we obtain
214
i=1
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
Zθ2
θ1
zFα′′ ,...,αn (z)
Re 1 + ′ 1
Fα1 ,...,αn (z)
n
X
=
dθ ≥
n
P
i=1
αi ρi −
n
P
αi [−βi π + ρi (θ2 − θ1 )]+ 1 −
i=1
αi ρi −
i=1
Since
n
X
n
X
i=1
n
X
i=1
n
X
!
αi + 1 (θ2 − θ1 ) − π
αi
!
(θ2 − θ1 ) =
αi βi .
i=1
αi + 1 > 0 so, minimum is for θ1 = θ2 we obtain that
i=1
Zθ2
θ1
zFα′′ ,...,αn (z)
Re 1 + ′ 1
Fα1 ,...,αn (z)
dθ > −π
then Fα1 ,...,αn (z) ∈ C.
Corollary 5. Let fi ∈ C (β, ρ), i = {1, 2, ..., n}. If
n
P
i=1
αi ≤ 1, then Fα1 ,α2 ,...,αn (z) ∈
C, z ∈ U, where Fα1 ,α2 ,...,αn is defined as in (4), and C is the class of close to convex
functions.
Proof. We considet in Theorem 4 β1 = ... = βn and ρ1 = ... = ρn .
Theorem 6. If fi ∈ CS ∗ (βi , ρi ), i = {1, 2, ..., n} and
n
P
αi βi ≤ 1, then Fn ∈ C,
i=1
where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. Since the proof is similar to the proof of theorems in (1), (3) and (4), it
will be omitted.
Corollary 7. If fi ∈ CS ∗ (β, ρ), i = {1, 2, ..., n} and
n
P
αi ≤ 1, then Fn ∈ C,
i=1
where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. We consider in Theorem 6 β1 = ... = βn and ρ1 = ... = ρn .
Acknowledgement: The work here is fully supported by UKM-GUP-TMK07-02-107, UKM.
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A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
References
[1] D. Breaz and N. Breaz, Two integral operators, Studia Universitatis BabesBolyai, Mathematica, 47:3(2002), 13-19.
[2] D. Breaz, S. Owa and N. Breaz, A new integral univalent operator, Acta
Universitatis Apulensis, No 16/2008, pp. 11-16.
[3] D. Breaz, A convexity property for an integral operator on the class Sp (β),
Journal of Inequalities and Applications, vol. 2008, Article ID 143869.
[4] D. Breaz, Certain Integral Operators On the Classes M(βi ) and N (βi ), Journal of Inequalities and Applications, vol. 2008, Article ID 719354.
[5] S. L. Shukla and V. Kumar, On The products of close-to-starlike and closeto-convex functions, Indian J. Pure appl. Math. 16(3): (1985), 279-290.
Aabed Mohammed
School of Mathematical Sceinces,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor D. Ehsan,
Malaysia
e-mail:[email protected]
Maslina Darus
School of Mathematical Sceinces,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor D. Ehsan,
Malaysia
e-mail:[email protected]
Daniel Breaz
“1 Decembrie 1918” University of Alba Iulia,
Faculty of Science,
Department of Mathematics-Informatics,
510009 Alba Iulia,
Romania
e-mail:[email protected]
216
No 19/2009
ON CLOSE-TO-CONVEX FOR CERTAIN INTEGRAL OPERATORS
Aabed Mohammed, Maslina Darus and Daniel Breaz
Abstract. In this paper, we consider some sufficient conditions for two integral
operators to be close-to-convex function defined in the open unit disk.
Keywords: Analytic functions, closed-to-convex, close-to-star, integral operators.
2000 Mathematics Subject Classification: 30C45
1. Introduction
Let U = {z ∈ C : |z| < 1} be the open unit disk and A denotes the class of
functions normalized by
f (z) = z +
∞
X
ak z k
(1)
k=2
which are analytic in the open unit disk U and satisfy the condition f (0) = f ′ (0)−1 =
0. We also denote by S the subclass of A consisting of functions which are also
univalent in U.
A function f ∈ A is the convex function of order ρ, 0 ≤ ρ < 1, if f satisfies the
following inequality
′′
zf (z)
Re
+ 1 > ρ, z ∈ U
f ′ (z)
and we denote this class by K(ρ).
Similarly, if f ∈ A satisfies the following inequality
′
zf (z)
> ρ, z ∈ U
Re
f (z)
for some ρ, 0 ≤ ρ < 1, then f is said to be starlike of order ρ and we denote this
class by S ∗ (ρ). We note that f ∈ K ⇔ zf ′ (z) ∈ S ∗ , z ∈ U. In particular case, the
209
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
classes K(0) = K and S ∗ = S ∗ are familiar classes of starlike and convex functions
in U.
A function f ∈ A is called close-to-convex if there exist a convex function g such
that
Re
f ′ (z)
> 0, z ∈ U.
g ′ (z)
(2)
Since f ∈ K ⇔ zf ′ (z) ∈ S ∗ , z ∈ U, we can replace (2) by the requirement that
Re
zf ′ (z)
> 0, z ∈ U,
h(z)
where h is a starlike function on U. Furthermore, f is closed-to-convex if and only
if
Zθ2
zf ′′ (z)
Re 1 + ′
f (z)
θ1
dθ > −π,
where 0 ≤ θ1 < θ2 ≤ 2π, z = reiθ and r < 1.
Let C denote the set of normalized close-to-convex functions on U it is clear that
K ⊂ S ∗ ⊂ C ⊂ S.
A function f ∈ A is called close-to-star in U if there exist a starlike g such that
Re
f (z)
> 0, z ∈ U.
g(z)
Also f is close-to-star in U if and only if
Zθ2
Re
zf ′ (z)
dθ > −π,
f (z)
θ1
where 0 ≤ θ1 < θ2 ≤ 2π, z = reiθ and r < 1. Let CS ∗ denote the class of close-tostar functions in U, it is known that the close-to-star functions are not necessarily
univalent in U.
Shukla and Kumar [5] introduced the following subclasses of C and CS ∗ .
A function f ∈ S belongs to the class C(β, ρ) of close-to-convex functions of order
β and type ρ if for some g ∈ S ∗ (ρ),
210
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
′
arg zf (z) < βπ , z ∈ U,
g(z)
2
where β ∈ [0, 1].
A function f ∈ S belongs to the class CS ∗ (β, ρ) of close-to-star functions of order
β and type ρ if for some g ∈ S ∗ (ρ),
arg f (z) < βπ , z ∈ U,
g(z)
2
where β ∈ [0, 1].
It is clear that C(0, ρ) ≡ K(ρ) and CS ∗ (0, ρ) ≡ S ∗ (ρ).
Also C(β, ρ) ⊂ C(1, 0) ≡ C and CS ∗ (β, ρ) ⊂ CS ∗ (1, 0) ≡ CS ∗ .
Now, we consider the following integral operators
Fn (z) =
Zz
f1 (t)
t
0
α 1
· ... ·
fn (t)
t
α n
dt,
(3)
and
Fα1 ,...,αn (z) =
Zz
0
α
α
f1′ (t) 1 · ... · fn′ (t) n dt,
(4)
where fj ∈ A and αj > 0, for all j ∈ {1, 2, ..., n}.
These operators are introduced by D.Breaz and N.Breaz [1] and studied by many
authors (see [2], [3], [4]).
In the present paper, we obtain some sufficient conditions for the above integral
operators to be in the class of close-to-convex function C.
Before embarking on the proof of our results, we need the following Lemmas
introduced by Shukla and Kumar [5].
Lemma 1. If f ∈ S ∗ (ρ), then
ρ(θ2 − θ1 ) ≤
Zθ2
Re
zf ′ (z)
dθ ≤ 2π(1 − ρ) + ρ(θ2 − θ1 )
f (z)
θ1
where z = reiθ and 0 ≤ θ1 ≤ θ2 ≤ 2π.
Lemma 2. If f ∈ C(β, ρ) then
211
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
−βπ + ρ(θ2 − θ1 ) ≤
Zθ2
zf ′′ (z)
Re 1 + ′
f (z)
θ1
dθ ≤ βπ + 2π(1 − ρ) + ρ(θ2 − θ1 )
where z = reiθ and 0 ≤ θ1 ≤ θ2 ≤ 2π.
Lemma 3. If f ∈ CS ∗ (β, ρ) then
−βπ + ρ(θ2 − θ1 ) ≤
Zθ2
Re
zf ′ (z)
dθ ≤ βπ + 2π(1 − ρ) + ρ(θ2 − θ1 )
f (z)
θ1
where z = reiθ and 0 ≤ θ1 ≤ θ2 ≤ 2π.
2.Main results
Theorem 1. Let fi ∈ S ∗ (ρi ), for i ∈ {1, 2, ..., n}. If
n
P
αi ≤ 1, then Fn (z) ∈ C,
i=1
z ∈ U, where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. It is clear that Fn′ (z) 6= 0 for z ∈ U. We calculate for Fn the derivatives
of the first and second order. Since
Fn (z) =
Zz
f1 (t)
t
0
α 1
· ... ·
fn (t)
t
α n
dt,
then
Fn′ (z)
=
f1 (z)
z
α 1
...
fn (z)
z
α n
.
Differentiating the above expression logarithmically, we have
′
n
fi (z) 1
Fn′′ (z) X
−
.
=
αi
Fn′ (z)
fi (z) z
i=1
By multiplying the above expression with z we obtain
′
n
zFn′′ (z) X
zfi (z)
=
−1 .
αi
Fn′ (z)
fi (z)
i=1
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A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
That is equivalent to
n
n
i=1
i=1
X
zF ′′ (z) X zfi′ (z)
1 + ′n
=
+1−
αi
αi .
Fn (z)
fi (z)
(5)
Taking real parts in (5) and integrating with respect to θ we get
Zθ2
θ1
zF ′′ (z)
Re 1 + ′n
Fn (z)
dθ =
Zθ2 X
n
αi Re
θ1 i=1
zfi′ (z)
fi (z)
1−
dθ +
n
X
αi
i=1
!
(θ2 − θ1 ) .
Since fi ∈ S ∗ (ρi ) then by applying Lemma 1, we obtain
Zθ2
θ1
Since
n
P
zF ′′ (z)
Re 1 + ′n
Fn (z)
αi ρi −
i=1
n
P
i=1
dθ ≥
n
X
αi ρi −
i=1
n
X
!
αi + 1 (θ2 − θ1 ) .
i=1
αi + 1 > 0 so, minimum is for θ2 = θ1 we obtain that
Zθ2
θ1
zF ′′ (z)
Re 1 + ′n
Fn (z)
dθ > −π,
then Fn (z) ∈ C.
Corollary 2. Let fi ∈ S ∗ (ρ), for i ∈ {1, 2, ..., n}. If
n
P
αi ≤ 1, then Fn (z) ∈ C,
i=1
z ∈ U, where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. We consider in Theorem 1, ρ1 = ρ2 = ... = ρn .
Theorem 3. Let fi ∈ CS ∗ , i = {1, 2, ..., n}. If
n
P
αi ≤ 1, then Fn (z) ∈ C,
i=1
z ∈ U, where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. Following the same steps as in Theorem 1, we obtain that
Zθ2
θ1
zF ′′ (z)
Re 1 + ′n
Fn (z)
dθ =
Zθ2 X
n
θ1
i=1
αi Re
zfi′ (z)
fi (z)
Since fi ∈ CS ∗ , then
213
dθ +
1−
n
X
i=1
αi
!
(θ2 − θ1 ) .
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
Zθ2
θ1
Since 1 −
zF ′′ (z)
Re 1 + ′n
Fn (z)
n
P
dθ > −π
n
X
αi +
1−
i=1
n
X
i=1
αi
!
(θ2 − θ1 ) .
αi > 0 so, minimum is for θ1 = θ2 we obtain that
i=1
Zθ2
θ1
zF ′′ (z)
Re 1 + ′n
Fn (z)
dθ > −π
Then Fn ∈ C.
Theorem 4. Let fi ∈ C (βi , ρi ), i = {1, 2, ..., n}. If
n
P
i=1
αi βi ≤ 1, then Fα1 ,α2 ,...,αn (z) ∈
C, z ∈ U, where Fα1 ,α2 ,...,αn is defined as in (4), and C is the class of close to convex
functions.
Proof. It is clear that Fα′ 1 ,...,αn (z) 6= 0 for z ∈ U. Since
Fα1 ,...,αn (z) =
Zz
0
α
α
f1′ (t) 1 · ... · fn′ (t) n dt.
Following the same steps as in Theorem 1, we obtain that
n
zFα′′1 ,...,αn (z) X
zfi′′ (z)
.
=
α
i
Fα′ 1 ,...,αn (z)
fi′ (z)
i=1
That is equivalent to
n
zFα′′ ,...,αn (z) X
1+ ′ 1
=
αi
Fα1 ,...,αn (z)
i=1
n
X
zfi′′ (z)
+
1
+
1
−
αi .
fi′ (z)
(6)
i=1
Taking real parts in (6) and integrating with respect to θ we get
Zθ2
θ1
zFα′′ ,...,αn (z)
Re 1 + ′ 1
Fα1 ,...,αn (z)
dθ =
Zθ2 X
n
θ1
i=1
αi Re
!
n
X
zfi′′ (z)
+ 1 dθ+ 1 −
αi (θ2 − θ1 ) .
fi′ (z)
Since fi ∈ C (βi , ρi ) then by applying Lemma 2, we obtain
214
i=1
A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
Zθ2
θ1
zFα′′ ,...,αn (z)
Re 1 + ′ 1
Fα1 ,...,αn (z)
n
X
=
dθ ≥
n
P
i=1
αi ρi −
n
P
αi [−βi π + ρi (θ2 − θ1 )]+ 1 −
i=1
αi ρi −
i=1
Since
n
X
n
X
i=1
n
X
i=1
n
X
!
αi + 1 (θ2 − θ1 ) − π
αi
!
(θ2 − θ1 ) =
αi βi .
i=1
αi + 1 > 0 so, minimum is for θ1 = θ2 we obtain that
i=1
Zθ2
θ1
zFα′′ ,...,αn (z)
Re 1 + ′ 1
Fα1 ,...,αn (z)
dθ > −π
then Fα1 ,...,αn (z) ∈ C.
Corollary 5. Let fi ∈ C (β, ρ), i = {1, 2, ..., n}. If
n
P
i=1
αi ≤ 1, then Fα1 ,α2 ,...,αn (z) ∈
C, z ∈ U, where Fα1 ,α2 ,...,αn is defined as in (4), and C is the class of close to convex
functions.
Proof. We considet in Theorem 4 β1 = ... = βn and ρ1 = ... = ρn .
Theorem 6. If fi ∈ CS ∗ (βi , ρi ), i = {1, 2, ..., n} and
n
P
αi βi ≤ 1, then Fn ∈ C,
i=1
where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. Since the proof is similar to the proof of theorems in (1), (3) and (4), it
will be omitted.
Corollary 7. If fi ∈ CS ∗ (β, ρ), i = {1, 2, ..., n} and
n
P
αi ≤ 1, then Fn ∈ C,
i=1
where Fn is defined as in (3), and C is the class of close to convex functions.
Proof. We consider in Theorem 6 β1 = ... = βn and ρ1 = ... = ρn .
Acknowledgement: The work here is fully supported by UKM-GUP-TMK07-02-107, UKM.
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A. Mohammed, M. Darus, D. Breaz - On close-to-convex for certain integral...
References
[1] D. Breaz and N. Breaz, Two integral operators, Studia Universitatis BabesBolyai, Mathematica, 47:3(2002), 13-19.
[2] D. Breaz, S. Owa and N. Breaz, A new integral univalent operator, Acta
Universitatis Apulensis, No 16/2008, pp. 11-16.
[3] D. Breaz, A convexity property for an integral operator on the class Sp (β),
Journal of Inequalities and Applications, vol. 2008, Article ID 143869.
[4] D. Breaz, Certain Integral Operators On the Classes M(βi ) and N (βi ), Journal of Inequalities and Applications, vol. 2008, Article ID 719354.
[5] S. L. Shukla and V. Kumar, On The products of close-to-starlike and closeto-convex functions, Indian J. Pure appl. Math. 16(3): (1985), 279-290.
Aabed Mohammed
School of Mathematical Sceinces,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor D. Ehsan,
Malaysia
e-mail:[email protected]
Maslina Darus
School of Mathematical Sceinces,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor D. Ehsan,
Malaysia
e-mail:[email protected]
Daniel Breaz
“1 Decembrie 1918” University of Alba Iulia,
Faculty of Science,
Department of Mathematics-Informatics,
510009 Alba Iulia,
Romania
e-mail:[email protected]
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