Paper-14-Acta23.2010. 114KB Jun 04 2011 12:03:10 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 23/2010
pp. 139-145
CONVOLUTION PROPERTIES OF HARMONIC UNIVALENT
FUNCTIONS PRESERVED BY SOME INTEGRAL OPERATOR
F. M. Al-Oboudi
Abstract. A complex valued function f = u + iv defined in a domain D ⊂ C,
is harmonic in D, if u and v are real harmonic. Such functions can be represented
as f (z) = h(z) + g(z), where h an g are analytic in D. In this paper we study
n f, n ∈ N =
some convolution properties preserved by the integral operator IH,λ
0
N ∪ {0}, λ > 0, where the functions f are univalent harmonic and sense-preserving
n
in the open unit disc E = {z : |z| < 1}, IH,λ
f (z) = Iλn h(z) + Iλn g(z), and Ihn h(z) =
∞
∞
X
X
ak
bk
k n
z
,
I
g(z)
=
z k . Relevant connections of
z+
λ
n
[1 + λ(k − 1)]
[1 + λ(k − 1)]n
k=2
k=1
the results presented here with those obtained in earlier works are pointed out.
2000 Mathematics Subject Classification: 30C45.
1. Introduction
Let A denote the class of analytic functions in the open unit disc E = {z : |z| <
1}, with the normalization
∞
X
f (z) = z +
ak z k .
k=2
Let K(α), Rα , and C(α) denote the classes of functions f ∈ A, which are, respectively, convex, prestarlike and close to convex of order α.
A continuous complex-valued function f = u + iv defined in a simply connected
domain D is said to be harmonic in D if both u and v are real harmonic in D.
There is a close inter-relation between analytic functions and harmonic functions.
For example, for real harmonic functions u and v there exist analytic functions U
and V so that u = Re U and v = Im V . Then
f (z) = h(z) + g(z),
where h and g are, respectively, the analytic functions (U + V )/2 and (U − V )/2.
In this case, the Jacobian of f = h + g is given by
Jf (z) = |h′ (z)|2 − |g ′ (z)|2 ,
139
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
The mapping z → f (z) is sense preserving and locally univalent in D if and
only if Jf > 0 in D. See also Clune and Sheil-Small [4]. The function f = h + g
is said to be harmonic univalent in D if the mapping z → f (z) is sense preserving
harmonic and univalent in D. We call h the analytic part and g the co-analytic part
of f = h + g.
Let SH denote the family of functions f = h + g which are harmonic univalent
and sense-preserving in E with the normalization
h(z) = z +
∞
X
ak z k , g(z) =
k=2
∞
X
bk z k .
(1.1)
k=1
Let KH and CH denote the subclasses of SH consisting of harmonic functions
which are, respectively. convex and close to convex in E.
Finally, we define convolution of two complex-valued harmonic functions f1 (z) =
∞
∞
∞
∞
X
X
X
X
k
k
k
z+
a1k z +
b1k z and f2 (z) = z +
b2k z k by
a2k z +
k=2
k=1
f1 (z) ∗ f2 (z) = z +
k=2
∞
X
k=1
k
a1k a2k z +
k=2
∞
X
b1k b2k z k .
k=1
The above convolution formula reduces to the Hadamard product if the coanalytic parts of f1 and f2 are zero.
n
In this paper we define and give some properties of the integral operator IH,λ
f, n ∈
∼
N0 = N ∪ {0}, λ > 0, constructed as the convolution f ∗ ψn of harmonic univalent
and sense-preserving functions f in the unit disc E with the prestarlike function ψn .
We mainly study some convolution properties preserved by this operator.
2. Definitions and Preliminary Results
n f and derive some of its basic
In the following we define the integral operator IH,λ
properties. We need the following definition.
∼
Definition 1. Let f ∈ SH , and assume ψ is analytic in E. Then (f ∗ ψ)(z) =
∼
(ψ ∗ f )(z) = (h ∗ ψ)(z) + (g ∗ ψ)(z).
n f, n ∈ N , λ > 0, is
Definition 2. Let f ∈ SH . Then the integral operator IH,λ
0
defined by
n
(2.1)
IH,λ
f (z) = Iλn h(z) + Iλn g(z),
140
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
where Iλn f, λ > 0 is the multiplier integral operator defined [3] on f ∈ A as follows
Iλ0 f (z) =f (z)
Iλ1 f (z)
=
Iλ2 f (z) =
Iλn f (z) =
1
λz
1
−1
λ
1
λz
1
−1
λ
1
λz
Z
z
Z
z
1
t λ −2 f (t)dt
0
0
···
Z
1
−1
λ
z
0
1
t λ −2 Iλ1 f (t)dt
1
t λ −2 Iλn−1 f (t)dt, n ∈ N.
(2.2)
n f is zero, then I n f ≡ I n f , [3]. When
Remark 1. If the co-analytic part of IH,λ
H,λ
λ
1
1
n
λ=
, γ > −1, Iλ f is Bernardi integral operator. When λ = 1, I1 f is Salagean
1+γ
integral operator [8].
n f satisfies the following
Remark 2. The integral operator IH,λ
n
n
(i) IH,λ
(DH,λ
f (z)) = f (z),
n f is harmonic differential operator defined by Li Shuai and Liu Peide [9].
where DH,λ
∼
n
f (z) = (f ∗ ψn )(z) where
(ii) IH,λ
ψn (z) = z +
∞
X
k=2
and
1
zk ,
[1 + λ(k − 1)]n
= (ψ ∗ ψ · · · ∗ ψ)(z)
|
{z
}
n-times
∞
X
1
zk ,
[1 + λ(k − 1)]
k=2
1
1
= z2 F1 1, ; 1 + ; z ,
λ
λ
ψ(z) = z +
(2.3)
(2.4)
a(a + 1) · b(b + 1) 2
a·b
z+
z +· · · , for any real
1·c
2! · c(c + 1)
or complex numbers a, b and c (c 6= 0, −1, −2, . . .), is the well-known hypergeometric
series which represents an analytic function E
where the function 2 F1 (a, b; c; z) = 1+
141
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
n
(iii) IH,λ
f (z) = z +
∞
X
k=2
∞
X
ak
bk
zk +
zk ,
n
[(1 + λ(k − 1)]
[1 + λ(k − 1)]n
k=1
n+1
n+1
n+1
n
f (z))z ].
(iv) IH,λ
f (z) = (1 − λ)IH,λ
f (z) + λ[z(IH,λ
f (z))z + z(IH,λ
n f (z), given by (2.1), is sense-preserving, but may not be
(v) The operator IH,λ
n
univalent in E.
I H,λ
f (z) = H(z) + G(z), then |G′ | = |(Iλn g)′ | =
Consider
1
1
|(g ∗ ψn )′ | = ψn ∗ g ′ < ψn ∗ h′ = |(h ∗ ψn )′ | = |(Iλn h)′ | = |H ′ |, which
z
z
n f (z) is sense-preserving. To show that I n f (z), may not
shows that IH,λ
H,λ
n f is
be univalent in E, consider the case where the co-analytic part of IH,λ
1
zero, and n = 1, λ = . The function f (z) = (1 + i)[(1 − z)−1+i − 1] =
2
(1 + i)(e(−1+i) ln(1−z) − 1), given by Campbell and V. Singh [5] is normalized
univalent in E, with this function f , equation (2.2) gives
2
I 1 f (z) =
2
z
1
Z
z
f (t)dt = 2i(1 + i)
0
(1 − z)i − 1
− 2(1 − i).
z
If we set zm = 1 − e−2πm , we find that I 01 f (zm ) = −2(1 + i) for m = 1, 2, . . .
2
Hence I 01 f (z) is infant-valent [5].
2
We will need the following lemmas.
Lemma 1. [6] Let ϕ and ψ be convex analytic in E. Then
(i) ϕ ∗ ψ is convex analytic in E.
(ii) ϕ ∗ f is close to convex analytic in E, if f is close to convex analytic in E.
Lemma 2. [7] (i) Let α ≤ 1 and f, g ∈ Rα . Then f ∗ g ∈ Rα .
(ii) For α < β ≤ 1, we have Rα ⊂ Rβ .
(iii) For α < 1, let f ∈ C(α) and g ∈ Rα . Then f ∗ g ∈ C(α).
Lemma 3. [4] Let h ∈ A and g ∈ A.
(i) if |g ′ (0)| < |h′ (0)| and h + ǫg is close to convex in E, for each ǫ(|ǫ| = 1), then
f = h + g ∈ CH .
(ii) If f = h + g is harmonic and locally univalent in E, and if h + ǫg is convex
analytic in E for some ǫ(|ǫ| = 1), then f = h + g ∈ CH .
142
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
Lemma 4. Let ψ be as in (2.4). Then
(i) ψ ∈ R(1− 1 ) .
λ
(ii)
ψ(z)
is convex univalent in E.
z
For (i) we refer to [7] and for (ii) see [2].
From Lemma 2 and Lemma 4(i) we obtain
Corollary 1. ψn ∈ R(1− 1 ) .
λ
3. Main Results
We now state and prove our main results.
′
′
Theorem 1. Let f =
h + g, where h and g are given byn (1.1). If |g (0)| < |h (0)|
1
and (h + ǫg) ∈ C 1 − λ , λ ≥ 1 for each ǫ(ǫ| = 1), then IH,λ f ∈ CH .
Proof. Iλn (h + ǫg) = (h + ǫg) ∗ ψn
= Iλn h + ǫIλn g.
(3.1)
1
Since (h + ǫg) ∈ C 1 −
, and ψn ∈ R(1− 1 ) , applying Lemma 2(ii), we obtain
λ
λ
1
1
. Since 1 − ≥ 0, for λ ≥ 1 then Iλn h + ǫIλn g) ∈ C(0),
(Iλn h + ǫIλn g) ∈ C 1 −
λ
λ
the class of analytic close to convex functions in E. Applying Lemma 3(i), we get
the required result.
n
Theorem 2. If λ ≤ 1, IH,λ
f = Iλn h + Iλn g is harmonic and locally univalent in
n
E, and if h + ǫg is convex analytic in E for some ǫ(|ǫ| = 1), then IH,λ
f ∈ CH .
1
≤ 0, then applying Lemma
λ
2(ii), we get ψn ∈ R0 ≡ K(0). Since h + ǫg ∈ K(0), then from (3.1) and Lemma
1(i), we deduce that Iλn h + ǫIλn g, is convex analytic in E. Applying Lemma 3(ii), we
get the desired result.
Proof. Since ψn ∈ R(1− 1 ) , and λ ≤ 1, implies 1 −
λ
Remark 1. For n = 0, Theorems 1 and 2 reduce to Clune and Sheil-Small
results, given in Lemma 3.
143
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
Theorem 3. Let f = h + g, where h and g are given by (1.1), such that
|g ′ (0)| < |h′ (0)| and h + ǫg is close to convex analytic in E, for each ǫ(|ǫ| = 1). If ϕ
is convex analytic in E, then for λ ≥ 1,
n
(ϕ + σϕ) ∗ IH,λ
f ∈ CH , |σ| = 1.
Proof. Let (ϕ + σϕ) ∗ (Iλn h(z) + Iλn g(z)) = ϕ ∗ Iλn h + σϕ ∗ Iλn g = H + G. Now
H + γG = ϕ ∗ Iλn h + γσϕ ∗ Iλn g
= ϕ ∗ (Iλn h + ǫIλn g), ǫ = γσ.
From the proof of Theorem 1, we see that (Iλn h(z) + ǫIλn g(z)), is close to convex
analytic function for λ ≥ 1 and for each ǫ (|ǫ| = 1). Applying Lemma 3(i), we
deduce that H + γG is close to convex analytic function for each γ(|γ| = 1). Next
we show that |G′ (0)| < |H ′ (0)|,
1
n ′
′
n ′
|G (0)| = |(σϕ ∗ Iλ g) |z=0 = ϕ ∗ σ(Iλ g)
z
z=0
1
= |(ϕ ∗ Iλn h)′ |z=0 = |H ′ (0)|.
< σϕ ∗ (Iλn h)′
z
z=0
n f ∈C .
By Theorem 1, we obtain H + G = (ϕ + σϕ) ∗ IH,λ
H
Remark 2. For n = 0, Theorem 3 reduces to the results of Ahuja and Jahangiri
[1].
Theorem 4. Let f = h + g, where h and g are given by (1.1). Then
Z
0
1
n f (z)
IH,λ
z
=
n−1
t−λ IH,λ
f (ztλ )dt.
z dt
z
Proof. From (2.4), ψ can be written as ψ(z) =
. Since h∗
(z) =
λ
1−ztλ
0 1−zt
Z
1
z
h(tλ z)
g(tλ z)
n−1
(z)
=
and
g
∗
,
then
(I
h∗ψ)(z)
=
zt−λ Iλn−1 h(tλ z)dt,
λ
tλ
1 − ztλ
tλ
0
Z 1
n−1
and (Iλ g ∗ ψ)(z) =
zt−λ Iλn−1 g(tλ z)dt. Therefore (2.1) gives
Z
z
0
n
IH,λ
f (z) =
Z
0
1
zt−λ (Iλn−1 h(tλ z) + Iλn−1 g(tλ z))dt.
144
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
Hence
n f (z)
IH,λ
z
=
Z
0
1
n−1
t−λ IH,λ
f (tλ z)dt.
References
[1] O.P. Ahuja and J.M. Janahgiri, Convolutions for special classes of harmonic
univalent functions, Applied Mathematics Letters 16 (2003), 905–909.
[2] F.M. Al-Oboudi and K.A. Al-Amoudi, Subordination results for classes of
analytic functions related to conic domains defined by a fractional operator, J. Math.
Anal. Appl., 55 (2009), 412–420.
[3] F. M. Al-Oboudi and Z.M. Al-Qahtani, On a subclass of analytic functions
defined by a new multiplier integral operator, Far East J. Math Sci. (FGMS), 25,
no. 1 (2007), 59–72.
[4] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci.
Fenn. Ser. A.I. Math., 9 (1984), 3–25.
[5] D.M. Campbell and V. Singh, Valence properties of the solution of a differential equation, Pacific J. Math., 84 (1979), 29–33.
[6] St. Ruscheweyh, and T. Sheil-Small, Hadamard products of schlicht functions
and the Polya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119–135.
[7] St. Ruscheweyh, Convolutions in Geometric Function Theory, Sem. Math.
Sup. 83, Presses Univ. de Montreal, 1982.
[8] G. S
¸ . Sˇalˇ
agean, Subclasses of univalent functions, Complex analysis - Proc.
5th Rom.-Finn. Semin., Bucharest 1981, Part 1, Lect. Notes Math. 1013, (1983),
362–372.
[9] LI Shuai and LIU Peide, A new class of harmonic univalent functions by the
generalized Salagean operaor, WUJNS. 12, no. 6(2007), 965–970.
[10] Z. Zongzhu and S. Owa, Convolution properties of a class of bounded analytic
functions, Bull. Austral. Math. Soc., 45, no. 1 (1992), 9–23.
F. Al-Oboudi
Department of Mathematical Sciences
College of Science, Princess Nora Bint Abdul Rahman University
Riyadh, Saudi Arabia
email:[email protected]
145
ISSN: 1582-5329
No. 23/2010
pp. 139-145
CONVOLUTION PROPERTIES OF HARMONIC UNIVALENT
FUNCTIONS PRESERVED BY SOME INTEGRAL OPERATOR
F. M. Al-Oboudi
Abstract. A complex valued function f = u + iv defined in a domain D ⊂ C,
is harmonic in D, if u and v are real harmonic. Such functions can be represented
as f (z) = h(z) + g(z), where h an g are analytic in D. In this paper we study
n f, n ∈ N =
some convolution properties preserved by the integral operator IH,λ
0
N ∪ {0}, λ > 0, where the functions f are univalent harmonic and sense-preserving
n
in the open unit disc E = {z : |z| < 1}, IH,λ
f (z) = Iλn h(z) + Iλn g(z), and Ihn h(z) =
∞
∞
X
X
ak
bk
k n
z
,
I
g(z)
=
z k . Relevant connections of
z+
λ
n
[1 + λ(k − 1)]
[1 + λ(k − 1)]n
k=2
k=1
the results presented here with those obtained in earlier works are pointed out.
2000 Mathematics Subject Classification: 30C45.
1. Introduction
Let A denote the class of analytic functions in the open unit disc E = {z : |z| <
1}, with the normalization
∞
X
f (z) = z +
ak z k .
k=2
Let K(α), Rα , and C(α) denote the classes of functions f ∈ A, which are, respectively, convex, prestarlike and close to convex of order α.
A continuous complex-valued function f = u + iv defined in a simply connected
domain D is said to be harmonic in D if both u and v are real harmonic in D.
There is a close inter-relation between analytic functions and harmonic functions.
For example, for real harmonic functions u and v there exist analytic functions U
and V so that u = Re U and v = Im V . Then
f (z) = h(z) + g(z),
where h and g are, respectively, the analytic functions (U + V )/2 and (U − V )/2.
In this case, the Jacobian of f = h + g is given by
Jf (z) = |h′ (z)|2 − |g ′ (z)|2 ,
139
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
The mapping z → f (z) is sense preserving and locally univalent in D if and
only if Jf > 0 in D. See also Clune and Sheil-Small [4]. The function f = h + g
is said to be harmonic univalent in D if the mapping z → f (z) is sense preserving
harmonic and univalent in D. We call h the analytic part and g the co-analytic part
of f = h + g.
Let SH denote the family of functions f = h + g which are harmonic univalent
and sense-preserving in E with the normalization
h(z) = z +
∞
X
ak z k , g(z) =
k=2
∞
X
bk z k .
(1.1)
k=1
Let KH and CH denote the subclasses of SH consisting of harmonic functions
which are, respectively. convex and close to convex in E.
Finally, we define convolution of two complex-valued harmonic functions f1 (z) =
∞
∞
∞
∞
X
X
X
X
k
k
k
z+
a1k z +
b1k z and f2 (z) = z +
b2k z k by
a2k z +
k=2
k=1
f1 (z) ∗ f2 (z) = z +
k=2
∞
X
k=1
k
a1k a2k z +
k=2
∞
X
b1k b2k z k .
k=1
The above convolution formula reduces to the Hadamard product if the coanalytic parts of f1 and f2 are zero.
n
In this paper we define and give some properties of the integral operator IH,λ
f, n ∈
∼
N0 = N ∪ {0}, λ > 0, constructed as the convolution f ∗ ψn of harmonic univalent
and sense-preserving functions f in the unit disc E with the prestarlike function ψn .
We mainly study some convolution properties preserved by this operator.
2. Definitions and Preliminary Results
n f and derive some of its basic
In the following we define the integral operator IH,λ
properties. We need the following definition.
∼
Definition 1. Let f ∈ SH , and assume ψ is analytic in E. Then (f ∗ ψ)(z) =
∼
(ψ ∗ f )(z) = (h ∗ ψ)(z) + (g ∗ ψ)(z).
n f, n ∈ N , λ > 0, is
Definition 2. Let f ∈ SH . Then the integral operator IH,λ
0
defined by
n
(2.1)
IH,λ
f (z) = Iλn h(z) + Iλn g(z),
140
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
where Iλn f, λ > 0 is the multiplier integral operator defined [3] on f ∈ A as follows
Iλ0 f (z) =f (z)
Iλ1 f (z)
=
Iλ2 f (z) =
Iλn f (z) =
1
λz
1
−1
λ
1
λz
1
−1
λ
1
λz
Z
z
Z
z
1
t λ −2 f (t)dt
0
0
···
Z
1
−1
λ
z
0
1
t λ −2 Iλ1 f (t)dt
1
t λ −2 Iλn−1 f (t)dt, n ∈ N.
(2.2)
n f is zero, then I n f ≡ I n f , [3]. When
Remark 1. If the co-analytic part of IH,λ
H,λ
λ
1
1
n
λ=
, γ > −1, Iλ f is Bernardi integral operator. When λ = 1, I1 f is Salagean
1+γ
integral operator [8].
n f satisfies the following
Remark 2. The integral operator IH,λ
n
n
(i) IH,λ
(DH,λ
f (z)) = f (z),
n f is harmonic differential operator defined by Li Shuai and Liu Peide [9].
where DH,λ
∼
n
f (z) = (f ∗ ψn )(z) where
(ii) IH,λ
ψn (z) = z +
∞
X
k=2
and
1
zk ,
[1 + λ(k − 1)]n
= (ψ ∗ ψ · · · ∗ ψ)(z)
|
{z
}
n-times
∞
X
1
zk ,
[1 + λ(k − 1)]
k=2
1
1
= z2 F1 1, ; 1 + ; z ,
λ
λ
ψ(z) = z +
(2.3)
(2.4)
a(a + 1) · b(b + 1) 2
a·b
z+
z +· · · , for any real
1·c
2! · c(c + 1)
or complex numbers a, b and c (c 6= 0, −1, −2, . . .), is the well-known hypergeometric
series which represents an analytic function E
where the function 2 F1 (a, b; c; z) = 1+
141
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
n
(iii) IH,λ
f (z) = z +
∞
X
k=2
∞
X
ak
bk
zk +
zk ,
n
[(1 + λ(k − 1)]
[1 + λ(k − 1)]n
k=1
n+1
n+1
n+1
n
f (z))z ].
(iv) IH,λ
f (z) = (1 − λ)IH,λ
f (z) + λ[z(IH,λ
f (z))z + z(IH,λ
n f (z), given by (2.1), is sense-preserving, but may not be
(v) The operator IH,λ
n
univalent in E.
I H,λ
f (z) = H(z) + G(z), then |G′ | = |(Iλn g)′ | =
Consider
1
1
|(g ∗ ψn )′ | = ψn ∗ g ′ < ψn ∗ h′ = |(h ∗ ψn )′ | = |(Iλn h)′ | = |H ′ |, which
z
z
n f (z) is sense-preserving. To show that I n f (z), may not
shows that IH,λ
H,λ
n f is
be univalent in E, consider the case where the co-analytic part of IH,λ
1
zero, and n = 1, λ = . The function f (z) = (1 + i)[(1 − z)−1+i − 1] =
2
(1 + i)(e(−1+i) ln(1−z) − 1), given by Campbell and V. Singh [5] is normalized
univalent in E, with this function f , equation (2.2) gives
2
I 1 f (z) =
2
z
1
Z
z
f (t)dt = 2i(1 + i)
0
(1 − z)i − 1
− 2(1 − i).
z
If we set zm = 1 − e−2πm , we find that I 01 f (zm ) = −2(1 + i) for m = 1, 2, . . .
2
Hence I 01 f (z) is infant-valent [5].
2
We will need the following lemmas.
Lemma 1. [6] Let ϕ and ψ be convex analytic in E. Then
(i) ϕ ∗ ψ is convex analytic in E.
(ii) ϕ ∗ f is close to convex analytic in E, if f is close to convex analytic in E.
Lemma 2. [7] (i) Let α ≤ 1 and f, g ∈ Rα . Then f ∗ g ∈ Rα .
(ii) For α < β ≤ 1, we have Rα ⊂ Rβ .
(iii) For α < 1, let f ∈ C(α) and g ∈ Rα . Then f ∗ g ∈ C(α).
Lemma 3. [4] Let h ∈ A and g ∈ A.
(i) if |g ′ (0)| < |h′ (0)| and h + ǫg is close to convex in E, for each ǫ(|ǫ| = 1), then
f = h + g ∈ CH .
(ii) If f = h + g is harmonic and locally univalent in E, and if h + ǫg is convex
analytic in E for some ǫ(|ǫ| = 1), then f = h + g ∈ CH .
142
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
Lemma 4. Let ψ be as in (2.4). Then
(i) ψ ∈ R(1− 1 ) .
λ
(ii)
ψ(z)
is convex univalent in E.
z
For (i) we refer to [7] and for (ii) see [2].
From Lemma 2 and Lemma 4(i) we obtain
Corollary 1. ψn ∈ R(1− 1 ) .
λ
3. Main Results
We now state and prove our main results.
′
′
Theorem 1. Let f =
h + g, where h and g are given byn (1.1). If |g (0)| < |h (0)|
1
and (h + ǫg) ∈ C 1 − λ , λ ≥ 1 for each ǫ(ǫ| = 1), then IH,λ f ∈ CH .
Proof. Iλn (h + ǫg) = (h + ǫg) ∗ ψn
= Iλn h + ǫIλn g.
(3.1)
1
Since (h + ǫg) ∈ C 1 −
, and ψn ∈ R(1− 1 ) , applying Lemma 2(ii), we obtain
λ
λ
1
1
. Since 1 − ≥ 0, for λ ≥ 1 then Iλn h + ǫIλn g) ∈ C(0),
(Iλn h + ǫIλn g) ∈ C 1 −
λ
λ
the class of analytic close to convex functions in E. Applying Lemma 3(i), we get
the required result.
n
Theorem 2. If λ ≤ 1, IH,λ
f = Iλn h + Iλn g is harmonic and locally univalent in
n
E, and if h + ǫg is convex analytic in E for some ǫ(|ǫ| = 1), then IH,λ
f ∈ CH .
1
≤ 0, then applying Lemma
λ
2(ii), we get ψn ∈ R0 ≡ K(0). Since h + ǫg ∈ K(0), then from (3.1) and Lemma
1(i), we deduce that Iλn h + ǫIλn g, is convex analytic in E. Applying Lemma 3(ii), we
get the desired result.
Proof. Since ψn ∈ R(1− 1 ) , and λ ≤ 1, implies 1 −
λ
Remark 1. For n = 0, Theorems 1 and 2 reduce to Clune and Sheil-Small
results, given in Lemma 3.
143
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
Theorem 3. Let f = h + g, where h and g are given by (1.1), such that
|g ′ (0)| < |h′ (0)| and h + ǫg is close to convex analytic in E, for each ǫ(|ǫ| = 1). If ϕ
is convex analytic in E, then for λ ≥ 1,
n
(ϕ + σϕ) ∗ IH,λ
f ∈ CH , |σ| = 1.
Proof. Let (ϕ + σϕ) ∗ (Iλn h(z) + Iλn g(z)) = ϕ ∗ Iλn h + σϕ ∗ Iλn g = H + G. Now
H + γG = ϕ ∗ Iλn h + γσϕ ∗ Iλn g
= ϕ ∗ (Iλn h + ǫIλn g), ǫ = γσ.
From the proof of Theorem 1, we see that (Iλn h(z) + ǫIλn g(z)), is close to convex
analytic function for λ ≥ 1 and for each ǫ (|ǫ| = 1). Applying Lemma 3(i), we
deduce that H + γG is close to convex analytic function for each γ(|γ| = 1). Next
we show that |G′ (0)| < |H ′ (0)|,
1
n ′
′
n ′
|G (0)| = |(σϕ ∗ Iλ g) |z=0 = ϕ ∗ σ(Iλ g)
z
z=0
1
= |(ϕ ∗ Iλn h)′ |z=0 = |H ′ (0)|.
< σϕ ∗ (Iλn h)′
z
z=0
n f ∈C .
By Theorem 1, we obtain H + G = (ϕ + σϕ) ∗ IH,λ
H
Remark 2. For n = 0, Theorem 3 reduces to the results of Ahuja and Jahangiri
[1].
Theorem 4. Let f = h + g, where h and g are given by (1.1). Then
Z
0
1
n f (z)
IH,λ
z
=
n−1
t−λ IH,λ
f (ztλ )dt.
z dt
z
Proof. From (2.4), ψ can be written as ψ(z) =
. Since h∗
(z) =
λ
1−ztλ
0 1−zt
Z
1
z
h(tλ z)
g(tλ z)
n−1
(z)
=
and
g
∗
,
then
(I
h∗ψ)(z)
=
zt−λ Iλn−1 h(tλ z)dt,
λ
tλ
1 − ztλ
tλ
0
Z 1
n−1
and (Iλ g ∗ ψ)(z) =
zt−λ Iλn−1 g(tλ z)dt. Therefore (2.1) gives
Z
z
0
n
IH,λ
f (z) =
Z
0
1
zt−λ (Iλn−1 h(tλ z) + Iλn−1 g(tλ z))dt.
144
F. Al-Oboudi - Convolution Properties of Harmonic Univalent Functions
Hence
n f (z)
IH,λ
z
=
Z
0
1
n−1
t−λ IH,λ
f (tλ z)dt.
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F. Al-Oboudi
Department of Mathematical Sciences
College of Science, Princess Nora Bint Abdul Rahman University
Riyadh, Saudi Arabia
email:[email protected]
145