ON A GENERALIZATION OF ALPHA CONVEXITY
KHALIDA INAYAT NOOR
Mathematics Department,
COMSATS Institute of Information Technology,
Islamabad, Pakistan.
EMail: khalidanoor@hotmail.com
Alpha Convexity
Khalida Inayat Noor
vol. 8, iss. 1, art. 16, 2007
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Received:
28 November, 2006
Accepted:
06 February, 2007
Communicated by:
N.E. Cho
2000 AMS Sub. Class.:
30C45, 30C50.
Key words:
Starlike, Convex, Strongly alpha-convex, Bounded boundary rotation.
Abstract:
In this paper, we introduce and study a class M̃k (α, β, γ), k ≥ 2 of analytic
functions defined in the unit disc. This class generalizes the concept of alphaconvexity and include several other known classes of analytic functions. Inclusion results, an integral representation and a radius problem is discussed for this
class.
Acknowledgements:
This research is supported by the Higher Education Commission, Pakistan,
through research grant No: 1-28/HEC/HRD/2005/90.
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Contents
1
Introduction
3
2
Main Results
5
Alpha Convexity
Khalida Inayat Noor
vol. 8, iss. 1, art. 16, 2007
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1.
Introduction
Let P̃ denote the class of functions of the form
(1.1)
p(z) = 1 + c1 z + c2 z 2 + · · · ,
which are analytic in the unit disc E = {z : |z| < 1}. Let P̃ (γ) be the subclass of P̃
consisting of functions p which satisfy the condition
(1.2)
|arg p(z)| ≤
πγ
,
2
Alpha Convexity
Khalida Inayat Noor
for some
γ(γ > 0),
z ∈ E.
We note that P̃ (1) = P is the class of analytic functions with positive real part. We
introduce the class P˜k (γ) as follows:
An analytic function p given by (1.1) belongs to P˜k (γ), for z ∈ E, if and only if
there exist p1 , p2 ∈ P̃ (γ) such that
k 1
k 1
(1.3)
p(z) =
+
p1 (z) −
−
p2 (z), k ≥ 2.
4 2
4 2
We now define the class M̃k (α, β, γ) as follows:
Definition 1.1. Let α ≥ 0, β ≥ 0 (α + β 6= 0) and let f be analytic in E with
0
(z)
f (0) = 0, f 0 (0) = 1 and f (z)f
6= 0. Then f ∈ M̃k (α, β, γ) if and only if , for
z
z ∈ E,
β (zf 0 (z))0
α zf 0 (z)
+
∈ P˜k (γ).
α + β f (z)
α + β f 0 (z)
We note that, for k = 2, β = (1 − α), we have the class M̃2 (α, 1 − α, γ) = M̃α (γ)
of strongly alpha-convex functions introduced and studied in [4].
We also have the following special cases.
vol. 8, iss. 1, art. 16, 2007
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(i) M̃2 (α, 0, 1) = S ? , M̃2 (0, β, 1) = C, where S ? and C are respectively the
well-known classes of starlike and convex functions. It is known [3] that
M̃α (γ) ⊂ S ? and M̃2 (α, 0, γ) coincides with the class of strongly starlike functions of order γ, see [1, 7, 8].
(ii) M̃k (α, 0, 1) = Rk , M̃k (0, β, 1) = Vk , where Rk is the class of functions of
bounded radius rotation and Vk is the class of functions of bounded boundary
rotation.
Alpha Convexity
Khalida Inayat Noor
Also M̃k (0, β, γ) = V˜k (γ) ⊂ Vk and M̃k (α, 0, γ) = R̃k (γ) ⊂ Rk .
vol. 8, iss. 1, art. 16, 2007
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2.
Main Results
Theorem 2.1. A function f ∈ M̃k (α, β, γ),
function F ∈ R̃k (γ) such that
"
(2.1)
f (z) =
α+β
α
Z
0
z
α, β > 0, if and only if, there exists a
(F (t))
t
β
# α+β
α+β
β
dt
.
Alpha Convexity
Khalida Inayat Noor
Proof. A simple calculation yields
vol. 8, iss. 1, art. 16, 2007
α zf 0 (z)
β (zf 0 (z))0
zF 0 (z)
+
=
.
α + β f (z)
α + β f 0 (z)
F (z)
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If the right hand side belongs to P˜k (γ) so does the left and conversely, and the result
follows.
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Theorem 2.2. Let f ∈ M̃k (α, β, γ). Then the function
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g(z) = f (z)
(2.2)
zf 0 (z)
f (z)
β
α+β
belongs to R̃k (γ) for z ∈ E.
Proof. Differentiating (2.2) logarithmically, we have
zg 0 (z)
α zf 0 (z)
β (zf 0 (z))0
=
+
,
g(z)
α + β f (z)
α + β f 0 (z)
and, since f ∈ M̃k (α, β, γ), we obtain the required result.
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Theorem 2.3. Let f ∈ M̃k (α, β, γ),
z ∈ E.
Proof. Let
zf 0 (z)
f (z)
β > 0, 0 < γ ≤ 1. Then f ∈ R̃k (γ) for
= p(z). Then
(zf 0 (z))0
zp0 (z)
=
p(z)
+
.
f 0 (z)
p(z)
Alpha Convexity
Therefore, for z ∈ E,
(2.3)
α zf 0 (z)
β (zf 0 (z))0
+
=
α + β f (z)
α + β f 0 (z)
p(z) +
β zp0 (z)
α + β p(z)
Khalida Inayat Noor
∈ P˜k (γ).
Let
(2.4)
vol. 8, iss. 1, art. 16, 2007
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α
z
β
z
φ(α, β) =
+
.
α + β 1 − z α + β (1 − z)2
Then, using (1.3) and (2.4), we have
φ(α, β)
k 1
φ(α, β)
k 1
φ(α, β)
p?
=
+
p1 ?
−
−
p2 ?
,
z
4 2
z
4 2
z
where ? denotes the convolution (Hadamard product). This gives us
β zp0 (z)
k 1
β zp01 (z)
p(z) +
=
+
p1 (z) +
α + β p(z)
4 2
α + β p1 (z)
k 1
β zp02 (z)
−
−
p2 (z) +
.
4 2
α + β p2 (z)
From (2.3), it follows that
pi +
β zp0i
α + β pi
∈ P̃ (γ),
i = 1, 2,
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and, using a result due to Nunokawa and Owa [6], we conclude that pi ∈ P̃ (γ) in
E, i = 1, 2. Consequently p ∈ P˜k (γ) and hence f ∈ R̃k (γ) for z ∈ E.
Theorem 2.4. Let, for (α1 + β 1 ) 6= 0,
α1
α
<
,
α1 + β 1
α+β
β1
β
<
α1 + β 1
α+β
and
0 ≤ γ < 1.
Then
M̃k (α, β, γ) ⊂ M˜K (α1 , β 1 , γ),
Alpha Convexity
z ∈ E.
Khalida Inayat Noor
vol. 8, iss. 1, art. 16, 2007
Proof. We can write
α1 zf 0 (z)
β 1 (zf 0 (z))0
+
α1 + β 1 f (z)
α1 + β 1 f 0 (z)
β 1 (α + β) zf 0 (z)
= 1−
β(α1 + β 1 ) f (z)
β 1 (α + β)
α zf 0 (z)
β (zf 0 (z))0
+
+
β(α1 + β 1 )
α + β f (z)
α + β f 0 (z)
β (α + β)
β (α + β)
H1 (z) + 1
H2 (z),
= 1− 1
β(α1 + β 1 )
β(α1 + β 1 )
where H1 , H2 ∈ P˜k (γ) by using Definition 1.1 and Theorem 2.3. Since 0 < γ ≤ 1,
the class P̃ (γ) is a convex set and consequently, by (1.3), the class P˜k (γ) is a convex
set. This implies H ∈ P˜k (γ) and therefore f ∈ M̃k (α1 , β 1 , γ). This completes the
proof.
Theorem 2.5. Let f ∈ M̃k (α, β, γ). Then
α
α+β
Z z
β
f
(t)
0
(2.5) h(z) =
dt belongs to
(f (t)) α+β
t
0
V˜k (γ) for
z ∈ E.
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Proof. From (2.5), we have
0
0
h (z) = (f (z))
β
α+β
f (z)
z
α
α+β
.
Now the proof is immediate when we differentiate both sides logarithmically and
use the fact that f ∈ M̃k (α, β, γ).
Alpha Convexity
In the following we study the converse case of Theorem 2.3 with γ = 1.
Theorem 2.6. Let f ∈ R̃k (1). Then f ∈ M̃k (α, β, 1), β > 0 for |z| < r(α, β),
where
1
β
(2.6)
r(α, β) = 1 − ρ2 2 − ρ, with ρ =
.
α+β
This result is best possible.
Proof. Since f ∈ R̃k (1),
zf 0 (z)
f (z)
∈ P˜k (1) = Pk , and
Khalida Inayat Noor
vol. 8, iss. 1, art. 16, 2007
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α zf 0 (z)
β (zf 0 (z))0
β zp0 (z)
+
=
p(z)
+
.
α + β f (z)
α + β f 0 (z)
α + β p(z)
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Let φ(α, β) be as given by (2.4). Now using (1.3) and convolution techniques, we
have
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β zp0 (z)
p(z) +
= p(z) ?
α + β p(z)
k
=
+
4
φ(α, β)
z
1
φ(α, β)
k 1
φ(α, β)
p1 (z) ?
−
−
p2 (z) ?
.
2
z
4 2
z
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o
n
φ(α,β)
˜
Since pi ∈ P2 (1) = P and it is known [2] that Re
> 12 for |z| < r(α, β), it
z
h
i
follows from a well known result, see [5] that pi ? φ(α,β)
∈ P for |z| < r(α, β), i =
z
1, 2. with r(α, β) given by (2.6). The function φ(α, β) given by (2.4) shows that the
radius r(α, β) is best possible.
Alpha Convexity
Khalida Inayat Noor
vol. 8, iss. 1, art. 16, 2007
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References
[1] D.A. BRANNAN AND W.E. KIRWAN, On some classes of bounded univalent
functions, J. London Math. Soc., 2(1) (1969), 431–443.
[2] J.L. LIU AND K. INAYAT NOOR, On subordination for certain analytic functions associated with Noor integral operator, Appl. Math. Computation, (2006),
in press.
Alpha Convexity
Khalida Inayat Noor
[3] S.S. MILLER, P.T. MOCANU AND M.O. READE, All α-convex functions are
univalent and starlike, Proc. Amer. Math. Soc., 37 (1973), 552–554.
vol. 8, iss. 1, art. 16, 2007
[4] K. INAYAT NOOR, On strongly alpha-convex and alpha-quasi-convex functions, J. Natural Geometry, 10(1996), 111–118.
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[5] K. INAYAT NOOR, Some properties of certain analytic functions, J. Natural
Geometry, 7 (1995), 11–20.
[6] M. NUNOKAWA AND S. OWA, On certain differential subordination, PanAmer.
Math. J., 3 (1993), 35–38.
[7] J. STANSKIEWICS, Some remarks concerning starlike functions, Bull. Acad.
Polon. Sci. Ser. Sci. Math. Astronom. Phys., 18 (1970), 143–146.
[8] J. STANSKIEWICS, On a family of starlike functions, Ann. Univ. Mariae-CurieSkl. Sect. A., 22(24) (1968/70), 175–181.
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