Volume 8 (2007), Issue 1, Article 16, 4 pp.
ON A GENERALIZATION OF ALPHA CONVEXITY
KHALIDA INAYAT NOOR
M ATHEMATICS D EPARTMENT,
COMSATS I NSTITUTE OF I NFORMATION T ECHNOLOGY,
I SLAMABAD , PAKISTAN .
khalidanoor@hotmail.com
Received 28 November, 2006; accepted 06 February, 2007
Communicated by N.E. Cho
A BSTRACT. 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 alpha-convexity and include
several other known classes of analytic functions. Inclusion results, an integral representation
and a radius problem is discussed for this class.
Key words and phrases: Starlike, Convex, Strongly alpha-convex, Bounded boundary rotation.
2000 Mathematics Subject Classification. 30C45, 30C50.
1. I NTRODUCTION
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
πγ
(1.2)
|arg p(z)| ≤
, for some γ(γ > 0), z ∈ E.
2
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:
This research is supported by the Higher Education Commission, Pakistan, through research grant No: 1-28/HEC/HRD/2005/90.
306-06
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K HALIDA I NAYAT N OOR
Definition 1.1. Let α ≥ 0, β ≥ 0 (α+β 6= 0) and let f be analytic in E with f (0) = 0, f 0 (0) =
0
(z)
1 and f (z)f
6= 0. Then f ∈ M̃k (α, β, γ) if and only if , for z ∈ E,
z
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.
(i) M̃2 (α, 0, 1) = S ? , M̃2 (0, β, 1) = C, where S ? and C are respectively the wellknown 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.
Also M̃k (0, β, γ) = V˜k (γ) ⊂ Vk and M̃k (α, 0, γ) = R̃k (γ) ⊂ Rk .
2. M AIN R ESULTS
Theorem 2.1. A function f ∈ M̃k (α, β, γ), α, β > 0, if and only if, there exists a function
F ∈ R̃k (γ) such that
β
# α+β
"
α+β
Z z
β
(F (t))
α+β
(2.1)
f (z) =
dt
.
α
t
0
Proof. A simple calculation yields
α zf 0 (z)
β (zf 0 (z))0
zF 0 (z)
+
=
.
α + β f (z)
α + β f 0 (z)
F (z)
If the right hand side belongs to P˜k (γ) so does the left and conversely, and the result follows.
Theorem 2.2. Let f ∈ M̃k (α, β, γ). Then the function
β
0 α+β
zf (z)
(2.2)
g(z) = f (z)
f (z)
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.
Theorem 2.3. Let f ∈ M̃k (α, β, γ),
Proof. Let
zf 0 (z)
f (z)
β > 0, 0 < γ ≤ 1. Then f ∈ R̃k (γ) for z ∈ E.
= p(z). Then
(zf 0 (z))0
zp0 (z)
=
p(z)
+
.
f 0 (z)
p(z)
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 16, 4 pp.
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A LPHA C ONVEXITY
3
Therefore, for z ∈ E,
(2.3)
β (zf 0 (z))0
α zf 0 (z)
+
=
α + β f (z)
α + β f 0 (z)
β zp0 (z)
p(z) +
α + β p(z)
∈ P˜k (γ).
Let
φ(α, β) =
(2.4)
α
z
β
z
+
.
α + β 1 − z α + β (1 − z)2
Then, using (1.3) and (2.4), we have
k 1
k 1
φ(α, β)
φ(α, β)
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,
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 + β1 ) 6= 0,
α1
α
<
,
α1 + β1
α+β
β1
β
<
α1 + β1
α+β
and
0 ≤ γ < 1.
Then
M̃k (α, β, γ) ⊂ M˜K (α1 , β1 , γ),
z ∈ E.
Proof. We can write
α1 zf 0 (z)
β1 (zf 0 (z))0
+
=
α1 + β1 f (z)
α1 + β1 f 0 (z)
β1 (α + β) zf 0 (z)
1−
β(α1 + β1 ) f (z)
β1 (α + β)
α zf 0 (z)
β (zf 0 (z))0
+
+
β(α1 + β1 )
α + β f (z)
α + β f 0 (z)
β1 (α + β)
β1 (α + β)
= 1−
H1 (z) +
H2 (z),
β(α1 + β1 )
β(α1 + β1 )
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 , β1 , γ). This completes the proof.
Theorem 2.5. Let f ∈ M̃k (α, β, γ). Then
α
Z z
β
f (t) α+β
0
α+β
dt belongs to
(2.5)
h(z) =
(f (t))
t
0
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 16, 4 pp.
V˜k (γ) for
z ∈ E.
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4
K HALIDA I NAYAT N OOR
Proof. From (2.5), we have
α
f (z) α+β
h (z) = (f (z))
.
z
Now the proof is immediate when we differentiate both sides logarithmically and use the fact
that f ∈ M̃k (α, β, γ).
0
0
β
α+β
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
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)
Let φ(α, β) be as given by (2.4). Now using (1.3) and convolution techniques, we have
φ(α, β)
β zp0 (z)
= p(z) ?
p(z) +
α + β p(z)
z
k 1
φ(α, β)
k 1
φ(α, β)
=
+
p1 (z) ?
−
−
p2 (z) ?
.
4 2
z
4 2
z
n
o
φ(α,β)
˜
Since pi ∈ P2 (1) = P and it is known [2] that Re
> 12 for |z| < r(α, β), it follows
i z
h
∈ P for |z| < r(α, β), i = 1, 2. with
from a well known result, see [5] that pi ? φ(α,β)
z
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.
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[4] K. INAYAT NOOR, On strongly alpha-convex and alpha-quasi-convex functions, J. Natural Geometry, 10(1996), 111–118.
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11–20.
[6] M. NUNOKAWA AND S. OWA, On certain differential subordination, PanAmer. Math. J., 3 (1993),
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[7] J. STANSKIEWICS, Some remarks concerning starlike functions, Bull. Acad. Polon. Sci. Ser. Sci.
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J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 16, 4 pp.
http://jipam.vu.edu.au/