Journal of Inequalities in Pure and
Applied Mathematics
ON SOME CLASSES OF ANALYTIC FUNCTIONS
KHALIDA INAYAT NOOR AND M.A. SALIM
Department of Mathematics and Computer Science
College of Science
United Arab Emirates University
P.O. Box 17551, Al-Ain
United Arab Emirates.
volume 5, issue 4, article 98,
2004.
Received 20 September, 2004;
accepted 09 October, 2004.
Communicated by: Th.M. Rassias
EMail: KhalidaN@uaeu.ac.ae
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
189-04
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Abstract
We define some classes of analytic functions related with the class of functions with bounded boundary rotation and study these classes with reference
to certain integral operators.
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words: Close-to-convex functions, Univalent functions, Bounded boundary rotation, Integral operator.
On Some Classes Of Analytic
Functions
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Khalida Inayat Noor and M.A.
Salim
3
5
7
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1.
Introduction
P
m
Let A denote the class of functions f of the form f (z) = z + ∞
m=2 am z
which are analytic in the unit disk E = {z : |z| < 1}. Let C, S ? , K and S be
the subclasses of A which are respectively convex, starlike, close-to-convex and
univalent in E. It is known that C ⊂ S ? ⊂ K ⊂ S. In [1], Kaplan showed that
f ∈ K if, and only if, for z ∈ E, 0 ≤ θ1 < θ2 ≤ 2π, 0 < r < 1,
Z θ2
reiθ f 00 (reiθ )
Re 1 +
dθ > −π, z = reiθ .
0 (reiθ )
f
θ1
Let Vk (k ≥ 2) be the class of locally univalent functions f ∈ A that map E
conformally onto a domain whose boundary rotation is at most kπ. It is well
known that V2 ≡ C and Vk ⊂ K for 2 ≤ k ≤ 4.
Definition 1.1. Let f ∈ A and f 0 (z) 6= 0. Then f ∈ Tk (λ), k ≥ 2, 0 ≤ λ < 1
if there exists a function g ∈ Vk such that, for z ∈ E
0 f (z)
Re
> λ.
g 0 (z)
The class Tk (0) = Tk was considered in [2, 3] and T2 (0) = K, the class of
close-to-convex functions.
0
Definition 1.2. Let f ∈ A and f (z)fz (z) 6= 0, z ∈ E. Then f ∈ Tk (a, γ, λ),
Re a ≥ 0, 0 ≤ γ ≤ 1 if, and only if, there exists a function g ∈ Tk (λ) such that
(1.1)
zf 0 (z) + af (z) = (a + 1)z(g 0 (z))γ ,
z ∈ E.
On Some Classes Of Analytic
Functions
Khalida Inayat Noor and M.A.
Salim
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We note that Tk (0, 1, λ) = Tk (λ) and T2 (0, 1, λ) = K(λ) ⊂ K, and it follows
that f ∈ Tk (a, γ, λ) if, and only if, there exists F ∈ Tk (∞, γ, λ) such that
Z
a + 1 z a−1
f (z) =
t F (t)dt.
za
0
On Some Classes Of Analytic
Functions
Khalida Inayat Noor and M.A.
Salim
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2.
Preliminary Results
Lemma 2.1 ([2]). Let f ∈ A. Then, for 0 ≤ θ1 < θ2 ≤ 2π, z = reiθ , 0 <
r < 1, f ∈ Tk if and only if
0
Z θ2
zf (z))0
k
Re
dθ > − π.
0
f (z)
2
θ1
Lemma 2.2. Let q(z) be analytic in E and of the form q(z) = 1 + b1 z + · · ·
for |z| = r ∈ (0, 1). Then, for a, c1 , θ1 , θ2 with a ≥ 1, Re(c1 ) ≥ 0, 0 ≤ θ1 <
θ2 ≤ 2π,
Z θ2
azq 0 (z)
Re q(z) +
dθ > −β1 π;
(0 < β1 ≤ 1)
c1 a + q(z)
θ1
implies
Z
On Some Classes Of Analytic
Functions
Khalida Inayat Noor and M.A.
Salim
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θ2
Re q(z)dθ > −β1 π,
iθ
z = re .
θ1
This result is a direct consequence of the one proved in [4] for β1 = 1.
From (1.1) and Lemma 2.1, we can easily have the following:
Lemma 2.3. A function f ∈ Tk (∞, γ, λ) if and only if, it may be represented
as f (z) = p(z) · zG0 (z), where G ∈ Vk and Re p(z) > λ, z ∈ E.
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Proof. Since f ∈ Tk (∞, γ, λ), we have
f (z) = z(g 0 (z))γ ,
g ∈ Tk (λ)
γ
0
= z [G1 (z)p1 (z)] , G1 ∈ Vk , Re p1 (z) > λ
= zG0 (z).p(z),
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where G0 (z) = (G01 (z))γ ∈ Vk and p(z) = (p1 (z))γ , Re p(z) > λ, since
0 ≤ γ ≤ 1.
The converse case follows along similar lines.
Using Lemma 2.1 and Lemma 2.3, we have:
Lemma 2.4.
(i) Let f ∈ Tk (0, γ, λ). Then, with z = reiθ , 0 ≤ θ1 < θ − 2 ≤ 2π,
Z θ2
kγ
(zf 0 (z))0
dθ > − , see also [3].
Re
0
f (z)
2
θ1
(ii) Let f ∈ Tk (∞, γ, λ). Then, for z = reiθ and 0 ≤ θ1 < θ2 ≤ 2π,
0 Z θ2
kγ
zf (z)
dθ > − .
Re
f (z)
2
θ1
On Some Classes Of Analytic
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Khalida Inayat Noor and M.A.
Salim
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3.
Main Results
Theorem 3.1. For 0 < α <
f, g ∈ Tk (∞, γ, λ), z ∈ E, let
(3.1)
1
,
1−λ+λβ
0 < β <
λ
,
1−λ
0 ≤ λ <
1
2
and
1
1+β
Z z 1
1
1− α
−2
1
F (z) =
β+
z
t α (f (t))β g(t)dt
.
α
0
Then F1 , with F = zF10 and 0 < γ < 1, k ≤ γ2 , is close-to-convex and hence
univalent in E.
On Some Classes Of Analytic
Functions
Khalida Inayat Noor and M.A.
Salim
Proof. We can write (3.1) as
Z z 1
1
1− α
−2
1
β+1
(3.2)
(F (z))
= β+
z
t α (f (t))β g(t)dt.
α
0
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Let
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zF 0 (z)
(zF10 (z))0
=
= (1 − λ)H(z) + λ,
F (z)
F10 (z)
(3.3)
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where H(z) is analytic in E and H(z) = 1 + c1 z + c2 z 2 + · · · .
We differentiate (3.2) logarithmically to obtain
0
(β + 1)
zF (z)
=
F (z)
1−
1
α
+ Rz
0
z
t
1
α −1
1
α −2
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β
(f (z)) g(z)
.
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(f (z))β (t)g(t)dt
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Using (3.2) and differentiating again, we have after some simplifications,
(1 − λ)zH
Now
Rz
0 0
1 −2
t α (f (t))β g(t)dt
+ (1 − λ)H(z)
1 −1
z α (f (z))β g(z)
β
zf 0 (z)
1
zg 0 (z)
=
·
+
·
− λ.
1 + β f (z)
β + 1 g(z)
1 −1
z α (f (z))β g(z)
Rz
0
1 −2
=
t α (f (t))β g(t)dt
1
zF 0 (z)
− 1 + (1 + β)
.
α
F (z)
Hence
−λ+
β
zf 0 (z)
1
zg 0 (z)
·
+
·
1 + β f (z)
β + 1 g(z)
On Some Classes Of Analytic
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Khalida Inayat Noor and M.A.
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(1 − λ)zH 0 (z)
= (1 − λ)H(z) +
(1 − λ)(1 + β)H(z) + ( α1 − 1) + λ(1 + β)
and we have
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(3.4)
1
β
1−λ 1+β
0
zf (z)
1
−λ +
f (z)
1+β
0
zg (z)
−λ
g(z)
1
zH 0 (z)
(1+β)(1−λ)
h 1
i.
= H(z) +
( α −1)
λ
H(z) + (1+β)(1−λ)
+ 1−λ
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Since f, g ∈ Tk (∞, γ, λ), so with z = reiθ , 0 ≤ θ1 < θ2 ≤ 2π,
0
Z θ2
1
zf (z)
β
Re
−λ
dθ
1 + β θ1
1 − λ f (z)
0
Z θ2
1
1
zg (z)
−kγ
+
Re
−λ
dθ >
π,
1 + β θ1
1 − λ g(z)
2
and, therefore,
Z
θ2

Re H(z) +
θ1

1
zH 0 (z)
(1+β)(1−λ)
H(z) +
Now using Lemma 2.2 with a =
0, we obtain the required result.
n
1
−1)
(α
(1+β)(1−λ)
1
(1+β)(1−λ)
+
λ
1−λ
≥ 1, c1 =
−kγ
o  dθ >
π.
2
1
α
On Some Classes Of Analytic
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Khalida Inayat Noor and M.A.
Salim
− 1 + (1 + β)λ ≥
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Theorem 3.2. Let f, g ∈ Tk (∞, γ, λ), α, c, δ and ν be positively real, 0 < α ≤
1
, c > α(1 − λ), (ν + δ) = α. Then the function F defined by
1−λ
Z z
α
α−c
(c−δ−ν)−1
δ
ν
(3.5)
[F (z)] = cz
t
(f (t)) (g(t)) dt
0
belongs to Tk (∞, γ, λ) for k ≤ γ2 , 0 < γ < 1.
Proof. First we show that there exists an analytic function F satisfying (3.5).
Let
G(z) = z −(ν+δ) (f (z))δ (g(z))ν
= 1 + d1 z + d2 z 2 + · · ·
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and choose the branches which equal 1 when z = 0. For
K(z) = z
(c−ν−δ)−1
δ
ν
(f (z)) (g(z)) = z
c−1
G(z),
we have
Z
c z
c
L(z) = c
K(t)dt = 1 +
d1 z + · · · .
z 0
1+c
Hence L is well-defined and analytic in E.
Now let
1
α
α1
α
F (z) = z L(z) = z [L(z)] ,
1
where we choose the branch of [L(z)] α which equals 1 when z = 0. Thus F is
analytic in E and satisfies (3.5).
Set
(3.6)
zF 0 (z)
= (1 − λ)h(z) + λ,
F (z)
and let
zf 0 (z)
= (1 − λ)h1 (z) + λ
f (z)
zg 0 (z)
= (1 − λ)h2 (z) + λ.
g(z)
Now, from (3.5), we have
h (c−δ−ν)−1
i
zF 0 (z)
(c−α)
α
δ
ν
(3.7) z
[F (z)] (c − α) + α
=c z
(f (z)) (g(z)) .
F (z)
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We differentiate (3.7) logarithmically and use (3.6) to obtain
zh0 (z)
α(1 − λ) h(z) +
+ (δ + ν − α)
(c − α) + α{λ + (1 − λ)h(z)}
zf 0 (z)
zg 0 (z)
=δ
+ν
− αλ
f (z)
g(z)
0
0
zf (z)
zg (z)
=δ
−λ +ν
−λ .
f (z)
g(z)
This gives us
h(z) +
On Some Classes Of Analytic
Functions
zh0 (z)
(c − α) + α{λ + (1 − λ)h(z)}
0
0
δ
zf (z)
ν
zg (z)
=
−λ +
−λ .
α(1 − λ) f (z)
α(1 − λ) g(z)
Khalida Inayat Noor and M.A.
Salim
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Since f, g ∈ Tk (∞, γ, λ), we have, for 0 ≤ θ1 < θ2 ≤ 2π, z = reiθ ,
Z θ2
zh0 (z)
dθ
Re h(z) +
(c − α) + α{λ + (1 − λ)h(z)}
θ1
Z θ2
Z
δ
ν θ2
=
Re h1 (z)dθ +
Re h2 (z)dθ
α θ1
α θ1
δ
γk
ν
γk
>
− π +
− π
α
2
α
2
γk
γk
δ+ν
− π = − π,
=
α
2
2
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where we have used Lemma 2.4.
Now using Lemma 2.2 with a =
1
α(1−λ)
> 1, for α <
1
1−λ
and
c1 = c − α + αλ = c − α(1 − λ) ≥ 0,
we obtain the required result.
On Some Classes Of Analytic
Functions
Khalida Inayat Noor and M.A.
Salim
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References
[1] W. KAPLAN, Close-to-convex Schlicht functions, Michigan Math. J.,
(1952), 169–185.
[2] K.I. NOOR, On a generalization of close-to-convexity, Intern. J. Math. &
Math. Sci., 6 (1983), 327–334.
[3] K.I. NOOR, Higher order close-to-convex functions, Math. Japonica, 37
(1192), 1–8.
[4] R. PARVATHAN AND S. RADHA, On certain classes of analytic functions,
Ann. Polon. Math., 49 (1988), 31–34.
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Khalida Inayat Noor and M.A.
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