Journal of Inequalities in Pure and
Applied Mathematics
ON CERTAIN CLASSES OF ANALYTIC FUNCTIONS
KHALIDA INAYAT NOOR
Mathematics Department
COMSATS Institute of Information Techonolgy
Islamabad, Pakistan
volume 7, issue 2, article 49,
2006.
Received 28 August, 2005;
accepted 21 October, 2005.
Communicated by: Th.M. Rassias
EMail: khalidanoor@hotmail.com
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
054-06
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Abstract
P
n
Let A be the class of functions f : f(z) = z + ∞
n=2 an z which are analytic in
the unit disk E. We introduce the class Bk (λ, α, ρ) ⊂ A and study some of their
interesting properties such as inclusion results and covering theorem. We also
consider an integral operator for these classes.
On Certain Classes of Analytic
Functions
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words: Analytic functions, Univalent, Functions with positive real part, Convex
functions, Convolution, Integral operator.
Khalida Inayat Noor
This research is supported by the Higher Education Commission, Pakistan, through
grant No: 1-28/HEC/HRD/2005/90.
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Contents
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
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1.
Introduction
Let A denote the class of functions
f : f (z) = z +
∞
X
an z n
n=2
which are analytic in the unit disk E = {z : |z| < 1} and let S ⊂ A be the class
of functions univalent in E.
Let Pk (ρ) be the class of functions p(z) analytic in E satisfying the properties p(0) = 1 and
Z 2π Re p(z) − ρ (1.1)
1 − ρ dθ ≤ kπ,
0
where z = rei θ, k ≥ 2 and 0 ≤ ρ < 1. This class has been introduced in [7].
We note that, for ρ = 0, we obtain the class Pk defined and studied in [8], and
for ρ = 0, k = 2, we have the well known class P of functions with positive
real part. The case k = 2 gives the class P (ρ) of functions with positive real
part greater than ρ.
From (1.1) we can easily deduce that p ∈ Pk (ρ) if, and only if, there exist
p1 , p2 ∈ P (ρ) such that, for E,
k 1
k 1
(1.2)
p(z) =
+
p1 (z) −
−
p2 (z).
4 2
4 2
P
P∞
m
m
Let f and g be analytic in E with f (z) = ∞
m=0 am z and g(z) =
m=0 bm z
On Certain Classes of Analytic
Functions
Khalida Inayat Noor
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in E. Then the convolution ? (or Hadamard Product) of f and g is defined by
(f ? g)(z) =
∞
X
am b m z m ,
m ∈ N0 = {0, 1, 2, . . .}.
m=0
Definition 1.1. Let f ∈ A. Then f ∈ Bk (λ, α, ρ) if and only if
α
α f (z)
zf 0 (z) f (z)
(1.3)
(1 − λ)
+λ
∈ Pk (ρ),
z
f (z)
z
z ∈ E,
where α > 0, λ > 0, k ≥ 2 and 0 ≤ ρ < 1. The powers are understood as
principal values.
For k = 2 and with different choices of λ, α and ρ, these classes have been
studied in [2, 3, 4, 10]. In particular B2 (1, α, ρ) is the class of Bazilevic functions studied in [1].
We shall need the following results.
Lemma 1.1 ([9]). If p(z) is analytic in E with p(0) = 1 and if λ is a complex
number satisfying Re λ ≥ 0, (λ 6= 0), then
Re[p(z) + λzp0 (z)] > β
(0 ≤ β < 1)
On Certain Classes of Analytic
Functions
Khalida Inayat Noor
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implies
Re p(z) > β + (1 − β)(2γ − 1),
where γ is given by
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Z
γ = γRe λ =
0
1
(1 + tRe λ )−1 dt.
J. Ineq. Pure and Appl. Math. 7(2) Art. 49, 2006
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Lemma 1.2 ([5]). Let c > 0, λ > 0, ρ < 1 and p(z) = 1 + b1 z + b2 z 2 + · · · be
analytic in E. Let Re[p(z) + cλzp0 (z)] > ρ in E, then
1
Re[p(z) + czp (z)] ≥ 2ρ − 1 + 2(1 − ρ) 1 −
λ
0
1
cλ
Z
0
1
1 −1
u cλ
du.
1+u
This result is sharp.
On Certain Classes of Analytic
Functions
Khalida Inayat Noor
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2.
Main Results
Theorem 2.1. Let λ, α > 0, 0 ≤ ρ < 1 and let f ∈ bk (λ, α, ρ). Then
Pk (ρ1 ), where ρ1 is given by
f (z)
z
α
∈
ρ1 = ρ + (1 − ρ)(2γ − 1),
(2.1)
and
1
Z
γ=
λ
1 + tα
−1
dt.
On Certain Classes of Analytic
Functions
0
Proof. Let
f (z)
z
α
= p(z) =
k 1
+
4 2
p1 (z) −
k 1
−
4 2
Khalida Inayat Noor
p2 (z).
Then p(z) = 1 + αa2 z + · · · is analytic in E, and
(2.2)
(f (z))α = z α p(z).
Differentiation of (2.2) and some computation give us
α
α
zf 0 (z) f (z)
λ
f (z)
(1 − λ)
+λ
= p(z) + zp0 (z).
z
f (z)
z
α
Since f ∈ Bk (λ, α, ρ), so {p(z) + αλ zp0 (z)} ∈ Pk (ρ) for z ∈ E. This implies
that
λ 0
Re pi (z) + zpi (z) > ρ, i = 1, 2.
α
Using Lemma 1.1, we see that Re{pi (z)} > ρ1 , where ρ1 is given by (2.1).
Consequently p ∈ Pk (ρ1 ) for z ∈ E, and the proof is complete.
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Corollary 2.2. Let f = zF10 and f ∈ B2 (λ, 1, ρ). Then F1 is univalent in E.
Proceeding as in Theorem 2.1 and using Lemma 1.2, we have the following.
Theorem 2.3. Let α > 0, λ > 0, 0 ≤ ρ < 1 and let f ∈ Bk (λ, α, ρ). Then
zf 0 (z) f (z) α
( z ) ∈ Pk (ρ2 ), where
f (z)
α −1
Z
1−ρ
1 α 1 uλ
+ 2(1 − ρ) 1 −
du.
ρ2 = 2ρ − 1 +
λ
λ λ 0 1+u
This result is sharp.
On Certain Classes of Analytic
Functions
Khalida Inayat Noor
For k = 2, we note that f is univalent, see [1].
Theorem 2.4. Let, for α > 0, λ > 0, 0 ≤ ρ < 1, f ∈ Bk (λ, α, ρ) and define
I(f ) : A −→ A as
(2.3)
1 α− λ1
I(f ) = F (z) =
z
λ
Z
z
t
1 −1−α
λ
α1
(f (z)) dt ,
α
z ∈ E.
0
Then F ∈ Bk (αλ, α, ρ1 ) for z ∈ E, where ρ1 is given by (2.1).
Proof. Differentiating (2.3), we have
α
α α
zF 0 (z) F (z)
f (z)
F (z)
(1 − αλ)
+ αλ
=
.
z
F (z)
z
z
Now, using Theorem 2.1, we obtain the required result.
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Theorem 2.5. Let
f : f (z) = z +
∞
X
an z n ∈ Bk (λ, α, ρ).
n=2
Then
|an | ≤
k(1 − ρ)
.
λ+α
The function fλ,α,ρ (z) defined as
fλ,α,ρ (z)
z
α
α
=
λ
Z
1
0
On Certain Classes of Analytic
Functions
α
k 1
1 + (1 − 2ρ)uz
+
u λ −1
4 2
1 − uz
α
k 1
1 − (1 − 2ρ)uz
−1
−
−
uλ
du
4 2
1 + uz
Khalida Inayat Noor
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shows that this inequality is sharp.
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Proof. Since f ∈ Bk (λ, α, ρ), so
(1 − λ) 1 +
∞
X
n=2
!α
an z n−1
+λ 1+
∞
X
!
nan z n−1
1+
n=2
= H(z) =
∞
X
!α
an z n−1
n=2
1+
∞
X
!
cn z n
∈ Pk (ρ).
n=1
It is known that |cn | ≤ k(1 − ρ) for all n and using this inequality, we prove the
required result.
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Different choices of k, λ, α and ρ yield several known results.
Theorem 2.6 (Covering Theorem). Let λ > 0 and 0 < ρ < 1. Let f = zF10 ∈
B2 (λ, 1, ρ). If D is the boundary of the image of E under F1 , then every point
λ+1
of D has a distance of at least (3+2λ−ρ)
from the origin.
Proof. Let F1 (z) 6= w0 , w0 6= 0. Then f1 (z) =
F1 is univalent. Let
f (z) = z +
∞
X
n
an z ,
w0 F1 (z)
w0 +F1 (z)
F1 (z) = z +
n=2
is univalent in E since
∞
X
bn z n .
n=2
On Certain Classes of Analytic
Functions
Khalida Inayat Noor
Then a2 = 2b2 . Also
1
f1 (z) = z + b2 +
w0
and so |b2 +
λ+1
.
3+2λ−ρ
1
|
w0
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2
z + ··· ,
≤ 2. Since, by Theorem 2.5, |b2 | ≤
1−ρ
,
1+λ
Contents
we obtain |w0 | ≥
Theorem 2.7. For each α > 0, Bk (λ1 , α, ρ) ⊂ Bk (λ2 , α, ρ) for 0 ≤ λ2 < λ1 .
Proof. For λ2 = 0, the proof is immediate. Let λ2 > 0 and let f ∈ Bk (λ1 , α, ρ).
Then there exist two functions h1 , h2 ∈ Pk (ρ) such that, from Definition 1.1 and
Theorem 2.1,
α
α
f (z)
zf 0 (z) f (z)
(1 − λ)
+ λ1
= h1 (z),
z
f (z)
z
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and
f (z)
z
α
= h2 (z).
Hence
(2.4) (1−λ2 )
f (z)
z
α
zf 0 (z)
+λ2
f (z)
f (z)
z
α
λ2
λ2
= h1 (z)+ 1 −
h2 (z).
λ1
λ1
Since the class Pk (ρ) is a convex set, see [6], it follows that the right hand side
of (2.4) belongs to Pk (ρ) and this proves the result.
On Certain Classes of Analytic
Functions
Khalida Inayat Noor
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References
[1] I.E. BAZILEVIC, On a class of integrability in quadratures of the
Loewner-Kuarev equation, Math. Sb., 37 (1955), 471–476.
[2] M.P. CHEN, On the regular functions satisfying Re f (z)
> α, Bull. Inst.
z
Math. Acad. Sinica, 3 (1975), 65–70.
[3] P.N. CHICHRA, New Subclass of the class of close-to-convex functions,
Proc. Amer. Math. Soc., 62 (1977), 37–43.
[4] S.S. DING, Y. LING AND G.J. BAO, Some properties of a class of analytic
functions, J. Math. Anal. Appl., 195 (1995), 71–81.
[5] L. MING SHENG, Properties for some subclasses of analytic functions,
Bull. Inst. Math. Acad. Sinica, 30 (2002), 9–26
[6] K. INAYAT NOOR, On subclasses of close-to-convex functions of higher
order, Internat. J. Math. and Math. Sci., 15 (1992), 279–290.
[7] K. PADMANABHAN AND R. PARVATHAM, Properties of a class of
functions with bounded boundary rotation, Ann. Polon. Math., 31 (1975),
311–323.
On Certain Classes of Analytic
Functions
Khalida Inayat Noor
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[8] B. PINCHUCK, Functions with bounded boundary rotation, Isr. J. Math.,
10 (1971), 7–16.
[9] S. PONNUSAMY, Differential subordination and Bazilevic functions,
Preprint.
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J. Ineq. Pure and Appl. Math. 7(2) Art. 49, 2006
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[10] S. OWA AND M. OBRADOVIC, Certain subclasses of Bazilevic functions
of type α, Internat. J. Math. and Math. Sci., 9 (1986), 97–105.
On Certain Classes of Analytic
Functions
Khalida Inayat Noor
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