MATEMATIQKI VESNIK
originalni nauqni rad
research paper
63, 1 (2011), 55–58
March 2011
RADIUS ESTIMATES OF A SUBCLASS
OF UNIVALENT FUNCTIONS
Maslina Darus and Rabha W. Ibrahim
Abstract. For analytic functions f normalized by f (0) = f 0 (0) − 1 = 0 in the open unit
z
disk U , a class Pα (λ) of f defined by |Dzα ( f (z)
)| ≤ λ, where Dzα denotes the fractional derivative
of order α, m ≤ α < m + 1, m ∈ N0 , is introduced. In this article, we study the problem when
1
f (rz) ∈ Pα (λ), 3 ≤ α < 4.
r
1. Introduction
Let H be the class of functions analytic in U := {z ∈ C : |z| < 1} and
H[a, n] be the subclass of H consisting of functions of the form f (z) = a + an z n +
an+1 z n+1 + · · · . Let A be the subclass of H consisting of functions of the form
f (z) = z +
∞
P
n=2
an z n ,
z ∈ U.
(1)
Let S be the subclass of A consisting of all univalent functions f (z) in U .
In [1], Srivastava and Owa, gave definitions for fractional operators (derivative
and integral) in the complex z-plane C as follows:
Definition 1.1. The fractional derivative Dzα of order α is defined, for a
function f (z), by
(
R z f (ζ)
d
1
0 ≤ α < 1,
Γ(1−α) dz 0 (z−ζ)α dζ,
α
Dz f (z) =
d m+1 α−m
( dz
)
Dz
f (z),
m ≤ α < m + 1, m ∈ N0 .
where the function f (z) is analytic in simply-connected region of the complex zplane C containing the origin and the multiplicity of (z − ζ)−α is removed by
requiring log(z − ζ) to be real when (z − ζ) > 0.
2010 AMS Subject Classification: 30C45.
Keywords and phrases: Analytic functions, univalent functions, Cauchy-Schwarz inequality,
fractional differential operator.
The work presented here was supported by UKM-ST-06-FRGS0107-2009.
56
M. Darus, R. W. Ibrahim
For f (z) ∈ A, we define the class Pα (λ) of f (z) if f (z) satisfies f (z)
6= 0,
z
(z ∈ U ) and
¯ ³ z ´¯
¯ α
¯
(2)
¯D z
¯ ≤ λ, z ∈ U,
f (z)
for some real λ > 0 and m ≤ α < m + 1, m ∈ N0 .
Obradović and Ponnusamy [2] have studied the subclass P2 (λ) for f (z) ∈ A
¯¡ z ¢00 ¯
¯
¯ ≤ λ, (z ∈ U ) for some real λ > 0.
satisfying f (z)
z 6= 0, (z ∈ U ) and
f (z)
f (z)
z
Recently, Kuroki et al. studied the subclass P3 (λ) for f (z) ∈ A satisfying
¯¡ z ¢000 ¯
¯ ≤ λ, (z ∈ U ) for some real λ > 0 (see [3]).
6= 0, (z ∈ U ) and ¯ f (z)
In this work, we study the problem when 1r f (rz) ∈ Pα (λ), 3 ≤ α < 4. For this
purpose, we need the following result.
Lemma 1.1. [4] If f (z) ∈ S and
∞
P
z
=1+
bn z n ,
f (z)
n=1
then
∞
P
(3)
(n − 1)|bn |2 ≤ 1.
n=1
2. Results
First we derive the following result.
P∞
z
Theorem 2.1. Let f ∈ A and f (z)
= 1 + n=1 bn z n 6= 0, (z ∈ U ). If f (z)
satisfies
∞
P
n(n − 1)(n − 2) · · · (n − m)|bn | ≤ λ,
n=m+1
(λ > 0, m ≤ α < m + 1, m ∈ N0 ), then f (z) ∈ Pα (λ).
Proof. By Definition 1.1, we observe
Z z
¯ ³ z ´¯ P∞
¯ α
¯
n=m+1 n(n − 1)(n − 2) · · · (n − m)|bn |
D
|(z − ζ)|m−α dζ
≤
¯ z
¯
f (z)
Γ(m + 1 − α)
0
P∞
n=m+1 n(n − 1)(n − 2) · · · (n − m)|bn |
≤
Γ(m + 2 − α)
∞
P
<
n(n − 1)(n − 2) · · · (n − m)|bn | ≤ λ.
n=m+1
Hence, f (z) ∈ Pα (λ).
P∞
z
Corollary 2.2. Let f ∈ A and f (z)
= 1 + n=1 bn z n 6= 0, (z ∈ U ). If f (z)
satisfies
∞
P
n(n − 1)(n − 2)(n − 3)|bn | ≤ λ,
n=4
(λ > 0, 3 ≤ α < 4), then f (z) ∈ Pα (λ).
57
Radius estimates of a subclass of univalent functions
Proof. By letting m = 3 in Theorem 2.1.
Theorem 2.3. Let f ∈ S and λ > 0. Then the function 1r f (rz), (r > 0,
z ∈ U ) belongs to the class Pα (λ) for 3 ≤ α < 4 and 0 < r ≤ r0 (λ), where r0 (λ) is
the smallest root of the equation
¡
F (r) := r2 A1 (r) − 11(1 − r2 )A2 (r) + 47(1 − r2 )2 A3 (r) − 97(1 − r2 )3 A4 (r)
¢
+ 96(1 − r2 )4 A5 (r) − 36(1 − r2 )5 A6 (r) − λ2 (1 − r2 )8 = 0, (4)
where
A1 (r) := 5 + 424r2 + 2989r4 + 3544r6 + 989r8 + 88r10 ,
A2 (r) := 5 + 197r2 + 668r4 + 268r6 + 14r8 + 9r10 ,
A3 (r) := 5 + 86r2 + 108r4 − 14r6 + 9r8 ,
A4 (r) := 5 + 23r2 − r4 − 3r6 ,
A5 (r) := 1 + 4r2 + r4 ,
A6 (r) := 1 + r2 ,
in the interval (0, 1).
Proof. Let f ∈ S. Since
then we have
z
f (z)
6= 0, (z ∈ U ), if we write
z
1
r f (rz)
=1+
∞
P
n=1
z
f (z)
=1+
P∞
n=1 bn z
n
,
(rn bn )z n ,
for 0 < r < 1. It follows from Lemma 1.1 that
∞
P
(n − 1)|bn |2 ≤
n=4
∞
P
(n − 1)|bn |2 ≤ 1.
n=1
To verify that 1r f (rz) ∈ Pα (λ) for 3 ≤ α < 4, we have to show that
∞
P
n=4
n(n − 1)(n − 2)(n − 3)|rn bn | ≤ λ,
(λ > 0,
3 ≤ α < 4)
(5)
by mean of Corollary 2.2. Now by the Cauchy-Schwarz inequality for the left-hand
side of (5), we have
∞
P
n=4
n(n − 1)(n − 2)(n − 3)|rn bn |
=
∞ ³
P
n2 (n − 1)(n − 2)2 (n − 3)2 |rn |2
´ 21 ³
n=4
≤
³P
∞
n2 (n − 1)(n − 2)2 (n − 3)2 |rn |2
n=4
≤
³P
∞
n=4
n2 (n − 1)(n − 2)2 (n − 3)2 r2n
(n − 1)|bn |2
´ 12 ³ P
∞
´ 12
´ 12
(n − 1)|bn |2
n=4
´ 21
58
M. Darus, R. W. Ibrahim
=
h ³
1
r2 A1 (r) − 11(1 − r2 )A2 (r) + 47(1 − r2 )2 A3 (r)
(1 − r2 )4
− 97(1 − r2 )3 A4 (r) + 96(1 − r2 )4 A5 (r) − 36(1 − r2 )5 A6 (r)
´i 21
.
Consequently, 1r f (rz) ∈ Pα (λ) for 3 ≤ α < 4 and 0 < r ≤ r0 (λ), where r0 (λ) is
the positive solution for the equation (4), which satisfies that F (0) = −λ2 < 0
and F (1) = A1 (1) > 0. Hence (4) has a solution r0 (λ) in the interval (0, 1). This
completes the proof of the theorem.
REFERENCES
[1] H.M. Srivastava, S. Owa, Univalent Functions, Fractional Calculus, and Their Applications,
Halsted Press, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1989.
[2] M. Obradović, S. Ponnusamy, Radius properties for subclasses of univalent functions, Analysis
25 (2005), 183–188.
[3] H. Kobashi, K. Kuroki, S. Owa, Notes on radius problems of certain univalent functions,
General Mathematics 17 (2009), 5–12.
[4] A.W. Goodman, Univalent Functions, Vols. I and II, Mariner, Tampa, Florida, 1983.
(received 10.01.2010; in revised form 13.10.2010)
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan
Malaysia, Bangi 43600, Selangor Darul Ehsan, Malaysia
E-mail: maslina@ukm.my, rabhaibrahim@yahoo.com