Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 312387, 8 pages
http://dx.doi.org/10.1155/2013/312387
Research Article
On Certain Subclass of Analytic Functions with Fixed Point
T. Al-Hawary,1 B. A. Frasin,2 and M. Darus1
1
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
43600 Bangi, Selangor, Malaysia
2
Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 130095, Jordan
Correspondence should be addressed to M. Darus; maslina@ukm.my
Received 14 October 2012; Revised 27 December 2012; Accepted 27 December 2012
Academic Editor: Yongkun Li
Copyright © 2013 T. Al-Hawary et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
𝑚
∗
𝑓(𝑧) and a subclass 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) are introduced for functions of the form 𝑓(𝑧) =
A new certain differential operator 𝑇𝛼,𝛽,𝛿,𝜆
∞
𝑛
(𝑧 − 𝑤) − ∑𝑛=2 𝑎𝑛 (𝑧 − 𝑤) which are univalent in the unit disc 𝑈 = {𝑧 ∈ C : |𝑧| < 1}. In this paper, we obtain coefficient inequalities,
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆).
distortion theorem, closure theorems, and class preserving integral operators of functions belonging to the class 𝑆𝑚,𝑤
1. Introduction
Let 𝐻(𝑈) be the set of functions which are regular in the unit
disc 𝑈 = {𝑧 ∈ C : |𝑧| < 1}, 𝐴 = {𝑓 ∈ 𝐻(𝑈) : 𝑓(0) =
𝑓󸀠 (0) − 1 = 0} and 𝑆 = {𝑓 ∈ 𝐴 : 𝑓 is univalent in 𝑈}.
We recall here the definitions of the well-known classes of
starlike and convex functions:
𝑆∗ = {𝑓 ∈ 𝐴 : Re (
𝑧𝑓󸀠 (𝑧)
) > 0, 𝑧 ∈ 𝑈} ,
𝑓 (𝑧)
𝑧𝑓󸀠󸀠 (𝑧)
) > 0, 𝑧 ∈ 𝑈} .
𝑆 = {𝑓 ∈ 𝐴 : Re (1 + 󸀠
𝑓 (𝑧)
(1)
𝑐
Let 𝑤 be a fixed point in 𝑈 and 𝐴(𝑤) = {𝑓 ∈ 𝐻(𝑈) :
𝑓(𝑤) = 𝑓󸀠 (𝑤) − 1 = 0}.
In [1], Kanas and Rønning introduced the following
classes:
𝑆𝑤 = {𝑓 ∈ 𝐴 (𝑤) : 𝑓 is univalent in 𝑈} ,
∗
= {𝑓 ∈ 𝑆𝑤 : Re (
𝑆𝑇𝑤 = 𝑆𝑤
(𝑧 − 𝑤) 𝑓󸀠 (𝑧)
) > 0, 𝑧 ∈ 𝑈} ,
𝑓 (𝑧)
𝑐
𝑆𝑉𝑤 = 𝑆𝑤
= {𝑓 ∈ 𝑆𝑤 : 1+Re (
(𝑧−𝑤) 𝑓󸀠󸀠 (𝑧)
) > 0, 𝑧 ∈ 𝑈} .
𝑓󸀠 (𝑧)
(2)
These classes are extensively studied by Acu and Owa [2].
∗
is defined by geometric property that the
The class 𝑆𝑤
image of any circular arc centered at 𝑤 is starlike with respect
𝑐
is defined by the
to 𝑓(𝑤) and the corresponding class 𝑆𝑤
property that the image of any circular arc centered at 𝑤 is
convex. We observe that the definitions are somewhat similar
to the ones for uniformly starlike and convex functions
introduced by Goodman in [3, 4], except that in this case the
point 𝑤 is fixed.
In fact, several subclasses of 𝑆 have been introduced by a
fixed geometric property of the image domain. It is interesting
to note that these subclasses play an important role in other
branches of mathematics. For example, starlike functions and
convex functions play an important role in the solution of
certain differential equations (see Robertson [5], Saitoh [6],
Owa et al. [7], Reade and Silverman [8], and Sokół and
Wiśniowska-Wajnryb [9]). One of the important problems
in geometric function theory are the extremal problems,
which pose an effective method for establishing the existence of analytic functions with certain natural properties.
Extremal problems (supremum Re{𝑓}) play an important role
in geometric function theory, for finding sharp estimates,
coefficient bounds, and an extremal function. The results
we obtained here may have potential application in other
branches of mathematics, both pure and applied. For example
the extremal problems are closely connected to Hele-Shaw
flow of fluid mechanics [10], exterior inverse problems in
potential theory, and many others.
2
Journal of Applied Mathematics
Now let us begin with our definitions as follows.
The function 𝑓(𝑧) in 𝑆𝑤 is said to be starlike functions of
order 𝛾 if and only if
Re {
(𝑧 − 𝑤) 𝑓󸀠 (𝑧)
}>𝛾
𝑓 (𝑧)
(𝑧 ∈ 𝑈)
(3)
for some 𝛾 (0 ≤ 𝛾 < 1). We denote by 𝑆𝑇𝑤 (𝛾) the class of all
starlike functions of order 𝛾. Similarly, a function 𝑓(𝑧) in 𝑆𝑤
is said to be convex of order 𝛾 if and only if
Re {1 +
(𝑧 − 𝑤) 𝑓󸀠󸀠 (𝑧)
}>𝛾
𝑓󸀠 (𝑧)
(𝑧 ∈ 𝑈)
(4)
for some 𝛾 (0 ≤ 𝛾 < 1). We denote by 𝑆𝑉𝑤 (𝛾) the class of all
convex functions of order 𝛾.
It is easy to see that 𝑆𝑤 denoting the subclass of 𝐴(𝑤) has
the series of expansion:
Remark 1. (i) When 𝑤 = 𝛿 = 0, we have the operator
introduced and studied by Darus and Ibrahim (see [11]).
(ii) When 𝑤 = 𝛼 = 𝛿 = 0 and 𝛽 = 1, we have the operator
introduced and studied by Al-Oboudi (see [12]).
(iii) When 𝛼 = 𝛿 = 0 and 𝜆 = 𝛽 = 1, we have the operator
introduced and studied by Acu and Owa (see [2]).
(iv) And when 𝛼 = 𝛿 = 𝑤 = 0 and 𝜆 = 𝛽 = 1, we have the
̆
operator introduced and studied by Sal̆ agean
(see [13]).
𝑚
, we
With the help of the differential operator 𝑇𝛼,𝛽,𝛿,𝜆
define the class 𝑆𝑚,𝑤 (𝛼, 𝛽, 𝛾, 𝛿, 𝜆) as follows.
Definition 2. A function 𝑓(𝑧) ∈ 𝑆𝑤 is said to be a member of
the class 𝑆𝑚,𝑤 (𝛼, 𝛽, 𝛾, 𝛿, 𝜆) if it satisfies
󵄨󵄨
󵄨󵄨
󸀠
𝑚
󵄨󵄨 (𝑧 − 𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
󵄨󵄨
𝑓 (𝑧))
󵄨󵄨
󵄨
− 1󵄨󵄨󵄨
󵄨󵄨
𝑚
󵄨󵄨
󵄨󵄨
𝑇
𝑓
(𝑧)
𝛼,𝛽,𝛿,𝜆
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󸀠
𝑚
󵄨󵄨 (𝑧 − 𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
󵄨󵄨
𝑓 (𝑧))
󵄨󵄨
󵄨
< 󵄨󵄨
+ 1 − 2𝛾󵄨󵄨󵄨
𝑚
󵄨󵄨
󵄨󵄨
𝑇
𝑓
(𝑧)
𝛼,𝛽,𝛿,𝜆
󵄨󵄨
󵄨󵄨
∞
𝑓 (𝑧) = (𝑧 − 𝑤) + ∑ 𝑎𝑛 (𝑧 − 𝑤)𝑛 .
(5)
𝑛=2
For the function 𝑓(𝑧) in the class 𝑆𝑤 , we define the
following new differential operator:
0
𝑇 𝑓 (𝑧) = 𝑓 (𝑧) ,
1
𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧) = (1 − 𝛽 (𝜆 − 𝛼)) 𝑓 (𝑧)
+ 𝛽 (𝜆 − 𝛼) (𝑧 − 𝑤) 𝑓󸀠 (𝑧) + 𝛿(𝑧 − 𝑤)2 𝑓󸀠󸀠 (𝑧) ,
2
1
𝑓 (𝑧) = (1 − 𝛽 (𝜆 − 𝛼)) (𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧))
𝑇𝛼,𝛽,𝛿,𝜆
for some 0 ≤ 𝛾 < 1, 𝛼 ≥ 0, 𝛽 ≥ 0, 𝛿 ≥ 0, 𝜆 ≥ 0, and 𝑚 ∈ N0
and for all 𝑧 ∈ 𝑈.
It is easy to check that 𝑆0,0 (0, 1, 𝛾, 0, 1) is the class of
starlike functions of order 𝛾 and 𝑆0,0 (0, 1, 0, 0, 1) gives the all
class of starlike functions.
Let 𝑇𝑤 denote the subclass of 𝐴(𝑤) consisting of functions
of the following form:
∞
𝑓 (𝑧) = (𝑧 − 𝑤) − ∑ 𝑎𝑛 (𝑧 − 𝑤)𝑛 ,
󸀠
1
𝑓 (𝑧))
+ 𝛽 (𝜆 − 𝛼) (𝑧 − 𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
𝑎𝑛 ≥ 0.
𝑛=2
󸀠󸀠
1
+ 𝛿(𝑧 − 𝑤)2 (𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧)) ,
(9)
(10)
∗
Further, we define the class 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) by
(6)
and for 𝑚 = 1, 2, 3, . . ..
(11)
In this paper, coefficient inequalities, distortion theorem,
and closure theorems of functions belonging to the class
∗
𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) are obtained. Finally, the class preserving
integral operators of the form
𝑚
𝑚−1
𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧) = (1 − 𝛽 (𝜆 − 𝛼)) (𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧))
󸀠
𝑚−1
𝑓 (𝑧))
+ 𝛽 (𝜆 − 𝛼) (𝑧 − 𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
󸀠󸀠
𝑚−1
+ 𝛿(𝑧 − 𝑤)2 (𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧))
∞
∗
𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) = 𝑆𝑚,𝑤 (𝛼, 𝛽, 𝛾, 𝛿, 𝜆) ∩ 𝑇𝑤 .
𝐹 (𝑧) =
= (𝑧 − 𝑤) + ∑ [1 + (𝑛 − 1) (𝛽 (𝜆 − 𝛼) + 𝑛𝛿)]
𝑚
𝑧
𝑐+1
𝑐−1
𝑓 (𝑡) 𝑑𝑡
𝑐 ∫ (𝑡 − 𝑤)
(𝑧 − 𝑤) 𝑤
(𝑐 > −1)
(12)
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) is considered.
for the class 𝑆𝑚,𝑤
𝑛=2
2. Coefficient Inequalities
× 𝑎𝑛 (𝑧 − 𝑤)𝑛 ,
(7)
for 𝛼 ≥ 0, 𝛽 ≥ 0, 𝛿 ≥ 0, 𝜆 ≥ 0, and 𝑚 ∈ N0 = N ∪ {0}.
It easily verified from (7) that
𝛽 (𝜆 − 𝛼) (𝑧 −
Theorem 3. A function 𝑓(𝑧) being defined by (10) is in
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) if and only if
𝑆𝑚,𝑤
󸀠
𝑚
𝑓 (𝑧))
𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
𝑚+1
𝑚
= 𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧) − (1 − 𝛽 (𝜆 − 𝛼)) 𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧)
2
− 𝛿(𝑧 − 𝑤)
󸀠󸀠
𝑚
(𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧)) .
Our first result provides a sufficient condition for a function,
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆).
regular in 𝑈, to be in 𝑆𝑚,𝑤
∞
(8)
𝑚
∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛 ≤ 1 − 𝛾
𝑛=2
(𝑚 ∈ N0 ) ,
(13)
Journal of Applied Mathematics
3
where 0 ≤ 𝛾 < 1. The result (13) is sharp for functions of the
following form:
𝑓𝑛 (𝑧) = (𝑧 − 𝑤)
−
1−𝛾
𝑛
(14)
𝑚 (𝑧−𝑤)
(𝑛−𝛾) [1+(𝑛−1) 𝛽 ((𝜆−𝛼)+𝑛𝛿)]
(𝑛 ≥ 2; 𝑚 ∈ N0 ) .
Proof. Suppose that (13) holds true for 0 ≤ 𝛾 < 1. consider
the expression
󸀠
𝑀 (𝑓, 𝑓 )
󵄨󵄨
󵄨󵄨
󸀠
𝑚
𝑚
𝑓 (𝑧)) − (𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧))󵄨󵄨󵄨
= 󵄨󵄨󵄨(𝑧 − 𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
󵄨
󵄨
󵄨󵄨
󵄨󵄨
󸀠
𝑚
𝑚
− 󵄨󵄨󵄨(𝑧 − 𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧)) + (1 − 2𝛾) (𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧))󵄨󵄨󵄨 .
󵄨
󵄨
(15)
For the converse, assume that
󵄨󵄨
󵄨󵄨
󸀠
𝑚
󵄨󵄨
󵄨󵄨 (𝑧 − 𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
𝑓 (𝑧))
󵄨
󵄨󵄨
− 1󵄨󵄨󵄨
󵄨󵄨
𝑚
󵄨󵄨
󵄨󵄨
𝑇
𝑓
(𝑧)
𝛼,𝛽,𝛿,𝜆
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󸀠
𝑚
󵄨󵄨 (𝑧 − 𝑤) (𝑇𝛼,𝛽,𝛿,𝜆
󵄨󵄨
𝑓 (𝑧))
󵄨󵄨
󵄨󵄨
/ 󵄨󵄨
+
1
−
2𝛾
󵄨󵄨
𝑚
󵄨󵄨
󵄨󵄨
𝑇
𝑓
(𝑧)
𝛼,𝛽,𝛿,𝜆
󵄨󵄨
󵄨󵄨
󵄨󵄨
∞
󵄨󵄨
𝑚
= 󵄨󵄨󵄨 (− ∑ (𝑛 − 1) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
󵄨󵄨
𝑛=2
󵄨
× 𝑎𝑛 (𝑧 − 𝑤)𝑛 )
(17)
× (2 (1 − 𝛾) (𝑧 − 𝑤)
Then for |𝑧 − 𝑤| = 𝑟 < 1, we have
∞
− ∑ (𝑛−2𝛾+1) [1+(𝑛−1) 𝛽 ((𝜆−𝛼)+𝑛𝛿)]
󸀠
𝑀 (𝑓, 𝑓 )
𝑛=2
󵄨󵄨 ∞
󵄨󵄨
𝑚
= 󵄨󵄨󵄨 ∑ (𝑛 − 1) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
󵄨󵄨𝑛=2
󵄨
󵄨󵄨
󵄨󵄨
× 𝑎𝑛 (𝑧 − 𝑤)𝑛 󵄨󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨
󵄨󵄨
− 󵄨󵄨󵄨 2 (1 − 𝛾) (𝑧 − 𝑤)
󵄨󵄨
󵄨
∞
− ∑ (𝑛−2𝛾+1) [1+(𝑛−1) 𝛽 ((𝜆−𝛼)+𝑛𝛿)]
−1 󵄨󵄨
󵄨󵄨
× 𝑎𝑛 (𝑧 − 𝑤) ) 󵄨󵄨󵄨
󵄨󵄨
󵄨
𝑛
< 1.
Since Re(𝑧 − 𝑤) ≤ |𝑧 − 𝑤| for all (𝑧 − 𝑤), it follows from (17)
that
𝑚
𝑛=2
𝑀 (𝑓, 𝑓󸀠 )
𝑚
∞
󵄨󵄨
󵄨󵄨
× 𝑎𝑛 (𝑧 − 𝑤)𝑛 󵄨󵄨󵄨 ,
󵄨󵄨
󵄨
Re { ( ∑ (𝑛 − 1) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
𝑚
𝑛=2
× 𝑎𝑛 (𝑧 − 𝑤)𝑛 )
∞
≤ ∑ (𝑛 − 1)
× (2 (1 − 𝛾) (𝑧 − 𝑤)
𝑛=2
𝑚
× [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛 𝑟𝑛
∞
− ∑ (𝑛 − 2𝛾 + 1) [1+(𝑛−1) 𝛽 ((𝜆−𝛼)+𝑛𝛿)]
∞
− 2 (1 − 𝛾) + ∑ (𝑛 + 2𝛾 − 1)
𝑛=2
𝑛=2
𝑚
𝑛
≤ ∑ 2 (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
−1
× 𝑎𝑛 (𝑧 − 𝑤) ) }
𝑛
× [1 + 𝜆 (𝑛 + 𝑝 − 2)] 𝑎𝑛 𝑟
∞
𝑚
𝑚
< 1.
(18)
𝑛=2
× 𝑎𝑛 − 2 (1 − 𝛾) ≤ 0.
(16)
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆).
So that 𝑓(𝑧) ∈ 𝑆𝑚,𝑤
Choose values of (𝑧 − 𝑤) on the real axis so that
𝑚
𝑚
(𝑧 − 𝑤)(𝑇𝛼,𝛽,𝛿,𝜆
𝑓(𝑧))󸀠 /𝑇𝛼,𝛽,𝛿,𝜆
𝑓(𝑧) is real. Upon clearing the
4
Journal of Applied Mathematics
denominator in (18) and letting 𝑟 → 1− through real values,
we obtain
∞
Thus, for |𝑧 − 𝑤| = 𝑟 < 1, and making use of (22), we have
∞
󵄨
󵄨󵄨
𝑛
󵄨󵄨𝑓 (𝑧)󵄨󵄨󵄨 ≤ |𝑧 − 𝑤| + ∑ 𝑎𝑛 |𝑧 − 𝑤|
𝑛=2
𝑚
∑ (𝑛 − 1) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛
∞
𝑛=2
≤ 𝑟 + 𝑟2 ∑ 𝑎𝑛
< 2 (1 − 𝛾)
𝑛=2
∞
≤𝑟+
𝑚
− ∑ (𝑛 − 2𝛾 + 1) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛 .
𝑛=2
(19)
1−𝛾
2
𝑚𝑟 ,
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)]
∞
(23)
󵄨
󵄨󵄨
𝑛
󵄨󵄨𝑓 (𝑧)󵄨󵄨󵄨 ≥ |𝑧 − 𝑤| − ∑ 𝑎𝑛 |𝑧 − 𝑤|
𝑛=2
This gives the required condition. Hence the theorem follows.
∞
≥ 𝑟 − 𝑟2 ∑ 𝑎𝑛
𝑛=2
Corollary 4. Let the function 𝑓(𝑧) be defined by (10) and
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆), then
𝑓(𝑧) ∈ 𝑆𝑤 . If 𝑓 ∈ 𝑆𝑚,𝑤
𝑎𝑛 ≤
1−𝛾
𝑚,
(𝑛 − 𝛾) [1 + (𝑛 − 1)𝛽((𝜆 − 𝛼) + 𝑛𝛿)]
≥𝑟−
1−𝛾
2
𝑚𝑟 .
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)]
Also from Theorem 3, it follows that
𝑛 ≥ 2.
(20)
𝑚
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)] ∞
∑ 𝑛𝑎𝑛
2
𝑛=2
∞
The result (20) is sharp for functions 𝑓𝑛 (𝑧) given by (14).
𝑚
≤ ∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛
(24)
𝑛=2
≤ 1 − 𝛾.
3. Distortion Theorem
A distortion property for functions in the class 𝑓
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) is contained in
𝑆𝑚,𝑤
∈
Theorem 5. Let the function 𝑓(𝑧) one has defined by (10) be
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆). Then for |𝑧 − 𝑤| = 𝑟 < 1, we
in the class 𝑓 ∈ 𝑆𝑚,𝑤
have
Hence
󵄨󵄨 󸀠 󵄨󵄨
󵄨󵄨𝑓 (𝑧)󵄨󵄨
󵄨
󵄨
∞
≤ 1 + ∑ 𝑛𝑎𝑛 |𝑧 − 𝑤|𝑛−1
𝑛=2
∞
≤ 1 + 𝑟 ∑ 𝑛𝑎𝑛
1−𝛾
2
𝑟−
𝑚𝑟
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)]
󵄨
󵄨
≤ 󵄨󵄨󵄨𝑓 (𝑧)󵄨󵄨󵄨
≤𝑟+
1−
𝑛=2
≤1+
1−𝛾
2
𝑚𝑟 ,
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)]
2 (1 − 𝛾)
(21)
𝑚𝑟
(25)
∞
≥ 1 − ∑ 𝑛𝑎𝑛 |𝑧 − 𝑤|𝑛−1
𝑛=2
∞
≥ 1 − 𝑟 ∑ 𝑛𝑎𝑛
𝑛=2
2 (1 − 𝛾)
𝑚 𝑟.
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)]
≥1−
∗
Proof. Since 𝑓 ∈ 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆), Theorem 3 readily yields
the inequality
2 (1 − 𝛾)
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)]
𝑚 𝑟.
This completes the proof of Theorem 5.
4. Radii of Starlikeness and Convexity
∞
1−𝛾
∑ 𝑎𝑛 ≤
𝑚.
(2 − 𝛾) [1 + 𝛽((𝜆 − 𝛼) + 2𝛿)]
𝑛=2
𝑚 𝑟,
󵄨󵄨 󸀠 󵄨󵄨
󵄨󵄨𝑓 (𝑧)󵄨󵄨
󵄨
󵄨
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)]
󵄨
󵄨
≤ 󵄨󵄨󵄨󵄨𝑓󸀠 (𝑧)󵄨󵄨󵄨󵄨
≤1+
2 (1 − 𝛾)
(2 − 𝛾) [1 + 𝛽 ((𝜆 − 𝛼) + 2𝛿)]
(22)
The radii of starlikeness and convexity for the class
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) is given by the following theorems.
𝑆𝑚,𝑤
Journal of Applied Mathematics
5
Theorem 6. If the function 𝑓(𝑧) defined by (10) is in the class
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆), then 𝑓(𝑧) is of order 𝜂(0 ≤ 𝜂 < 1) in |𝑧 −
𝑆𝑚,𝑤
𝑤| < 𝑟1 , where
Theorem 7. If the function 𝑓(𝑧) defined by (10) is in the class
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆), then 𝑓(𝑧) is convex of order 𝜂 (0 ≤ 𝜂 < 1)
𝑆𝑚,𝑤
in |𝑧 − 𝑤| < 𝑟2 , where
1/𝑛
𝑚
(1 − 𝜂) (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
𝑟1 = inf {
} .
𝑛≥2
(𝑛 − 𝜂) (1 − 𝛾)
(26)
𝑚
(1−𝜂)(𝑛−𝛾)[1+(𝑛−1)𝛽((𝜆−𝛼)+𝑛𝛿)]
𝑟2 = inf {
}
𝑛≥2
𝑛(𝑛+𝜂−2)(1−𝛾)
1/(𝑛−1)
(𝑛 ≥ 2) .
(33)
The result is sharp for the functions 𝑓𝑛 (𝑧) given by (14).
Proof. It suffices to prove that
The result is sharp for the functions 𝑓𝑛 (𝑧) given by (14).
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 (𝑧 − 𝑤) 𝑓󸀠 (𝑧)
󵄨󵄨 ≤ 1 − 𝜂,
󵄨󵄨
−
1
󵄨󵄨
󵄨󵄨
𝑓
(𝑧)
󵄨
󵄨
(27)
for |𝑧 − 𝑤| < 𝑟1 . We have
󵄨
󵄨󵄨
󵄨󵄨 (𝑧 − 𝑤) 𝑓󸀠󸀠 (𝑧) 󵄨󵄨󵄨
󵄨󵄨 ≤ 1 − 𝜂,
󵄨󵄨
󵄨󵄨
𝑓󸀠 (𝑧)
󵄨󵄨󵄨
󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 (𝑧 − 𝑤) 𝑓󸀠 (𝑧)
󵄨󵄨
󵄨󵄨
−
1
󵄨󵄨
󵄨󵄨
𝑓
(𝑧)
󵄨
󵄨
󵄨󵄨 − ∑∞ (𝑛 − 1) 𝑎 (𝑧 − 𝑤)𝑛 󵄨󵄨
󵄨
󵄨󵄨
𝑛
𝑛=2
󵄨
= 󵄨󵄨󵄨󵄨
∞
𝑛 󵄨󵄨
󵄨󵄨 (𝑧 − 𝑤) − ∑𝑛=2 𝑎𝑛 (𝑧 − 𝑤) 󵄨󵄨
≤
Proof. By using the technique employed in the proof of
Theorem 6, we can show that
(28)
𝑛
∑∞
𝑛=2 (𝑛 − 1) 𝑎𝑛 |𝑧 − 𝑤|
∞
𝑛 .
1 − ∑𝑛=2 𝑎𝑛 |𝑧 − 𝑤|
(34)
for |𝑧 − 𝑤| < 𝑟2 , with the aid of Theorem 3. Thus we have the
assertion of Theorem 7.
5. Closure Theorems
Let the functions 𝑓𝑗 (𝑧), 𝑗 = 1, 2, . . . , 𝑙, be defined by
Hence (28) holds true if
∞
∑ (𝑛 − 1) 𝑎𝑛 |𝑧 − 𝑤|𝑛
∞
𝑛=2
(29)
∞
≤ (1 − 𝜂) (1 − ∑ 𝑎𝑛 |𝑧 − 𝑤|𝑛 )
𝑛=2
𝑓𝑗 (𝑧) = (𝑧 − 𝑤) − ∑ 𝑎𝑛,𝑗 (𝑧 − 𝑤)𝑛 ,
𝑛=2
(𝑎𝑛,𝑗 ≥ 0)
(35)
for 𝑧 ∈ 𝑈.
or
∞
𝑛−𝜂
𝑎𝑛 |𝑧 − 𝑤|𝑛 ≤ 1
∑
1
−
𝜂
𝑛=2
(30)
Theorem 8. Let the functions 𝑓𝑗 (𝑧) be defined by (35) in
∗
the class 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) for every 𝑗 = 1, 2, . . . , 𝑙. Then the
function 𝐺(𝑧) defined by
with the aid of (13), (30) is true if
∞
𝐺 (𝑧) = (𝑧 − 𝑤) − ∑ 𝑏𝑛 (𝑧 − 𝑤)𝑛 ,
𝑛−𝜂
|𝑧 − 𝑤|𝑛
1−𝜂
𝑛=2
(31)
𝑚
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
≤
.
1−𝛾
𝑏𝑛 =
|𝑧 − 𝑤|
𝑚
1/𝑛
(𝑛 ≥ 2) .
Proof. Since 𝑓𝑗 (𝑧)
Theorem 3 that
∞
(32)
(36)
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆), where
is a member of the class 𝑆𝑚,𝑤
Solving (31) for |𝑧 − 𝑤|, we obtain
(1 −𝜂) (𝑛 −𝛾) [1 + (𝑛 −1) 𝛽 ((𝜆−𝛼)+𝑛𝛿)]
}
≤{
(𝑛−𝜂) (1−𝛾)
(𝑏𝑛 ≥ 0)
∈
1 𝑙
∑𝑎
𝑙 𝑗=1 𝑛,𝑗
(𝑛 ≥ 2) .
(37)
∗
𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆), it follows from
𝑚
∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛,𝑗 ≤ 1 − 𝛾
𝑛=2
This completes the proof of Theorem 6.
(38)
6
Journal of Applied Mathematics
for every 𝑗 = 1, 2, . . . , 𝑙. Hence,
Theorem 10. Let
∞
𝑓0 (𝑧) = 𝑧 − 𝑤,
𝑚
∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑏𝑛
𝑓𝑛 (𝑧) = (𝑧 − 𝑤)
𝑛=2
∞
= ∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
𝑚
𝑛=2
(𝑛 ≥ 2; 𝑚 ∈ N0 ) .
(43)
}
{1 𝑙
× { ∑ 𝑎𝑛,𝑗 }
𝑙
}
{ 𝑗=1
=
𝑙
∗
Then 𝑓(𝑧) ∈ 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) if and only if it can be expressed
in the form
∞
1
∑ ( ∑ (𝑛 − 𝛾)
𝑙 𝑗=1 𝑛=2
∞
𝑓 (𝑧) = ∑ 𝜆 𝑛 𝑓𝑛 (𝑧) ,
𝑚
where 𝜆 𝑛 ≥ 0 and ∑∞
𝑛=1 𝜆 𝑛 = 1.
Proof. Suppose that
1 𝑙
∑ (1 − 𝛾) = 1 − 𝛾
𝑙 𝑗=1
∞
𝑓 (𝑧) = ∑ 𝜆 𝑛 𝑓𝑛 (𝑧) ,
(39)
which (in view of Theorem 5) implies that 𝐺(𝑧)
∗
𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆).
∈
where 𝜆 𝑛 ≥ 0 (𝑛 ≥ 1) and ∑∞
𝑛=1 𝜆 𝑛 = 1. Then
𝑓 (𝑧) = ∑ 𝜆 𝑛 𝑓𝑛 (𝑧)
𝑛=1
∞
Proof. Suppose that the functions 𝑓𝑗 (𝑧) (𝑗 = 1, 2) defined by
∗
(35) be in the class 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆), it is sufficient to prove
that the function
𝐻 (𝑧) = 𝜇𝑓1 (𝑧) + (1 − 𝜇) 𝑓2 (𝑧)
is also in the class
(0 ≤ 𝜇 ≤ 1)
= 𝜆 1 𝑓1 (𝑧) + ∑ 𝜆 𝑛 𝑓𝑛 (𝑧)
𝑛=2
(40)
∞
1−𝛾
𝑚
𝑛=2 (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
−∑
Since, for 0 ≤ 𝜇 ≤ 1,
∞
𝑛=2
(46)
= (𝑧 − 𝑤)
× 𝜆 𝑛 (𝑧 − 𝑤)𝑛 .
𝐻 (𝑧) = (𝑧 − 𝑤) − ∑ {𝜇𝑎𝑛,1 + (1 − 𝜇) 𝑎𝑛,2 } (𝑧 − 𝑤)𝑛 , (41)
Since
we observe that
∞
∞
∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
(45)
𝑛=1
∞
∗
Theorem 9. The class 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) is closed under convex
linear combination.
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆).
𝑆𝑚,𝑤
(44)
𝑛=1
× [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛,𝑗 )
≤
1−𝛾
𝑛
𝑚 (𝑧 − 𝑤)
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
−
∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
𝑚
𝑛=2
𝑛=2
⋅
× {𝜇𝑎𝑛,1 + (1 − 𝜇) 𝑎𝑛,2 }
∞
𝑚
1−𝛾
𝑚 𝜆𝑛
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
(47)
∞
𝑚
= (1 − 𝛾) ∑ 𝜆 𝑛
= 𝜇 ∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛,1
𝑛=2
𝑛=2
∞
= (1 − 𝛾) (1 − 𝜆 0 ) ≤ 1 − 𝛾,
+ (1 − 𝜇) ∑ (𝑛 − 𝛾)
𝑛=2
𝑚
× [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛,2
≤ 𝜇 (1 − 𝛾) + (1 − 𝜇) (1 − 𝛾) = (1 − 𝛾) .
(42)
∗
Hence 𝐻(𝑧) ∈ 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆). This completes the proof of
Theorem 9.
it follows from Theorem 3 that the function 𝑓(𝑧) ∈
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆). Conversely, let us suppose that 𝑓(𝑧) ∈
𝑆𝑚,𝑤
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆). Since
𝑆𝑚,𝑤
𝑎𝑛 ≤
1−𝛾
𝑚
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
(𝑛 ≥ 2; 𝑚 ∈ N0 ) .
(48)
Journal of Applied Mathematics
7
Setting
Since
∞
∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
𝑚
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
𝜆𝑛 =
𝑎𝑛 ,
1−𝛾
(𝑛 ≥ 2; 𝑚 ∈ N0 ) ,
𝑛=2
1−𝛾
𝑚 𝜆𝑛
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
⋅
(49)
∞
(54)
∞
𝜆1 = 1 − ∑ 𝜆𝑛,
= (1 − 𝛾) ∑ 𝜆 𝑛
𝑛=2
𝑛=2
∗
it follows that 𝑓(𝑧) = 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆). This completes the
proof of the theorem.
6. Extreme Points
= (1 − 𝛾) (1 − 𝜆 0 ) ≤ 1 − 𝛾,
it follows from Theorem 3 that the function 𝑓(𝑧) ∈
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆). Conversely, let us suppose that 𝑓(𝑧) ∈
𝑆𝑚,𝑤
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆). Since
𝑆𝑚,𝑤
𝑎𝑛 ≤
Theorem 11. Let
1−𝛾
𝑚
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
(55)
(𝑛 ≥ 2; 𝑚 ∈ N0 ) .
𝑓0 (𝑧) = 𝑧 − 𝑤,
Setting
𝑓𝑛 (𝑧) = (𝑧 − 𝑤)
1−𝛾
−
𝑚
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
× (𝑧 − 𝑤)𝑛
Then 𝑓(𝑧) ∈
in the form
𝑚
(50)
𝑚
𝜆𝑛 =
(𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
𝑎𝑛 ,
1−𝛾
(𝑛 ≥ 2; 𝑚 ∈ N0 ) ,
(𝑛 ≥ 2; 𝑚 ∈ N0 ) .
∞
𝜆1 = 1 − ∑ 𝜆𝑛,
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆) if and only if it can be expressed
𝑆𝑚,𝑤
𝑛=2
it follows that 𝑓(𝑧) =
proof of theorem.
∞
𝑓 (𝑧) = ∑ 𝜆 𝑛 𝑓𝑛 (𝑧) ,
(51)
𝑛=1
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆).
𝑆𝑚,𝑤
Proof. Suppose that
7. Integral Operators
𝐹 (𝑧) =
∞
(52)
𝑛=1
𝑧
𝑐+1
𝑐−1
𝑓 (𝑡) 𝑑𝑡
𝑐 ∫ (𝑡 − 𝑤)
(𝑧 − 𝑤) 𝑤
∗
also belongs to the class 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆).
∞
𝑚
∑ (𝑛 − 𝛾) [1 + (𝑛 − 1)𝛽((𝜆 − 𝛼) + 𝑛𝛿)] 𝑏𝑛
∞
𝑓 (𝑧) = ∑ 𝜆 𝑛 𝑓𝑛 (𝑧)
𝑛=2
𝑛=1
∞
= ∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)]
∞
= 𝜆 1 𝑓1 (𝑧) + ∑ 𝜆 𝑛 𝑓𝑛 (𝑧)
(53)
∞
1 −𝛾
𝑚
𝑛=2 (𝑛 −𝛾) [1 +(𝑛 −1) 𝛽 ((𝜆 −𝛼) +𝑛𝛿)]
× 𝜆 𝑛 (𝑧 − 𝑤)𝑛 .
𝑚
𝑛=2
𝑛=2
−∑
(57)
Proof. From (57), it follows that 𝐹(𝑧) = (𝑧 − 𝑤) −
𝑛
∑∞
𝑛=2 𝑏𝑛 (𝑧 − 𝑤) , where 𝑏𝑛 = ((𝑐 + 1)/(𝑐 + 𝑛))𝑎𝑛 . Therefore
where 𝜆 𝑛 ≥ 0 (𝑛 ≥ 1) and ∑∞
𝑛=1 𝜆 𝑛 = 1. Then
= (𝑧 − 𝑤)
This completes the
Theorem 12. If the function 𝑓(𝑧) defined by (10) is in the class
∗
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆), and let 𝑐 be a real number such that 𝑐 > −1.
𝑆𝑚,𝑤
Then the function 𝐹(𝑧) defined by
where 𝜆 𝑛 ≥ 0 and ∑∞
𝑛=1 𝜆 𝑛 = 1.
𝑓 (𝑧) = ∑ 𝜆 𝑛 𝑓𝑛 (𝑧) ,
(56)
×(
∞
𝑐+1
)𝑎
𝑐+𝑛 𝑛
(58)
𝑚
≤ ∑ (𝑛 − 𝛾) [1 + (𝑛 − 1) 𝛽 ((𝜆 − 𝛼) + 𝑛𝛿)] 𝑎𝑛
𝑛=2
≤ 1 − 𝛾,
8
Journal of Applied Mathematics
∗
since 𝑓(𝑧) ∈ 𝑆𝑚,𝑤
(𝛼, 𝛽, 𝛾, 𝛿, 𝜆). Hence by Theorem 3, 𝐹(𝑧) ∈
∗
𝑆𝑚,𝑤 (𝛼, 𝛽, 𝛾, 𝛿, 𝜆).
Acknowledgment
The work here was fully supported by LRGS/TD/2011/
UKM/ICT/03/02 and UKM-DLP-2011-050.
References
[1] S. Kanas and F. Rønning, “Uniformly starlike and convex
functions and other related classes of univalent functions,”
Annales Universitatis Mariae Curie-Skłodowska A, vol. 53, pp.
95–105, 1999.
[2] M. Acu and S. Owa, “On a subclass of n-starlike functions,”
International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 17, pp. 2841–2846, 2005.
[3] A. W. Goodman, “On uniformly starlike functions,” Journal of
Mathematical Analysis and Applications, vol. 155, no. 2, pp. 364–
370, 1991.
[4] A. W. Goodman, “On uniformly convex functions,” Annales
Polonici Mathematici, vol. 56, no. 1, pp. 87–92, 1991.
[5] M. S. Robertson, “Extremal problems for analytic functions
with positive real part and applications,” Transactions of the
American Mathematical Society, vol. 106, pp. 236–253, 1963.
󸀠󸀠
[6] H. Saitoh, “Univalence and starlikeness of solutions of 𝑊 +
𝑎𝑊󸀠 + 𝑏𝑊 = 0,” Annales Universitatis Mariae Curie-Skłodowska
A, vol. 53, pp. 209–216, 1999.
[7] S. Owa, H. Saitoh, H. M. Srivastava, and R. Yamakawa,
“Geometric properties of solutions of a class of differential
equations,” Computers & Mathematics with Applications, vol. 47,
no. 10-11, pp. 1689–1696, 2004.
[8] M. O. Reade and H. Silverman, “Radii problems for linear properties of univalent functions,” Houston Journal of Mathematics,
vol. 17, no. 2, pp. 227–235, 1991.
[9] J. Sokół and A. Wiśniowska-Wajnryb, “On certain problem in
the class of k-starlike functions,” Computers & Mathematics with
Applications, vol. 62, no. 12, pp. 4733–4741, 2011.
[10] P. Curt and D. Fericean, “A special class of univalent functions
in Hele-Shaw flow problems,” Abstract and Applied Analysis, vol.
2011, Article ID 948236, 10 pages, 2011.
[11] M. Darus and R. W. Ibrahim, “On subclasses for generalized
operators of complex order,” Far East Journal of Mathematical
Sciences, vol. 33, no. 3, pp. 299–308, 2009.
[12] F. M. Al-Oboudi, “On univalent functions defined by a generalized Sălăgean operator,” International Journal of Mathematics
and Mathematical Sciences, vol. 2004, no. 27, pp. 1429–1436,
2004.
[13] G. Sălăgean, “Subclasses of univalent functions,” in Lecture
Notes in Mathematics, pp. 362–372, Springer, Heidelberg, Germany, 1983.
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