American Journal of Mathematics and Statistics 2014, 4(3): 147-155
DOI: 10.5923/j.ajms.20140403.03
On a New Class of Multivalent Functions with Negative
Coefficient Defined by Hadamard Product Involving a
Linear Operator
Waggas Galib Atshan1,*, Ali Hussein Battor2, Amal Mohammed Dereush2
1
Department of Mathematics, College of Computer Science and Mathematics, University of Al-Qadisiya, Diwaniya, Iraq
2
Department of Mathematics, College of Education for Girls, University of Kufa, Najaf, Iraq
Abstract In this paper, we have introduced and studied a new class of multivalent functions in the open unit disk
𝑈𝑈 = {𝑧𝑧 ∈ ℂ: |𝑧𝑧| < 1}, we obtain some interesting properties, like, coefficient inequality, distortion bounds, closure theorems,
radii of starlikeness, convexity and close-to-convexity, weighted mean, neighborhoods and partial sums.
Keywords
Multivalent Function, Convolution, Distortion, Neighborhoods, Partial Sums, Weighted Mean, Linear
Operator
1. Introduction
Let 𝐺𝐺 denote the class of all functions of the from
𝑛𝑛
𝑓𝑓(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 + ∑∞
𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 𝑧𝑧 ,
(𝑝𝑝 ∈ 𝑁𝑁 = {1,2, … })
(1)
which are analytic and multivalent in the open unit disk 𝑈𝑈.
Let 𝑆𝑆𝑚𝑚 denote the subclass of 𝐺𝐺consisting of functions of
the from
𝑛𝑛
𝑓𝑓(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 − ∑∞
𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 𝑧𝑧 ,
(𝑎𝑎𝑛𝑛 ≥ 0, 𝑝𝑝 ∈ 𝑁𝑁)
𝑔𝑔(𝑧𝑧) = 𝑧𝑧 −
𝑛𝑛
∑∞
𝑛𝑛=𝑝𝑝+1 𝑏𝑏𝑛𝑛 𝑧𝑧 ,
(𝑏𝑏𝑛𝑛 ≥ 0, 𝑝𝑝 ∈ 𝑁𝑁)
(3)
we define the convolution (or Hadamard product) of
𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔 by
𝑛𝑛
(𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 − ∑∞
𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 𝑏𝑏𝑛𝑛 𝑧𝑧 .
(4)
A function 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 is said to be p-valentlystarlike of order
𝜇𝜇 if and only if
𝑅𝑅𝑅𝑅 �
𝑧𝑧𝑧𝑧′ (𝑧𝑧)
𝑓𝑓(𝑧𝑧)
� > 𝜇𝜇,
(0 ≤ 𝜇𝜇 < 𝑝𝑝; 𝑧𝑧 ∈ 𝑈𝑈).
(5)
(0 ≤ 𝜇𝜇 < 𝑝𝑝; 𝑧𝑧 ∈ 𝑈𝑈).
(6)
A function 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 is said to be p-valently convex of
order 𝜇𝜇 if and only if
𝑅𝑅𝑅𝑅 �1 +
𝑧𝑧𝑧𝑧′′ (𝑧𝑧)
𝑓𝑓′ (𝑧𝑧)
� > 𝜇𝜇,
* Corresponding author:
waggashnd@gmail.com (Waggas Galib Atshan)
Published online at http://journal.sapub.org/ajms
Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved
𝑅𝑅𝑅𝑅 �
𝑓𝑓′ (𝑧𝑧)
𝑧𝑧 𝑝𝑝 −1
be
p-valently
(0 ≤ 𝜇𝜇 < 𝑝𝑝; 𝑧𝑧 ∈ 𝑈𝑈).
� > 𝜇𝜇,
(7)
Definition 1 [8]: Let 𝛾𝛾, 𝛽𝛽, 𝑚𝑚, ∈ ℛ, 𝛾𝛾 ≥ 0, 𝛽𝛽 ≥ 0, 𝑚𝑚 ≥
0, 𝑝𝑝 ∈ 𝑁𝑁 and
∞
𝑓𝑓(𝑧𝑧) = 𝑧𝑧 𝑛𝑛 − � 𝑎𝑎𝑛𝑛 𝑧𝑧 𝑛𝑛 .
(2)
which are analytic and multivalent in the open unit disk 𝑈𝑈.
For the function 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 given by (2) and 𝑔𝑔 ∈ 𝑆𝑆𝑚𝑚
defined by
𝑝𝑝
A function 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 is said to
close-to-convex of order 𝜇𝜇 if and only if
𝑛𝑛=𝑝𝑝+1
Then we define the linear operator
𝛾𝛾,𝛽𝛽
𝐷𝐷𝑝𝑝,𝑚𝑚 : 𝐺𝐺 → 𝐺𝐺 𝑏𝑏𝑏𝑏
𝛾𝛾,𝛽𝛽
𝐷𝐷𝑝𝑝,𝑚𝑚 𝑓𝑓(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 − ∑∞
𝑛𝑛=𝑝𝑝+1 �1 +
(𝑛𝑛−𝑝𝑝)𝛾𝛾 𝑚𝑚
(𝑝𝑝+𝛽𝛽 )
� 𝑎𝑎𝑛𝑛 𝑧𝑧 𝑛𝑛 , 𝑧𝑧 ∈ 𝑈𝑈. (8)
Definition 2: Let 𝑔𝑔 be a fixed function defined by (3).
The function 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 given by (2) is said to be in the class
𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈) if and only if
�
𝛾𝛾 ,𝛽𝛽
′
𝛾𝛾 ,𝛽𝛽
′′
�𝐷𝐷𝑝𝑝 ,𝑚𝑚 (𝑓𝑓∗𝑔𝑔)(𝑧𝑧)� +𝑧𝑧��𝐷𝐷𝑝𝑝 ,𝑚𝑚 (𝑓𝑓∗𝑔𝑔)(𝑧𝑧)� −𝑝𝑝 2 𝑧𝑧 𝑝𝑝 −2 �
𝛾𝛾 ,𝛽𝛽
′
𝛾𝛾 ,𝛽𝛽
′′
𝛼𝛼�𝐷𝐷𝑝𝑝 ,𝑚𝑚 (𝑓𝑓∗𝑔𝑔)(𝑧𝑧)� −𝑧𝑧𝑧𝑧 ��𝐷𝐷𝑝𝑝 ,𝑚𝑚 (𝑓𝑓∗𝑔𝑔)(𝑧𝑧)� −𝑝𝑝 2 𝑧𝑧 𝑝𝑝 −2 �
� < 𝜆𝜆, (9)
where 0 < 𝛼𝛼 < 1, 0 ≤ 𝜈𝜈 < 1, 0 < 𝜆𝜆 < 1, 𝛾𝛾, 𝛽𝛽, 𝑚𝑚 ∈ ℜ, 𝛾𝛾 ≥
0, 𝛽𝛽 ≥ 0, 𝑚𝑚 ≥ 0, 𝑝𝑝 ∈ 𝑁𝑁.
Some of the following properties studied for other class in
[1], [2], [3], [4], [6] and [7].
2. Coefficient Inequalities
Theorem 1: Let 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 . Then 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈)
if and only if
Waggas Galib Atshan et al.: On a New Class of Multivalent Functions with Negative
Coefficient Defined by Hadamard Product Involving a Linear Operator
148
∑∞
𝑛𝑛=𝑝𝑝+1 �1 +
(𝑛𝑛−𝑝𝑝)𝛾𝛾 𝑚𝑚
(𝑝𝑝+𝛽𝛽 )
(10)
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)] 𝑎𝑎𝑛𝑛 𝑏𝑏𝑛𝑛 ≤ 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈),
where 0 < 𝛼𝛼 < 1, 0 ≤ 𝜈𝜈 < 1, 0 < 𝜆𝜆 < 1, 𝛾𝛾, 𝛽𝛽, 𝑚𝑚 ∈ ℜ, 𝛾𝛾 ≥ 0, 𝛽𝛽 ≥ 0, 𝑚𝑚 ≥ 0, 𝑝𝑝 ∈ 𝑁𝑁.
The result is sharp for the function
𝑓𝑓(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 −
𝑝𝑝𝑝𝑝 (𝛼𝛼+𝜈𝜈)
(𝑛𝑛 −𝑝𝑝 )𝛾𝛾 𝑚𝑚
�1+ (𝑝𝑝 ) � [𝑛𝑛(𝜆𝜆(𝜈𝜈 (𝑛𝑛 −1)−𝛼𝛼)+𝑛𝑛)]𝑏𝑏𝑛𝑛
+𝛽𝛽
Proof: Suppose that the inequality (10) holds true and |𝑧𝑧| = 1. Then we have
′
𝛾𝛾,𝛽𝛽
′′
𝛾𝛾,𝛽𝛽
𝛾𝛾,𝛽𝛽
𝑧𝑧 𝑛𝑛 .
(11)
′
𝛾𝛾,𝛽𝛽
′′
��𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� + 𝑧𝑧 ��𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� − 𝑝𝑝2 𝑧𝑧 𝑝𝑝−2 �� − 𝜆𝜆 �𝛼𝛼 �𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� − 𝑧𝑧𝑧𝑧 ��𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� − 𝑝𝑝2 𝑧𝑧 𝑝𝑝−2 ��
∞
∞
𝑚𝑚
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
= �− � �1 +
� 𝑛𝑛2 𝑎𝑎𝑛𝑛 𝑏𝑏𝑛𝑛 𝑧𝑧 𝑛𝑛 � − 𝜆𝜆 �𝑝𝑝(𝛼𝛼 + 𝜈𝜈)𝑧𝑧 𝑝𝑝−1 − � �1 +
� �𝛼𝛼𝛼𝛼 − 𝜈𝜈(𝑛𝑛2 − 𝑛𝑛)� 𝑎𝑎𝑛𝑛 𝑏𝑏𝑛𝑛 𝑧𝑧 𝑛𝑛 �
(𝑝𝑝 + 𝛽𝛽)
(𝑝𝑝 + 𝛽𝛽)
𝑛𝑛=𝑝𝑝+1
∞
≤ � �1 +
𝑛𝑛=𝑝𝑝+1
𝑛𝑛=𝑝𝑝+1
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑎𝑎𝑛𝑛 𝑏𝑏𝑛𝑛 − 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈) ≤ 0,
(𝑝𝑝 + 𝛽𝛽)
by hypothesis.
Hence, by maximum modulus principle, 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈).
Conversely, suppose that 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈). Then from (9), we have
�
′
𝛾𝛾,𝛽𝛽
′
𝛾𝛾,𝛽𝛽
𝛾𝛾,𝛽𝛽
′′
𝛼𝛼 �𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� − 𝑧𝑧𝑧𝑧 ��𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� − 𝑝𝑝2 𝑧𝑧 𝑝𝑝−2 �
Since 𝑅𝑅𝑅𝑅(𝑧𝑧) ≤ |𝑧𝑧| for all 𝑧𝑧(𝑧𝑧 ∈ 𝑈𝑈), we get
𝑅𝑅𝑅𝑅 �
′′
𝛾𝛾,𝛽𝛽
�𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� + 𝑧𝑧 ��𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� − 𝑝𝑝2 𝑧𝑧 𝑝𝑝−2 �
(𝑛𝑛 −𝑝𝑝 )𝛾𝛾 𝑚𝑚
2
𝑛𝑛
− ∑∞
𝑛𝑛 =𝑝𝑝 +1 �1+ (𝑝𝑝 +𝛽𝛽 ) � 𝑛𝑛 𝑎𝑎 𝑛𝑛 𝑏𝑏𝑛𝑛 𝑧𝑧
(𝑛𝑛 −𝑝𝑝 )𝛾𝛾 𝑚𝑚
2
𝑛𝑛
𝑝𝑝 −1
− ∑∞
𝑛𝑛 =𝑝𝑝 +1 �1+ (𝑝𝑝 +𝛽𝛽 ) � [𝛼𝛼𝛼𝛼 −𝜈𝜈(𝑛𝑛 −𝑛𝑛)]𝑎𝑎 𝑛𝑛 𝑏𝑏𝑛𝑛 𝑧𝑧 +𝑝𝑝(𝛼𝛼+𝜈𝜈)𝑧𝑧
′′
𝛾𝛾,𝛽𝛽
We choose the value of 𝑧𝑧 on the real axis, so that �𝐷𝐷𝑝𝑝,𝑚𝑚 (𝑓𝑓 ∗ 𝑔𝑔)(𝑧𝑧)� is real.
Letting 𝑧𝑧 → 1− through real values, we obtain inequality (10).
Finally, sharpness follows if we take
𝑓𝑓(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 −
𝑝𝑝𝑝𝑝 (𝛼𝛼+𝜈𝜈)
(𝑛𝑛 −𝑝𝑝 )𝛾𝛾 𝑚𝑚
�1+ (𝑝𝑝 ) � [𝑛𝑛(𝜆𝜆(𝜈𝜈 (𝑛𝑛−1)−𝛼𝛼)+𝑛𝑛)]𝑏𝑏𝑛𝑛
+𝛽𝛽
Corollary 1: Let 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈). Then
𝑎𝑎𝑛𝑛 ≤
𝑝𝑝𝑝𝑝 (𝛼𝛼 +𝜈𝜈)
(𝑛𝑛 −𝑝𝑝 )𝛾𝛾 𝑚𝑚
�1+ (𝑝𝑝 ) � [𝑛𝑛(𝜆𝜆(𝜈𝜈 (𝑛𝑛−1)−𝛼𝛼)+𝑛𝑛)]𝑏𝑏𝑛𝑛
+𝛽𝛽
� < 𝜆𝜆.
� < 𝜆𝜆.
𝑧𝑧 𝑛𝑛 .
(12)
(13)
, 𝑛𝑛 = 𝑝𝑝 + 1, 𝑝𝑝 + 2, ….
(14)
3. Growth and Distortion Theorems
Theorem 2: Let the function 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈). Then
|𝑧𝑧|𝑝𝑝−1 −
Proof:
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
|𝑧𝑧|𝑝𝑝+1 ≤ |𝑓𝑓(𝑧𝑧)| ≤
𝑚𝑚
𝛾𝛾
� �(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑏𝑏𝑝𝑝+1
�1 +
(𝑝𝑝 + 𝛽𝛽)
≤ |𝑧𝑧|𝑝𝑝−1 +
�1+(𝑝𝑝
𝑝𝑝𝑝𝑝 (𝛼𝛼+𝜈𝜈)
𝑚𝑚
𝛾𝛾
� �(𝑝𝑝+1)�𝜆𝜆(𝜈𝜈𝜈𝜈 −𝛼𝛼)+(𝑝𝑝+1)��𝑏𝑏𝑝𝑝 +1
+𝛽𝛽 )
|𝑧𝑧|𝑝𝑝+1 .
∞
∞
∞
𝑛𝑛=𝑝𝑝+1
𝑛𝑛=𝑝𝑝+1
𝑛𝑛=𝑝𝑝+1
|𝑓𝑓(𝑧𝑧)| = �𝑧𝑧 𝑝𝑝 + � 𝑎𝑎𝑛𝑛 𝑧𝑧 𝑛𝑛 � ≤ |𝑧𝑧|𝑝𝑝 + � 𝑎𝑎𝑛𝑛 |𝑧𝑧|𝑛𝑛 ≤ |𝑧𝑧|𝑝𝑝 + |𝑧𝑧|𝑝𝑝+1 � 𝑎𝑎𝑛𝑛 .
(15)
American Journal of Mathematics and Statistics 2014, 4(3): 147-155
149
By Theorem 1, we get
∑∞
𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 ≤
Thus
|𝑓𝑓(𝑧𝑧)| ≤ |𝑧𝑧|𝑝𝑝 +
also
.
(16)
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
|𝑧𝑧|𝑝𝑝+1 ,
𝑚𝑚
𝛾𝛾
� �(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑏𝑏𝑝𝑝+1
�1 +
(𝑝𝑝 + 𝛽𝛽)
|𝑓𝑓(𝑧𝑧)| ≥
≥ |𝑧𝑧|𝑝𝑝 −
𝑝𝑝𝑝𝑝 (𝛼𝛼+𝜈𝜈)
𝑚𝑚
𝛾𝛾
�1+(𝑝𝑝 +𝛽𝛽 )� �(𝑝𝑝+1)�𝜆𝜆(𝜈𝜈𝜈𝜈 −𝛼𝛼)+(𝑝𝑝+1)��𝑏𝑏𝑝𝑝 +1
|𝑧𝑧|𝑝𝑝
∞
− � 𝑎𝑎𝑛𝑛
𝑛𝑛=𝑝𝑝+1
|𝑧𝑧|𝑛𝑛
≥
|𝑧𝑧|𝑝𝑝
−
|𝑧𝑧|𝑝𝑝+1
∞
� 𝑎𝑎𝑛𝑛
𝑛𝑛=𝑝𝑝+1
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
|𝑧𝑧|𝑝𝑝+1 ,
𝑚𝑚
𝛾𝛾
� �(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑏𝑏𝑝𝑝+1
�1 +
(𝑝𝑝 + 𝛽𝛽)
and the proof is complete.
Theorem 3: Let 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈). Then
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
|𝑧𝑧|𝑝𝑝 ≤ |𝑓𝑓′(𝑧𝑧)|
𝑚𝑚
𝛾𝛾
� �(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑏𝑏𝑝𝑝+1
�1 +
(𝑝𝑝 + 𝛽𝛽)
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
|𝑧𝑧|𝑝𝑝 .
≤ 𝑝𝑝|𝑧𝑧|𝑝𝑝−1 +
𝑚𝑚
𝛾𝛾
� �(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑏𝑏𝑝𝑝+1
�1 +
(𝑝𝑝 + 𝛽𝛽)
𝑝𝑝|𝑧𝑧|𝑝𝑝−1 −
Proof: Notice that
∞
𝑚𝑚
𝛾𝛾
� �(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑏𝑏𝑝𝑝+1 � 𝑛𝑛𝑎𝑎𝑛𝑛
�1 +
(𝑝𝑝 + 𝛽𝛽)
from Theorem 1, thus
≤ ∑∞
𝑛𝑛=𝑝𝑝+1 �1 +
𝑛𝑛=𝑝𝑝+1
(𝑛𝑛−𝑝𝑝)𝛾𝛾 𝑚𝑚
(𝑝𝑝+𝛽𝛽 )
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑏𝑏𝑛𝑛 ≤ 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈),
∞
∞
|𝑓𝑓′(𝑧𝑧)| = �𝑝𝑝𝑧𝑧 𝑝𝑝−1 + � 𝑛𝑛𝑎𝑎𝑛𝑛 𝑧𝑧 𝑛𝑛−1 � ≤ 𝑝𝑝|𝑧𝑧|𝑝𝑝−1 + � 𝑛𝑛𝑎𝑎𝑛𝑛 |𝑧𝑧|𝑛𝑛−1
𝑛𝑛=𝑝𝑝+1
On the other hand
≤ 𝑝𝑝|𝑧𝑧|𝑝𝑝−1 +
�1+(𝑝𝑝
𝑝𝑝𝑝𝑝 (𝛼𝛼+𝜈𝜈)
𝑛𝑛=𝑝𝑝+1
𝑚𝑚
𝛾𝛾
� �(𝑝𝑝+1)�𝜆𝜆(𝜈𝜈𝜈𝜈 −𝛼𝛼)+(𝑝𝑝+1)��𝑏𝑏𝑝𝑝 +1
+𝛽𝛽 )
∞
∞
𝑛𝑛=𝑝𝑝+1
𝑛𝑛=𝑝𝑝+1
|𝑧𝑧|𝑝𝑝 .
|𝑓𝑓′(𝑧𝑧)| = �𝑝𝑝𝑧𝑧 𝑝𝑝−1 + � 𝑛𝑛𝑎𝑎𝑛𝑛 𝑧𝑧 𝑛𝑛−1 � ≥ 𝑝𝑝|𝑧𝑧|𝑝𝑝−1 − � 𝑛𝑛𝑎𝑎𝑛𝑛 |𝑧𝑧|𝑛𝑛−1
≥ 𝑝𝑝|𝑧𝑧|𝑝𝑝−1 +
Combining (18) and (19), we get the result.
(17)
𝑝𝑝𝑝𝑝 (𝛼𝛼+𝜈𝜈)
𝑚𝑚
𝛾𝛾
�1+(𝑝𝑝 )� �(𝑝𝑝+1)�𝜆𝜆(𝜈𝜈𝜈𝜈 −𝛼𝛼)+(𝑝𝑝+1)��𝑏𝑏𝑝𝑝 +1
+𝛽𝛽
|𝑧𝑧|𝑝𝑝 .
(18)
(19)
Closure Theorems:
Theorem 4: Let the function 𝑓𝑓𝑖𝑖 defined by
𝑛𝑛
𝑓𝑓𝑖𝑖 (𝑧𝑧) = 𝑧𝑧 𝑝𝑝 − ∑∞
𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛,𝑖𝑖 𝑧𝑧 ,
�𝑎𝑎𝑛𝑛,𝑖𝑖 ≥ 0, 𝑝𝑝 ∈ 𝑁𝑁, 𝑖𝑖 = 1,2, … , 𝑚𝑚�,
be in the class 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈) for every𝑖𝑖 = 1,2, … , 𝑚𝑚. Then the function ℎ defined by
(20)
Waggas Galib Atshan et al.: On a New Class of Multivalent Functions with Negative
Coefficient Defined by Hadamard Product Involving a Linear Operator
150
𝑝𝑝
∞
ℎ(𝑧𝑧) = 𝑧𝑧 − � 𝑐𝑐𝑛𝑛 𝑧𝑧 𝑛𝑛 ,
(𝑐𝑐𝑛𝑛 ≥ 0, 𝑝𝑝 ∈ 𝑁𝑁),
𝑛𝑛=𝑝𝑝+1
also belongs to the class 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈), where
𝑐𝑐𝑛𝑛 =
𝑚𝑚
1
� 𝑎𝑎𝑛𝑛,,𝑖𝑖 ,
𝑚𝑚
(𝑛𝑛 ≥ 𝑝𝑝 + 1).
𝑖𝑖=1
Proof: Since 𝑓𝑓𝑖𝑖 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈), then by Theorem 1, we have
∑∞
𝑛𝑛=𝑝𝑝+1 �1 +
for every 𝑖𝑖 = 1,2, … , 𝑚𝑚.
Hence
(𝑛𝑛−𝑝𝑝)𝛾𝛾 𝑚𝑚
(𝑝𝑝+𝛽𝛽 )
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑎𝑎𝑛𝑛,𝑖𝑖 𝑏𝑏𝑛𝑛 ≤ 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
∞
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑐𝑐𝑛𝑛 𝑏𝑏𝑛𝑛
� �1 +
(𝑝𝑝 + 𝛽𝛽)
𝑛𝑛=𝑝𝑝+1
∞
= � �1 +
𝑛𝑛=𝑝𝑝+1
=
(21)
𝑚𝑚
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
1
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑏𝑏𝑛𝑛 � � 𝑎𝑎𝑛𝑛,𝑖𝑖 �
(𝑝𝑝 + 𝛽𝛽)
𝑚𝑚
𝑚𝑚
∞
𝑖𝑖=1
𝑛𝑛=𝑝𝑝+1
𝑖𝑖=1
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
1
� � � �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑎𝑎𝑛𝑛,𝑖𝑖 𝑏𝑏𝑛𝑛 � ≤ 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈).
(𝑝𝑝 + 𝛽𝛽)
𝑚𝑚
By Theorem 1, it follows that ℎ ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈).
Theorem 5: Let the function 𝑓𝑓𝑖𝑖 (𝑧𝑧), defined by (20) be in the class 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈), for every 𝑖𝑖 = 1,2, … , 𝑚𝑚. Then
the function ℎ(𝑧𝑧) defined by
𝑚𝑚
ℎ(𝑧𝑧) = � 𝑑𝑑𝑖𝑖 𝑓𝑓𝑖𝑖 (𝑧𝑧) 𝑎𝑎𝑎𝑎𝑑𝑑
is also in the class𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈).
Proof: By definition of ℎ(𝑧𝑧), we have
𝑖𝑖=1
𝑚𝑚
𝑖𝑖=1
∞
(𝑑𝑑𝑖𝑖 ≥ 0)
𝑚𝑚
ℎ(𝑧𝑧) = �� 𝑑𝑑𝑖𝑖 � 𝑧𝑧 − � �� 𝑑𝑑𝑖𝑖 𝑎𝑎𝑛𝑛,𝑖𝑖 � 𝑧𝑧 𝑛𝑛 .
𝑖𝑖=1
𝑝𝑝
𝑚𝑚
� 𝑑𝑑𝑖𝑖 = 1,
𝑛𝑛=𝑝𝑝+1 𝑖𝑖=1
Since 𝑓𝑓𝑖𝑖 (𝑧𝑧) are in the class 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈), for every 𝑖𝑖 = 1,2, … , 𝑚𝑚, we obtain
∞
� �1 +
𝑛𝑛=𝑝𝑝+1
𝑚𝑚
𝑚𝑚
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑏𝑏𝑛𝑛 �� 𝑑𝑑𝑖𝑖 𝑎𝑎𝑛𝑛,𝑖𝑖 �
(𝑝𝑝 + 𝛽𝛽)
𝑚𝑚
𝑖𝑖=1
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
= � 𝑑𝑑𝑖𝑖 ��1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑎𝑎𝑛𝑛,𝑖𝑖 𝑏𝑏𝑛𝑛 � ≤ 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈) � 𝑑𝑑𝑖𝑖 = 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈).
(𝑝𝑝 + 𝛽𝛽)
𝑖𝑖=1
𝑖𝑖=1
4. Radii of Starlikeness, Convexity and Close-to-Convexity
In the following theorems, we obtain the radii of starlikeness, convexity and close-to-convexity for the class
𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈).
Theorem 6: If 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈), then 𝑓𝑓(𝑧𝑧) is p-valentlystarlike of order 𝜇𝜇 (0 ≤ 𝜇𝜇 < 𝑝𝑝), in the dick |𝑧𝑧| < 𝑅𝑅1 ,
where
1
𝑛𝑛−𝑝𝑝
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
⎡(𝑝𝑝 − 𝜇𝜇) �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]⎤
(𝑝𝑝 + 𝛽𝛽)
⎥
𝑅𝑅1 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑛𝑛 ⎢
.
(𝑛𝑛 − 𝜇𝜇)𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
⎢
⎥
⎣
⎦
Proof: It is sufficient to show that
American Journal of Mathematics and Statistics 2014, 4(3): 147-155
𝑧𝑧𝑓𝑓 ′ (𝑧𝑧)
− 𝑝𝑝� ≤ 𝑝𝑝 − 𝜇𝜇,
𝑓𝑓(𝑧𝑧)
𝑛𝑛−𝑝𝑝
∑∞
𝑧𝑧𝑓𝑓 ′ (𝑧𝑧)
𝑛𝑛=𝑝𝑝+1(𝑛𝑛 − 𝑝𝑝)𝑎𝑎𝑛𝑛 |𝑧𝑧|
− 𝑝𝑝� ≤
.
∞
𝑛𝑛−𝑝𝑝
𝑓𝑓(𝑧𝑧)
1 − ∑𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 |𝑧𝑧|
�
Thus
𝑧𝑧𝑓𝑓 ′ (𝑧𝑧)
− 𝑝𝑝� ≤ 𝑝𝑝 − 𝜇𝜇,
𝑓𝑓(𝑧𝑧)
�
if
Hence, by Theorem 1, (22) will be true if
or if
(0 ≤ 𝜇𝜇 < 𝑝𝑝),
�
for |𝑧𝑧| < 𝑅𝑅1 , we have
151
∑∞
𝑛𝑛=𝑝𝑝+1
|𝑧𝑧|𝑛𝑛−𝑝𝑝
(𝑛𝑛 − 𝑝𝑝)𝑎𝑎𝑛𝑛
(𝑝𝑝 − 𝜇𝜇)
≤
(𝑛𝑛−𝑝𝑝)𝑎𝑎 𝑛𝑛 |𝑧𝑧|𝑛𝑛 −𝑝𝑝
(𝑝𝑝−𝜇𝜇 )
≤ 1.
(22)
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]
(𝑝𝑝 + 𝛽𝛽)
,
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
�1 +
1
𝑛𝑛 −𝑝𝑝
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
⎡(𝑝𝑝 − 𝜇𝜇) �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]⎤
(𝑝𝑝 + 𝛽𝛽)
⎥ .
|𝑧𝑧| ≤ ⎢
(𝑛𝑛 − 𝑝𝑝)𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
⎢
⎥
⎣
⎦
Setting |𝑧𝑧| = 𝑅𝑅1 , we get the desired result.
Theorem 7: If 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈). Then 𝑓𝑓(𝑧𝑧) is p-valently convex of order 𝜇𝜇 (0 ≤ 𝜇𝜇 < 𝑝𝑝) in the disk |𝑧𝑧| < 𝑅𝑅2 ,
where
1
𝑛𝑛 −𝑝𝑝
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
⎡𝑝𝑝(𝑝𝑝 − 𝜇𝜇) �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]⎤
(𝑝𝑝 + 𝛽𝛽)
⎥ .
𝑅𝑅2 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑛𝑛 ⎢
𝑛𝑛(𝑛𝑛 − 𝑝𝑝)𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
⎢
⎥
⎣
⎦
Proof: It is sufficient to show that
for |𝑧𝑧| < 𝑅𝑅2 , we have
Thus
�1 +
�1 +
𝑧𝑧𝑓𝑓 ′′ (𝑧𝑧)
− 𝑝𝑝� ≤ 𝑝𝑝 − 𝜇𝜇, (0 ≤ 𝜇𝜇 < 𝑝𝑝)
𝑓𝑓 ′ (𝑧𝑧)
𝑛𝑛−𝑝𝑝
∑∞
𝑧𝑧𝑓𝑓 ′′ (𝑧𝑧)
𝑛𝑛=𝑝𝑝+1 𝑛𝑛(𝑛𝑛 − 𝑝𝑝)𝑎𝑎𝑛𝑛 |𝑧𝑧|
−
𝑝𝑝�
≤
.
𝑛𝑛−𝑝𝑝
𝑓𝑓 ′ (𝑧𝑧)
𝑝𝑝 − ∑∞
𝑛𝑛=𝑝𝑝+1 𝑛𝑛𝑎𝑎𝑛𝑛 |𝑧𝑧|
�1 +
if
Hence by Theorem 1, (23) will be true if
and hence
|𝑧𝑧|𝑛𝑛−𝑝𝑝
𝑛𝑛(𝑛𝑛 − 𝑝𝑝)𝑎𝑎𝑛𝑛
𝑝𝑝(𝑝𝑝 − 𝜇𝜇)
𝑧𝑧𝑓𝑓 ′′ (𝑧𝑧)
− 𝑝𝑝� ≤ 𝑝𝑝 − 𝜇𝜇,
𝑓𝑓 ′ (𝑧𝑧)
∑∞
𝑛𝑛=𝑝𝑝+1
≤
�1 +
𝑛𝑛(𝑛𝑛−𝑝𝑝)𝑎𝑎 𝑛𝑛 |𝑧𝑧|𝑛𝑛 −𝑝𝑝
𝑝𝑝(𝑝𝑝−𝜇𝜇 )
≤ 1.
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]
(𝑝𝑝 + 𝛽𝛽)
,
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
(23)
152
Waggas Galib Atshan et al.: On a New Class of Multivalent Functions with Negative
Coefficient Defined by Hadamard Product Involving a Linear Operator
1
𝑛𝑛−𝑝𝑝
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
⎡𝑝𝑝(𝑝𝑝 − 𝜇𝜇) �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]⎤
(𝑝𝑝 + 𝛽𝛽)
⎥ .
|𝑧𝑧| ≤ ⎢
𝑛𝑛(𝑛𝑛 − 𝑝𝑝)𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
⎢
⎥
⎣
⎦
Setting |𝑧𝑧| = 𝑅𝑅2 , we get the desired result.
Theorem 8: Let the function 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈). Then 𝑓𝑓(𝑧𝑧) is p-valently close-to-convex of order 𝜇𝜇 (0 ≤ 𝜇𝜇 < 𝑝𝑝)
in the disk |𝑧𝑧| < 𝑅𝑅3 , where
1
𝑛𝑛−𝑝𝑝
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
⎡𝑝𝑝(𝑝𝑝 − 𝜇𝜇) �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]⎤
(𝑝𝑝 + 𝛽𝛽)
⎥ .
𝑅𝑅3 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑛𝑛 ⎢
𝑛𝑛(𝑛𝑛 − 𝑝𝑝)𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
⎢
⎥
⎣
⎦
Proof: It is sufficient to show that
for |𝑧𝑧| < 𝑅𝑅3 , we have
𝑓𝑓 ′ (𝑧𝑧)
− 𝑝𝑝� ≤ 𝑝𝑝 − 𝜇𝜇,
𝑧𝑧 𝑝𝑝−1
∞
𝑓𝑓 ′ (𝑧𝑧)
� 𝑝𝑝−1 − 𝑝𝑝� ≤ � 𝑛𝑛𝑎𝑎𝑛𝑛 |𝑧𝑧|𝑛𝑛−𝑝𝑝 .
𝑧𝑧
𝑛𝑛=𝑝𝑝+1
Thus
𝑓𝑓 ′ (𝑧𝑧)
� 𝑝𝑝−1 − 𝑝𝑝� ≤ 𝑝𝑝 − 𝜇𝜇,
𝑧𝑧
if
hence, by Theorem 1, (24) will be true if
and hence
(0 ≤ 𝜇𝜇 < 𝑝𝑝)
�
∑∞
𝑛𝑛=𝑝𝑝+1
𝑛𝑛𝑎𝑎 𝑛𝑛 |𝑧𝑧|𝑛𝑛 −𝑝𝑝
𝑝𝑝−𝜇𝜇
𝑚𝑚
≤ 1,
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
𝑛𝑛|𝑧𝑧|𝑛𝑛−𝑝𝑝 �1 + (𝑝𝑝 + 𝛽𝛽) � [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]
≤
,
(𝑝𝑝 − 𝜇𝜇)
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
(24)
1
𝑛𝑛−𝑝𝑝
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
⎡(𝑝𝑝 − 𝜇𝜇) �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]⎤
(𝑝𝑝 + 𝛽𝛽)
⎥ .
|𝑧𝑧| ≤ ⎢
𝑛𝑛𝑛𝑛𝑛𝑛(𝛼𝛼 + 𝜈𝜈)
⎢
⎥
⎣
⎦
Setting |𝑧𝑧| = 𝑅𝑅3 , we get the desired result.
5. Weighted Mean
Definition 3: Let 𝑓𝑓1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓2 be in the class𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈). Then the weighted mean 𝑤𝑤𝑞𝑞 of 𝑓𝑓1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓2 is given by
1
𝑤𝑤𝑞𝑞 = [(1 − 𝑞𝑞)𝑓𝑓1 (𝑧𝑧) + (1 + 𝑞𝑞)𝑓𝑓2 (𝑧𝑧)],
0 < 𝑞𝑞 < 1.
2
Theorem 9: Let 𝑓𝑓1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓2 be in the class 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈). Then the weighted mean 𝑤𝑤𝑞𝑞 of 𝑓𝑓1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓2 is also in the
class 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈).
Poof: By Definition 3, we have
American Journal of Mathematics and Statistics 2014, 4(3): 147-155
1
𝑤𝑤𝑞𝑞 = [(1 − 𝑞𝑞)𝑓𝑓1 (𝑧𝑧) + (1 + 𝑞𝑞)𝑓𝑓2 (𝑧𝑧)]
2
∞
∞
𝑛𝑛=𝑝𝑝+1
𝑛𝑛=𝑝𝑝+1
153
1
= �(1 − 𝑞𝑞) �𝑧𝑧 𝑝𝑝 − � 𝑎𝑎𝑛𝑛,1 𝑧𝑧 𝑛𝑛 � + (1 + 𝑞𝑞) �𝑧𝑧 𝑝𝑝 − � 𝑎𝑎𝑛𝑛,2 𝑧𝑧 𝑛𝑛 ��
2
1
𝑛𝑛
= 𝑧𝑧 𝑝𝑝 − ∑∞
𝑛𝑛=𝑝𝑝+1 �(1 − 𝑞𝑞)𝑎𝑎𝑛𝑛,1 + (1 + 𝑞𝑞)𝑎𝑎𝑛𝑛,2 �𝑧𝑧 .
2
(25)
Since 𝑓𝑓1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓2 are in the class 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈) so by Theorem 1, we get
∞
� �1 +
𝑛𝑛=𝑝𝑝+1
and
∞
� �1 +
𝑛𝑛=𝑝𝑝+1
Hence
�1 +
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑎𝑎𝑛𝑛,1 𝑏𝑏𝑛𝑛 ≤ 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
(𝑝𝑝 + 𝛽𝛽)
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑎𝑎𝑛𝑛,2 𝑏𝑏𝑛𝑛 ≤ 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈),
(𝑝𝑝 + 𝛽𝛽)
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
1
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)] � �(1 − 𝑞𝑞)𝑎𝑎𝑛𝑛,1 + (1 + 𝑞𝑞)𝑎𝑎𝑛𝑛,2 �� 𝑏𝑏𝑛𝑛
(𝑝𝑝 + 𝛽𝛽)
2
∞
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
1
= (1 − 𝑞𝑞) � �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑎𝑎𝑛𝑛,1 𝑏𝑏𝑛𝑛
(𝑝𝑝 + 𝛽𝛽)
2
𝑛𝑛=𝑝𝑝+1
∞
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
1
+ (1 + 𝑞𝑞) � �1 +
� [𝑛𝑛(𝜆𝜆(𝜈𝜈(𝑛𝑛 − 1) − 𝛼𝛼) + 𝑛𝑛)]𝑎𝑎𝑛𝑛,2 𝑏𝑏𝑛𝑛
(𝑝𝑝 + 𝛽𝛽)
2
𝑛𝑛=𝑝𝑝+1
1
1
≤ (1 − 𝑞𝑞)𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈) + (1 + 𝑞𝑞)𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈) = 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈).
2
2
Therefore 𝑤𝑤𝑞𝑞 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈).
The proof is complete.
6. Neighborhoods and Partial Sums
Now we define the (𝑛𝑛 − 𝛿𝛿) −neighborhoods of the function 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 by
(26)
For identity function𝑒𝑒(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 , (𝑝𝑝 ∈ 𝑁𝑁)
(27)
∞
𝑛𝑛
𝑁𝑁𝑛𝑛,𝛿𝛿 (𝑓𝑓) = �𝑔𝑔 ∈ 𝑆𝑆𝑚𝑚 : 𝑔𝑔(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 − ∑∞
𝑛𝑛=𝑝𝑝+1 𝑏𝑏𝑛𝑛 𝑧𝑧 𝑎𝑎𝑎𝑎𝑎𝑎 ∑𝑛𝑛=𝑝𝑝+1 𝑛𝑛|𝑎𝑎𝑛𝑛 − 𝑏𝑏𝑛𝑛 | ≤ 𝛿𝛿, 0 ≤ 𝛿𝛿 < 1�.
∞
𝑛𝑛
𝑁𝑁𝑛𝑛,𝛿𝛿 (𝑒𝑒) = �𝑔𝑔 ∈ 𝑆𝑆𝑚𝑚 : 𝑔𝑔(𝑧𝑧) = 𝑧𝑧 𝑝𝑝 − ∑∞
𝑛𝑛=𝑝𝑝+1 𝑏𝑏𝑛𝑛 𝑧𝑧 𝑎𝑎𝑎𝑎𝑎𝑎 ∑𝑛𝑛=𝑝𝑝+1 𝑛𝑛|𝑏𝑏𝑛𝑛 | ≤ 𝛿𝛿 ,0 ≤ 𝛿𝛿 < 1�.
The concept of neighborhoods was first introduced by Goodman [5] and then generalized by Ruscheweyh [9].
Definition 4: A function 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 is said to be in the class 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈), if there exist a function
𝑔𝑔 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈) such that
𝑓𝑓(𝑧𝑧)
− 1� < 𝑝𝑝 − 𝜂𝜂
𝑔𝑔(𝑧𝑧)
�
Theorem 10: If 𝑔𝑔 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈) and
𝜂𝜂 = 𝑝𝑝 −
𝛾𝛾
𝑚𝑚
(𝑧𝑧 ∈ 𝑈𝑈, 0 ≤ 𝜂𝜂 < 1).
𝛿𝛿�1+(𝑝𝑝 +𝛽𝛽 )� �(𝑝𝑝+1)�𝜆𝜆(𝜈𝜈𝜈𝜈 −𝛼𝛼)+(𝑝𝑝+1)��𝑎𝑎 𝑝𝑝 +1
(𝑛𝑛 −𝑝𝑝 )𝛾𝛾 𝑚𝑚
� �(𝑝𝑝+1)�𝜆𝜆(𝜈𝜈𝜈𝜈 −𝛼𝛼)+(𝑝𝑝+1)��𝑎𝑎 𝑝𝑝 +1 −𝑝𝑝𝑝𝑝 (𝛼𝛼+𝜈𝜈)
(𝑝𝑝 +𝛽𝛽 )
(𝑝𝑝+1)�1+
Then 𝑁𝑁𝑛𝑛,𝛿𝛿 (𝑔𝑔) ⊂ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈).
Proof: Let 𝑓𝑓 ∈ 𝑁𝑁𝑛𝑛,𝛿𝛿 (𝑔𝑔). Then we have from (26) that
.
(28)
Waggas Galib Atshan et al.: On a New Class of Multivalent Functions with Negative
Coefficient Defined by Hadamard Product Involving a Linear Operator
154
∞
� 𝑛𝑛|𝑎𝑎𝑛𝑛 − 𝑏𝑏𝑛𝑛 | ≤ 𝛿𝛿,
𝑛𝑛=𝑝𝑝+1
which readily implies the following coefficient inequality
∞
� |𝑎𝑎𝑛𝑛 − 𝑏𝑏𝑛𝑛 | ≤
𝑛𝑛=𝑝𝑝+1
Next, since 𝑔𝑔 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈), we have from Theorem 1
∞
� 𝑏𝑏𝑛𝑛 ≤
𝑛𝑛=𝑝𝑝+1
so that
�1 +
𝛿𝛿
.
𝑝𝑝 + 1
𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
𝑚𝑚
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾
� �(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑎𝑎𝑝𝑝+1
(𝑝𝑝 + 𝛽𝛽)
,
∑∞
𝑓𝑓(𝑧𝑧)
𝑛𝑛=𝑝𝑝+1 |𝑎𝑎𝑛𝑛 − 𝑏𝑏𝑛𝑛 |
− 1� ≤
1
− ∑∞
𝑔𝑔(𝑧𝑧)
𝑛𝑛=𝑝𝑝+1 𝑏𝑏𝑛𝑛
�
𝑚𝑚
𝛾𝛾
�
�(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑎𝑎𝑝𝑝+1
𝛿𝛿
(𝑝𝑝 + 𝛽𝛽)
≤
= 𝑝𝑝 − 𝜂𝜂.
𝑝𝑝 + 1
(𝑛𝑛 − 𝑝𝑝)𝛾𝛾 𝑚𝑚
�1 +
� �(𝑝𝑝 + 1)�𝜆𝜆(𝜈𝜈𝜈𝜈 − 𝛼𝛼) + (𝑝𝑝 + 1)��𝑎𝑎𝑝𝑝+1 − 𝑝𝑝𝑝𝑝(𝛼𝛼 + 𝜈𝜈)
(𝑝𝑝 + 𝛽𝛽)
�1 +
Then by Definition 3, 𝑓𝑓 ∈ 𝐻𝐻𝑚𝑚 (𝛾𝛾, 𝛽𝛽, 𝑚𝑚, 𝜆𝜆, 𝛼𝛼, 𝜈𝜈) for every 𝜂𝜂 given by (28).
Now, we introduce the partial sums.
Theorem 11: Let 𝑓𝑓 ∈ 𝑆𝑆𝑚𝑚 be given by (2) and define the 𝑆𝑆1 (𝑧𝑧)𝑎𝑎𝑎𝑎𝑎𝑎 𝑆𝑆𝜄𝜄 (𝑧𝑧) by
𝑆𝑆1 (𝑧𝑧) = 𝑧𝑧 𝑝𝑝
and
𝑝𝑝+𝜄𝜄−1
𝑆𝑆 𝜄𝜄 (𝑧𝑧) = 𝑧𝑧 𝑝𝑝 − ∑𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 𝑧𝑧 𝑛𝑛 ,
suppose also that
Thus, we have
∑∞
𝑛𝑛=𝑝𝑝+1 𝑑𝑑𝑛𝑛 𝑎𝑎𝑛𝑛
≤ 1, �𝑑𝑑𝑛𝑛 =
𝑅𝑅𝑅𝑅 �
and
𝑅𝑅𝑅𝑅 �
(𝑛𝑛 −𝑝𝑝 )𝛾𝛾 𝑚𝑚
� [𝑛𝑛(𝜆𝜆(𝜈𝜈 (𝑛𝑛−1)−𝛼𝛼)+𝑛𝑛)]𝑏𝑏𝑛𝑛
+𝛽𝛽 )
�1+ (𝑝𝑝
𝑓𝑓(𝑧𝑧)
𝑆𝑆 𝜄𝜄 (𝑧𝑧)
𝑓𝑓(𝑧𝑧)
𝑝𝑝𝑝𝑝 (𝛼𝛼+𝜈𝜈)
�>1−
𝑆𝑆 𝜄𝜄 (𝑧𝑧)
�>1−
(30)
𝜄𝜄 > 𝑝𝑝 + 1,
1
𝑑𝑑 𝑛𝑛
(31)
(32)
𝑑𝑑 𝑛𝑛
1+𝑑𝑑 𝑛𝑛
�.
.
(33)
Each of the bounds in (32) and (33) is the best possible for 𝑝𝑝 ∈ 𝑁𝑁.
Proof: For the coefficients 𝑑𝑑𝑛𝑛 given by (31), it is not difficult to verity that
𝑑𝑑𝑛𝑛+1 > 𝑑𝑑𝑛𝑛 > 1,
Therefore, by using the hypothesis (30), we have
By setting
and applying (34), we find that
𝑛𝑛 = 𝑝𝑝 + 1, 𝑝𝑝 + 2, ….
∞
∞
∑∞
𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 + 𝑑𝑑𝜄𝜄 ∑𝑛𝑛=𝑝𝑝+𝜄𝜄 𝑎𝑎𝑛𝑛 ≤ ∑𝑛𝑛=𝑝𝑝+1 𝑑𝑑𝑛𝑛 𝑎𝑎𝑛𝑛 ≤ 1.
𝑔𝑔1 = 𝑑𝑑𝜄𝜄 �
𝑛𝑛−𝑝𝑝
𝑑𝑑𝜄𝜄 ∑∞
𝑓𝑓(𝑧𝑧)
1
𝑛𝑛=𝑝𝑝+𝜄𝜄 𝑎𝑎𝑛𝑛 𝑧𝑧
− �1 − �� = 1 −
𝑝𝑝+𝜄𝜄−1
𝑆𝑆𝜄𝜄(𝑧𝑧)
𝑑𝑑𝜄𝜄
1 − ∑𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 𝑧𝑧 𝑛𝑛−𝑝𝑝
𝑑𝑑𝜄𝜄 ∑∞
𝑔𝑔1 (𝑧𝑧) − 1
𝑛𝑛=𝑝𝑝+𝜄𝜄 𝑎𝑎𝑛𝑛
�≤
≤ 1.
�
𝑝𝑝+𝜄𝜄−1
𝑔𝑔1 (𝑧𝑧) + 1
2 − 2 ∑𝑛𝑛=𝑝𝑝+1 𝑎𝑎𝑛𝑛 − 𝑑𝑑𝜄𝜄 ∑∞
𝑛𝑛=𝑝𝑝+𝜄𝜄 𝑎𝑎𝑛𝑛
This prove (32). Therefore, 𝑅𝑅𝑅𝑅�𝑔𝑔1 (𝑧𝑧)� > 0, and we obtain
(34)
American Journal of Mathematics and Statistics 2014, 4(3): 147-155
𝑅𝑅𝑅𝑅 �
155
𝑓𝑓(𝑧𝑧)
1
�>1− .
𝑆𝑆1 (𝑧𝑧)
𝑑𝑑𝑛𝑛
Now, in the same manner, we prove the assertion(33), by setting
and this completes the proof.
𝑔𝑔2 (𝑧𝑧) = (1 + 𝑑𝑑𝑛𝑛 ) �
𝑑𝑑𝑛𝑛
𝑆𝑆𝜄𝜄 (𝑧𝑧)
−
�,
𝑓𝑓(𝑧𝑧) 1 + 𝑑𝑑𝑛𝑛
involving a linear operator, Int. Journal of Math. Analysis Vol
7. 2013, no. 24, 1193-1206.
REFERENCES
[5]
A. W. Goodman, Univalent functions and non-analytic curves,
Proc. Amer. Math. Soc., 8(3)(1957), 598-601.
[1]
W. G. Atshan, On a certain class of multivalent functions
defined by Hadamard product, Journal of Asian Scientific
Research, 3(9)(2013), : 891-902.
[6]
S. M. Khairua and M. More, Some analytic and multivalent
functions defined by subordination property. Gen. Math. 17(3)
(2009) 105-124.
[2]
W. G. Atshan and A. S. Joudah, Subclass of meromorphic
univalent functions defined by Hadamard product with
multiplier transformation, Int. Math. Forum, 6(46) (2011),
2279-2292.
[7]
H. Mahzoon, New subclass of multivalent functions defined
by differential subordination. Appl. Math. Sciences, 6(95)
(2012), 4701.
[3]
W. G. Atshan and S. R. Kulkarni, On application of
differential subordination for certain subclass of
meromorphically p-valent functions with positive coefficients
defined by linear operator, J. Ineq. Pute Appl. Math.,
10(2)(2009), Articale 53, 11 pages.
[8]
H. Mahzoon and S. Latha, Neighborhoods of multivalent
functions, Int. Journal of Math. Anlysis, 3(30) (2009),
15011-1507.
[9]
S. Rucheweyh, Neighborhoods of univalent functions, Proc.
Amer. Math. Soc. 81(1981), 521-527.
[4]
W. G. Atshan, H. J. Mustafa and E. k. Mouajueed, Subclass of
multivalent functions defined by Hadamard product