Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 127615, 7 pages
http://dx.doi.org/10.1155/2013/127615
Research Article
Fekete-Szegy Inequality for a Subclass of
𝑝-Valent Analytic Functions
Mohsan Raza,1 Muhammad Arif,2 and Maslina Darus3
1
Department of Mathematics, Government College University Faisalabad, Faisalabad, Punjab 38000, Pakistan
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa 23200, Pakistan
3
School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, 43600 Bangi,
Selangor D. Ehsan, Malaysia
2
Correspondence should be addressed to Muhammad Arif; marifmaths@yahoo.com
Received 18 January 2013; Accepted 11 April 2013
Academic Editor: Mina Abd-El-Malek
Copyright © 2013 Mohsan Raza 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.
The main object of this paper is to study Fekete-Szegő problem for the class of 𝑝-valent functions. Fekete-Szegő inequality of several
classes is obtained as special cases from our results. Applications of the results are also obtained on the class defined by convolution.
1. Introduction and Preliminaries
Let 𝐴 𝑝 denote the class of functions 𝑓(𝑧) of the form
∞
𝑓 (𝑧) = 𝑧𝑝 + ∑ 𝑎𝑛 𝑧𝑛 ,
(𝑝 ∈ N = {1, 2, 3, . . .}) ,
𝑛=𝑝+1
(1)
which are analytic in the open unit disk 𝐸. Also, 𝐴 1 = 𝐴, the
usual class of analytic functions defined in the open unit disk
𝐸 = {𝑧 : |𝑧| < 1}. Let 𝑓(𝑧) and 𝑔(𝑧) be analytic in 𝐸. We say
that the function 𝑓 is subordinate to the function 𝑔 and write
𝑓(𝑧) ≺ 𝑔(𝑧), if and only if there exists Schwarz function 𝑤,
analytic in 𝐸 such that 𝑤(0) = 0, |𝑤(𝑧)| < 1 for 𝑧 ∈ 𝐸, and
𝑓(𝑧) = 𝑔(𝑤(𝑧)). In particular, if 𝑔 is univalent in 𝐸, then we
have the following equivalence:
𝑓 (𝑧) ≺ 𝑔 (𝑧) ⇐⇒ 𝑓 (0) = 𝑔 (0) ,
𝑓 (𝐸) ⊂ 𝑔 (𝐸) .
(2)
For any two analytic functions 𝑓(𝑧) of the form (1) and
𝑔(𝑧) with
𝑧𝑓󸀠 (𝑧)
≺ 𝜙 (𝑧) ,
𝑝𝑓 (𝑧)
𝑧 ∈ 𝐸.
(5)
The class 𝑆𝑝∗ (𝜙) was defined and studied by Ali et al. [1]. They
obtained the Fekete-Szegő inequality for functions in the class
𝑆𝑝∗ (𝜙). The class 𝑆1∗ (𝜙) coincides with the class 𝑆∗ (𝜙) discussed
by Ma and Minda [2]. Owa [3] introduced a subclass of 𝑝valently Bazilevic functions 𝐻𝑝 (𝐴, 𝐵, 𝛼, 𝛽). A function 𝑓 ∈
𝐴 𝑝 is said to be in the class 𝐻𝑝 (𝐴, 𝐵, 𝛼, 𝛽) if and only if
(1 − 𝛽) (
𝑓(𝑧) 𝛼
𝑧𝑓󸀠 (𝑧) 𝑓 (𝑧) 𝛼 1 + 𝐴𝑧
+
𝛽
,
)
) ≺
(
𝑧𝑝
𝑝𝑓 (𝑧) 𝑧𝑝
1 + 𝐵𝑧
𝑧 ∈ 𝐸,
(6)
∞
𝑔 (𝑧) = 𝑧𝑝 + ∑ 𝑏𝑛 𝑧𝑛 ,
𝑧 ∈ 𝐸,
𝑛=𝑝+1
(3)
the convolution (Hadamard product) is given by
𝑝
Let 𝜙(𝑧) be an analytic function with positive real part on 𝐸
with 𝜙(0) = 1, 𝜙󸀠 (0) > 0, which maps the unit disk 𝐸 onto
a region starlike with respect to 1 which is symmetric with
respect to the real axis. Denote by 𝑆𝑝∗ (𝜙) the class of functions
𝑓 analytic in 𝐸 for which
∞
𝑛
(𝑓 ∗ 𝑔) (𝑧) = 𝑧 + ∑ 𝑎𝑛 𝑏𝑛 𝑧 ,
𝑛=𝑝+1
𝑧 ∈ 𝐸.
(4)
where −1 ≤ 𝐵 < 𝐴 ≤ 1, 𝛼 ≥ 0, and 0 ≤ 𝛽 ≤ 1. We now define
the following subclass of analytic functions.
Definition 1. Let 𝜙(𝑧) be a univalent starlike function with
respect to 1 which maps the unit disk 𝐸 onto a region in the
right half-plane which is symmetric with respect to the real
2
Journal of Applied Mathematics
axis with 𝜙(0) = 1 and 𝜙󸀠 (0) > 0. A function 𝑓 ∈ 𝐴 𝑝 is in the
class 𝑉𝑝,𝑏,𝛼,𝛽 (𝜙) if
1−
𝛼
𝑓 (𝑧)
2 2
+ { (1 − 𝛽) ( 𝑝 )
𝑏 𝑏
𝑧
𝑧𝑓󸀠 (𝑧) 𝑓 (𝑧) 𝛼
+𝛽
) } ≺ 𝜙 (𝑧) ,
(
𝑝𝑓 (𝑧) 𝑧𝑝
We need the following results to obtain our main results.
Lemma 3 (see [1]). Let Ω be the class of analytic functions 𝑤,
normalized by 𝑤(0) = 0, satisfying condition |𝑤(𝑧)| < 1. If 𝑤 ∈
Ω and 𝑤(𝑧) = 𝑤1 𝑧 + 𝑤2 𝑧2 + ⋅ ⋅ ⋅ , 𝑧 ∈ 𝐸, then
(7)
−𝑡,
{
{
󵄨󵄨
2 󵄨󵄨
󵄨󵄨𝑤2 − 𝑡𝑤1 󵄨󵄨 ≤ {1,
󵄨 {
󵄨
{𝑡,
where 0 ≤ 𝛽 ≤ 1, 𝛼 ≥ 0, and 𝑏 > 0.
Definition 2. A function 𝑓 ∈ 𝐴 𝑝 is in the class 𝑉𝑝,𝑏,𝛼,𝛽,𝑔 (𝜙) if
𝛼
(𝑓 ∗ 𝑔) (𝑧)
2 2
)
1 − + { (1 − 𝛽) (
𝑏 𝑏
𝑧𝑝
󸀠
+𝛽
𝛼
𝑧(𝑓 ∗ 𝑔) (𝑧) (𝑓 ∗ 𝑔) (𝑧)
(
) } ≺ 𝜙 (𝑧) ,
𝑧𝑝
𝑝 (𝑓 ∗ 𝑔) (𝑧)
(8)
where 0 ≤ 𝛽 ≤ 1, 𝛼 ≥ 0, and 𝑏 > 0. In other words, a function 𝑓 ∈ 𝐴 𝑝 is in the class 𝑉𝑝,𝑏,𝛼,𝛽,𝑔 (𝜙) if (𝑓 ∗ 𝑔)(𝑧) ∈
𝑉𝑝,𝑏,𝛼,𝛽 (𝜙).
We have the following special cases.
(i) 𝑉𝑝,2,1,1 (𝜙) coincides with the class
and studied by Ali et al. [1].
𝑆𝑝∗ (𝜙) introduced
(ii) For 𝑝 = 1, 𝑏 = 2, and 𝛽 = 1, we have the class 𝐵𝛼 (𝜙)
introduced and studied by Ravichandran et al. [4].
(iii) For 𝑏 = 2 and 𝜙(𝑧) = (1 + 𝐴𝑧)/(1 + 𝐵𝑧), the class
𝑉𝑝,2,𝛼,𝛽 (𝜙) reduces to 𝐻𝑝 (𝐴, 𝐵, 𝛼, 𝛽) introduced and
studied by Owa [3].
(iv) For 𝜙(𝑧) = (1 + (1 − 2𝛾)𝑧)/(1 − 𝑧), the class 𝑉𝑝,2,𝛼,1 (𝜙)
reduces to the class 𝐵𝑝 (𝛼, 𝛾) defined as
𝑧𝑓 (𝑧) 𝑓 (𝑧)
) ) > 𝛾,
𝐵𝑝 (𝛼, 𝛾) = {𝑓 ∈ 𝐴 𝑝 : Re (
(
𝑝𝑓 (𝑧) 𝑧𝑝
(9)
(v) For 𝜙(𝑧) = (1 + (1 − 2𝛾)𝑧)/(1 − 𝑧), the class 𝑉𝑝,2,𝛼,0 (𝜙)
is defined as
𝑓 (𝑧) 𝛼
) > 𝛾, 0 ≤ 𝛾 < 1, 𝑧 ∈ 𝐸} .
𝑧𝑝
(10)
(vi) 𝑉1,2,0,1 (𝜙) = 𝑆∗ (𝜙) is investigated by Ma and Minda
[2].
(vii) For 𝜙(𝑧) = (1 + (1 − 2𝛾)𝑧)/(1 − 𝑧), the class 𝑉1,2,1,0 (𝜙)
reduces to the class
𝐵𝛾 = {𝑓 ∈ 𝐴 1 : Re
studied by Chen [5].
󵄨
󵄨󵄨
󵄨󵄨𝑤2 − 𝑡𝑤12 󵄨󵄨󵄨 + (1 + 𝑡) 󵄨󵄨󵄨󵄨𝑤1 󵄨󵄨󵄨󵄨2 ≤ 1,
󵄨
󵄨
󵄨󵄨
󵄨
󵄨󵄨𝑤2 − 𝑡𝑤12 󵄨󵄨󵄨 + (1 − 𝑡) 󵄨󵄨󵄨󵄨𝑤1 󵄨󵄨󵄨󵄨2 ≤ 1,
󵄨
󵄨
𝑓 (𝑧)
> 𝛾} .
𝑧
−1 < 𝑡 ≤ 0,
0 < 𝑡 < 1.
(13)
Lemma 4 (see [6, (7), page 10]). If 𝑤 ∈ Ω and 𝑤(𝑧) = 𝑤1 𝑧 +
𝑤2 𝑧2 + ⋅ ⋅ ⋅ , 𝑧 ∈ 𝐸, then
󵄨
󵄨󵄨
󵄨󵄨𝑤2 − 𝑡𝑤12 󵄨󵄨󵄨 ≤ max {1; |𝑡|} ,
󵄨
󵄨
(14)
for any complex number 𝑡. The result is sharp for the functions
𝑤(𝑧) = 𝑧2 or 𝑤(𝑧) = 𝑧.
Lemma 5 (see [7]). If 𝑤 ∈ Ω, then for any real number 𝑞1 and
𝑞2 , the following sharp estimate holds:
󵄨
󵄨󵄨
󵄨󵄨𝑤3 + 𝑞1 𝑤1 𝑤2 + 𝑞2 𝑤13 󵄨󵄨󵄨 ≤ 𝐻 (𝑞1 , 𝑞2 ) ,
󵄨
󵄨
(15)
where
𝐻 (𝑞1 , 𝑞2 )
0 ≤ 𝛾 < 1, 𝑧 ∈ 𝐸} .
{𝑓 ∈ 𝐴 𝑝 : Re (
(12)
For 𝑡 ≤ −1 or 𝑡 ≥ 1, the equality holds, if and only if 𝑤(𝑧) = 𝑧
or one of its rotation. For −1 ≤ 𝑡 ≤ 1, the equality holds, if
𝑤(𝑧) = 𝑧2 or one of its rotation. the equality holds for 𝑡 = −1,
if and only if 𝑤(𝑧) = 𝑧((𝜆 + 𝑧)/(1 + 𝜆𝑧)) (0 ≤ 𝜆 ≤ 1) or one
of its rotation, while for 𝑡 = 1, the equality holds, if and only if
𝑤(𝑧) = −𝑧((𝜆 + 𝑧)/(1 + 𝜆𝑧)) (0 ≤ 𝜆 ≤ 1) or one of its rotation.
The above upper bound for −1 < 𝑡 < 1 is sharp, and it can be
improved as follows:
𝛼
󸀠
𝑡 < −1,
−1 ≤ 𝑡 ≤ 1,
𝑡 > 1.
(11)
1,
{
{
{
{
(𝑞1 , 𝑞2 ) ∈ 𝐷1 ∪ 𝐷2 ,
{
{
{
󵄨󵄨 󵄨󵄨
{
{
𝑞
󵄨
{
󵄨 2 󵄨󵄨 ,
{
{
7
{
{
{
{ (𝑞1 , 𝑞2 ) ∈ ⋃ 𝐷𝑘 ,
{
{
{
𝑘=3
{
{
1/2
{
󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1
{
2
󵄨
󵄨
{
󵄨
󵄨
󵄨
󵄨
{
) ,
(󵄨𝑞 󵄨 + 1) ( 󵄨󵄨 󵄨󵄨
{
{
{ 3 󵄨 1󵄨
3 (󵄨󵄨𝑞1 󵄨󵄨 + 1 + 𝑞2 )
={
{
{ (𝑞1 , 𝑞2 ) ∈ 𝐷8 ∪ 𝐷9 ,
{
1/2
{
{
𝑞12 − 4
𝑞12 − 4
{
1
{
{
(
)
(
,
)
𝑞
{
{
3 2 𝑞12 − 4𝑞2
3 (𝑞2 − 1)
{
{
{
{
{
(𝑞1 , 𝑞2 ) ∈ 𝐷10 ∪ 𝐷11 \ {±2, 1} ,
{
{
{
1/2
󵄨󵄨 󵄨󵄨
{
{
󵄨󵄨𝑞1 󵄨󵄨 − 1
2 󵄨󵄨 󵄨󵄨
{
{
) ,
(󵄨𝑞 󵄨 − 1) ( 󵄨󵄨 󵄨󵄨
{
{
{
3 󵄨 1󵄨
3 (󵄨󵄨𝑞1 󵄨󵄨 − 1 − 𝑞2 )
{
{
{ (𝑞1 , 𝑞2 ) ∈ 𝐷12 .
(16)
Journal of Applied Mathematics
3
The extremal function up to the rotations is of the form
𝑤 (𝑧) = 𝑧3 ,
𝑤 (𝑧) = 𝑤0 (𝑧) =
𝑤 (𝑧) = 𝑧,
𝑧 ([(1 − 𝜆) 𝜀2 + 𝜆𝜀1 ] − 𝜀1 𝜀2 𝑧)
,
1 − [(1 − 𝜆) 𝜀1 + 𝜆𝜀2 ] 𝑧
𝑤 (𝑧) = 𝑤1 (𝑧) =
𝑧 (𝑡1 − 𝑧)
,
1 − 𝑡1 𝑧
𝑤 (𝑧) = 𝑤2 (𝑧) =
𝑧 (𝑡2 + 𝑧)
,
1 + 𝑡2 𝑧
󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨
󵄨󵄨𝜀1 󵄨󵄨 = 󵄨󵄨𝜀2 󵄨󵄨 = 1,
𝜃0
,
2
𝑡0 = (
3 (𝑞2 − 1) (𝑞12 − 4𝑞2 )
𝑡1 = (
𝑏 ± 𝑎 (17)
𝜆=
,
2𝑏
𝜃0
,
2
2𝑞2 (𝑞12 + 2) − 3𝑞12
1/2
)
,
1/2
󵄨󵄨 󵄨󵄨
󵄨󵄨𝑞1 󵄨󵄨 + 1
) ,
󵄨󵄨 󵄨󵄨
3 (󵄨󵄨𝑞1 󵄨󵄨 + 1 + 𝑞2 )
𝜃0 𝑞1
=
(
2
2
𝑞2 (𝑞12 + 8) − 2 (𝑞12 + 2)
2𝑞2 (𝑞12
+ 2) −
󵄨 󵄨
𝐷9 = { (𝑞1 , 𝑞2 ) : 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≥ 2,
3𝑞12
󵄨 󵄨 󵄨 󵄨
2 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1)
1 2
≤ 𝑞2 ≤
(𝑞 + 8)} ,
󵄨 󵄨
12 1
𝑞12 + 2 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 4
󵄨 󵄨
𝐷11 = { (𝑞1 , 𝑞2 ) : 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≥ 4,
󵄨 󵄨 󵄨 󵄨
󵄨 󵄨 󵄨 󵄨
2 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1)
2 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 − 1)
≤
𝑞
},
≤
󵄨
󵄨
󵄨 󵄨
2
𝑞12 + 2 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 4
𝑞12 − 2 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 4
󵄨 󵄨
𝐷12 = { (𝑞1 , 𝑞2 ) : 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≥ 4,
󵄨 󵄨 󵄨 󵄨
2 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 − 1)
2 󵄨 󵄨
≤ 𝑞2 ≤ (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 − 1)} .
󵄨󵄨 󵄨󵄨
2
3
𝑞1 − 2 󵄨󵄨𝑞1 󵄨󵄨 + 4
1/2
󵄨󵄨 󵄨󵄨
󵄨󵄨𝑞1 󵄨󵄨 − 1
𝑡2 = ( 󵄨󵄨 󵄨󵄨
) ,
3 (󵄨󵄨𝑞1 󵄨󵄨 − 1 − 𝑞2 )
cos
2 󵄨 󵄨
4 󵄨 󵄨
3
󵄨 󵄨
− (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1) ≤ 𝑞2 ≤ (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1) − (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1)} ,
3
27
󵄨 󵄨
𝐷10 = { (𝑞1 , 𝑞2 ) : 2 ≤ 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≤ 4,
𝜀1 = 𝑡0 − 𝑒−𝑖𝜃0 /2 (𝑎 ∓ 𝑏) ,
𝑏 = √ 1 − 𝑡02 sin2
(18)
2. Main Results
).
The sets 𝐷𝑘 , 𝑘 = 1, 2, . . . , 12 are defined as follows:
Theorem 6. Let 𝜙(𝑧) = 1 + 𝐵1 𝑧 + 𝐵2 𝑧2 + 𝐵3 𝑧3 + ⋅ ⋅ ⋅ , where 𝐵𝑛 ,
𝑠 are real with 𝐵1 > 0 and 𝐵2 ≥ 0. Let
2
𝜎1 =
󵄨 󵄨 1 󵄨 󵄨
𝐷1 = {(𝑞1 , 𝑞2 ) : 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≤ , 󵄨󵄨󵄨𝑞2 󵄨󵄨󵄨 ≤ 1} ,
2
𝐷2 = {(𝑞1 , 𝑞2 ) :
1 󵄨󵄨 󵄨󵄨
≤ 󵄨𝑞 󵄨 ≤ 2,
2 󵄨 1󵄨
󵄨 󵄨 󵄨 󵄨
2 󵄨󵄨𝑞 󵄨󵄨 (󵄨󵄨𝑞 󵄨󵄨 + 1)
2 󵄨 󵄨
− (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1) ≤ 𝑞2 ≤ 󵄨2 1 󵄨 󵄨󵄨󵄨 1 󵄨󵄨󵄨
},
3
𝑞1 + 2 󵄨󵄨𝑞1 󵄨󵄨 + 4
𝜀2 = −𝑒−𝑖𝜃0 /2 (𝑖𝑎 ± 𝑏) ,
𝑎 = 𝑡0 cos
𝐷8 = {(𝑞1 , 𝑞2 ) :
1 󵄨󵄨 󵄨󵄨
≤ 󵄨𝑞 󵄨 ≤ 2,
2 󵄨 1󵄨
4 󵄨󵄨 󵄨󵄨
3
󵄨 󵄨
(󵄨𝑞 󵄨 + 1) − (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1) ≤ 𝑞2 ≤ 1} ,
27 󵄨 1 󵄨
× {2 (𝐵2 − 𝐵1 ) − 𝑏𝑝𝐵12
1 2
󵄨 󵄨
𝐷6 = {(𝑞1 , 𝑞2 ) : 2 ≤ 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≤ 4, 𝑞2 ≥
(𝑞 + 8)} ,
12 1
2 󵄨 󵄨
󵄨 󵄨
𝐷7 = {(𝑞1 , 𝑞2 ) : 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≥ 4, 𝑞2 ≥ (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 − 1)} ,
3
(𝛼 − 1) (𝛼𝑝 + 2𝛽)
2(𝛼𝑝 + 𝛽)
2
},
2
𝜎2 =
(𝛼𝑝 + 𝛽)
𝑏𝑝𝐵12 (𝛼𝑝 + 2𝛽)
× {2 (𝐵2 + 𝐵1 ) − 𝑏𝑝𝐵12
󵄨 󵄨 1 󵄨 󵄨
𝐷3 = {(𝑞1 , 𝑞2 ) : 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≤ , 󵄨󵄨󵄨𝑞2 󵄨󵄨󵄨 ≤ −1} ,
2
2 󵄨 󵄨
󵄨 󵄨 1
𝐷4 = {(𝑞1 , 𝑞2 ) : 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≥ , 𝑞2 ≤ − (󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 + 1)} ,
2
3
󵄨 󵄨
𝐷5 = {(𝑞1 , 𝑞2 ) : 󵄨󵄨󵄨𝑞1 󵄨󵄨󵄨 ≤ 2, 𝑞2 ≥ 1} ,
(𝛼𝑝 + 𝛽)
𝑏𝑝𝐵12 (𝛼𝑝 + 2𝛽)
(𝛼 − 1) (𝛼𝑝 + 2𝛽)
2(𝛼𝑝 + 𝛽)
2
},
2
𝜎3 =
(𝛼𝑝 + 𝛽)
𝑏𝑝𝐵12 (𝛼𝑝 + 2𝛽)
× {2𝐵2 − 𝑏𝑝𝐵12
Φ (𝑝, 𝛼, 𝛽, 𝜇) =
(𝛼 − 1) (𝛼𝑝 + 2𝛽)
2
2(𝛼𝑝 + 𝛽)
(𝛼𝑝 + 2𝛽) (2𝜇 + 𝛼 − 1)
2
2(𝛼𝑝 + 𝛽)
},
.
(19)
4
Journal of Applied Mathematics
If 𝑓(𝑧) is of the form (1) and belongs to the class 𝑉𝑝,𝑏,𝛼,𝛽 (𝜙), then
󵄨󵄨
2 󵄨󵄨
󵄨󵄨𝑎𝑝+2 − 𝜇𝑎𝑝+1
󵄨󵄨
󵄨
󵄨
𝑏𝑝𝐵12
𝑏𝑝
{
−
(𝐵
Φ (𝑝, 𝛼, 𝛽, 𝜇)) ,
{
2
{
{
2
2 (𝛼𝑝 + 2𝛽)
{
{
{
{
𝜇 < 𝜎1 ,
{
{
{
{
{
{ 𝑏𝑝𝐵1
,
𝜎1 ≤ 𝜇 ≤ 𝜎2 ,
≤{
{
2 (𝛼𝑝 + 2𝛽)
{
{
{
{
{
𝑏𝑝𝐵12
𝑏𝑝
{
{
{
−
(𝐵
−
Φ (𝑝, 𝛼, 𝛽, 𝜇)) ,
2
{
{
2
{
{ 2 (𝛼𝑝 + 2𝛽)
𝜇 > 𝜎2 .
{
where 𝐻(𝑞1 , 𝑞2 ) is defined in Lemma 3 and
𝑞1 =
2𝐵2 𝑏𝑝𝐵1 (1 − 𝛼) (𝛼𝑝 + 3𝛽)
+
,
𝐵1
2 (𝛼𝑝 + 𝛽) (𝛼𝑝 + 2𝛽)
𝑞2 =
𝑏𝑝𝐵1 2 (𝛼 − 1) (2𝛼 − 1) (𝛼𝑝 + 3𝛽)
𝐵3
+(
)
3
𝐵1
2
6(𝛼𝑝 + 𝛽)
(20)
+
Proof. Since 𝑓 ∈ 𝑉𝑝,𝑏,𝛼,𝛽 (𝜙), therefore we have for a Schwarz
function
𝑤 (𝑧) = 𝑤1 𝑧 + 𝑤2 𝑧2 + ⋅ ⋅ ⋅ ,
1−
2
𝐵 (𝛼𝑝 + 𝛽)
1
(2 (1 − 2 )
𝑏𝑝𝐵1
𝐵1 (𝛼𝑝 + 2𝛽)
𝑏𝑝𝐵1
󵄨2
󵄨
+
(2𝜇 + 𝛼 − 1) ) 󵄨󵄨󵄨󵄨𝑎𝑝+1 󵄨󵄨󵄨󵄨
2
(21)
𝑏𝑝𝐵1
,
2 (𝛼𝑝 + 2𝛽)
𝑧𝑓󸀠 (𝑧) 𝑓(𝑧) 𝛼
𝑓(𝑧) 𝛼
2 2
(
) }
+ {(1 − 𝛽) ( 𝑝 ) + 𝛽
𝑏 𝑏
𝑧
𝑝𝑓 (𝑧) 𝑧𝑝
=1+
2
1
(𝛼𝑝 + 𝛽) 𝑎𝑝+1 𝑧 +
(𝛼𝑝 + 2𝛽)
𝑏𝑝
𝑏𝑝
2
} 𝑧2 +
× {2𝑎𝑝+2 + (𝛼 − 1) 𝑎𝑝+1
𝐵 (𝛼𝑝 + 𝛽)
1
(2 (1 + 2 )
𝑏𝑝𝐵1
𝐵1 (𝛼𝑝 + 2𝛽)
𝑏𝑝𝐵1
󵄨2
󵄨
−
(2𝜇 + 𝛼 − 1) ) 󵄨󵄨󵄨󵄨𝑎𝑝+1 󵄨󵄨󵄨󵄨
2
+
(22)
(28)
(𝛼 − 1) (𝛼 − 2) 3
𝑎𝑝+1 } 𝑧3 + ⋅ ⋅ ⋅ .
6
Also, we have
𝜙 (𝑤 (𝑧)) = 1 + 𝐵1 𝑤1 𝑧 + (𝐵1 𝑤2 + 𝐵1 𝑤12 ) 𝑧2
𝑏𝑝𝐵1
.
2 (𝛼𝑝 + 2𝛽)
+ (𝐵1 𝑤3 + 2𝐵2 𝑤1 𝑤2 + 𝐵3 𝑤13 ) 𝑧3 + ⋅ ⋅ ⋅ .
(29)
Comparing the coefficients of 𝑧, 𝑧2 , 𝑧3 and after simple calculations, we obtain
For any complex number 𝜇,
󵄨󵄨
2 󵄨󵄨
󵄨󵄨𝑎𝑝+2 − 𝜇𝑎𝑝+1
󵄨󵄨
󵄨
󵄨
󵄨󵄨 𝑏𝑝𝐵
𝑏𝑝𝐵1
𝐵 󵄨󵄨󵄨
󵄨
1
max {1, 󵄨󵄨󵄨
Φ (𝑝, 𝛼, 𝛽, 𝜇) − 2 󵄨󵄨󵄨} .
󵄨󵄨 2
𝐵1 󵄨󵄨
2 (𝛼𝑝 + 2𝛽)
(23)
𝑎𝑝+1 =
𝑏𝑝𝐵1 𝑤1
,
2 (𝛼𝑝 + 𝛽)
𝑎𝑝+2 =
𝑏𝑝𝐵1
2 (𝛼𝑝 + 2𝛽)
× {𝑤2 − (
Also,
𝑏𝑝𝐵1
󵄨󵄨
󵄨
󵄨󵄨𝑎𝑝+3 󵄨󵄨󵄨 ≤
󵄨
󵄨 2 (𝛼𝑝 + 3𝛽) 𝐻 (𝑞1 , 𝑞2 ) ,
2 (𝛼𝑝 + 3𝛽)
𝑏𝑝
× {𝑎𝑝+3 + (𝛼 − 1) 𝑎𝑝+1 𝑎𝑝+2
2
≤
𝑧𝑓󸀠 (𝑧) 𝑓(𝑧) 𝛼
𝑓(𝑧) 𝛼
2 2
) }
+ {(1 − 𝛽) ( 𝑝 ) + 𝛽
(
𝑏 𝑏
𝑧
𝑝𝑓 (𝑧) 𝑧𝑝
(27)
Now,
1−
󵄨󵄨
2 󵄨󵄨
󵄨󵄨𝑎𝑝+2 − 𝜇𝑎𝑝+1
󵄨󵄨
󵄨
󵄨
≤
(26)
= 𝜙 (𝑤 (𝑧)) .
and for 𝜎3 ≤ 𝜇 ≤ 𝜎2 ,
+
𝑧∈𝐸
such that
󵄨󵄨
2 󵄨󵄨
󵄨󵄨𝑎𝑝+2 − 𝜇𝑎𝑝+1
󵄨󵄨
󵄨
󵄨
≤
𝑏𝑝𝐵2 (1 − 𝛼) (𝛼𝑝 + 3𝛽)
.
2 (𝛼𝑝 + 𝛽) (𝛼𝑝 + 2𝛽)
These results are sharp.
Furthermore, for 𝜎1 ≤ 𝜇 ≤ 𝜎3 ,
+
(25)
𝑎𝑝+3 =
(24)
𝑏𝑝𝐵1 (𝛼 − 1) (𝛼𝑝 + 2𝛽) 𝐵2
− ) 𝑤12 } ,
2
2
𝐵1
2(𝛼𝑝 + 𝛽)
𝑏𝑝𝐵1
{𝑤 + 𝑞1 𝑤1 𝑤2 + 𝑞2 𝑤13 } ,
2 (𝛼𝑝 + 2𝛽) 3
(30)
Journal of Applied Mathematics
5
where 𝑞1 and 𝑞2 are defined in (25). It can be easily followed
from (30) that
2
=
𝑎𝑝+2 − 𝜇𝑎𝑝+1
𝑏𝑝𝐵1
{𝑤 − ]𝑤12 } ,
2 (𝛼𝑝 + 2𝛽) 2
(31)
2.1. Application of Theorem 6 to
the Function Defined by Convolutions
where
]=
𝑏𝑝𝐵1
𝐵
Φ (𝑝, 𝛼, 𝛽, 𝜇) − 2 .
2
𝐵1
Corollary 10. For 𝑏 = 2, 𝑝 = 1, 𝛼 = 0, and 𝛽 = 1, the results
from (20) to (22) coincide with the results obtained by Ma and
Minda [2] for the class 𝑆∗ (𝜙).
(32)
The results from (20) to (22) are obtained by using Lemma 3,
(23) by using Lemma 4, and (24) by using Lemma 5. To show
that these results are sharp, we define the functions 𝐾𝜙𝑛 (𝑧),
𝐹𝜆 (𝑧), and 𝐺𝜆 (𝑧) such that
Theorem 11. Let 𝜙(𝑧) = 1 + 𝐵1 𝑧 + 𝐵2 𝑧2 + 𝐵3 𝑧3 + ⋅ ⋅ ⋅ , where
𝐵𝑛 , 𝑠 are real with 𝐵1 > 0 and 𝐵2 ≥ 0. Let
𝜎1 =
󸀠
𝐺𝜆 (0) =
𝐺𝜆󸀠
(33)
− 𝑏𝑝𝐵12
(0) − 1 = 0,
with
󸀠
𝑧𝐾𝜙𝑛
(𝑧) 𝐾𝜙𝑛 (𝑧) 𝛼
𝐾𝜙𝑛 (𝑧) 𝛼
2 2
( 𝑝 ) }
1 − + {(1 − 𝛽) ( 𝑝 ) + 𝛽
𝑏 𝑏
𝑧
𝑝𝐾𝜙𝑛 (𝑧)
𝑧
𝜎2 =
𝑧 (𝑧 + 𝜆)
).
1 + 𝜆𝑧
𝜎3 =
It is clear that the functions 𝐾𝜙𝑛 , 𝐹𝜆 , 𝐺𝜆 ∈ 𝑉𝑝,𝑏,𝛼,𝛽 (𝜙). Let
𝐾𝜙 : = 𝐾𝜙2 . If 𝜇 < 𝜎1 or 𝜇 > 𝜎2 , then the equality occurs
for the function 𝐾𝜙 or one of its rotations. For 𝜎1 < 𝜇 < 𝜎2 ,
the equality is attained, if and only if 𝑓 is 𝐾𝜙3 or one of
its rotations. When 𝜇 = 𝜎1 , then the equality holds for the
function 𝐹𝜆 or one of its rotations. If 𝜇 = 𝜎2 , then the equality
is obtained for the function 𝐺𝜆 or one of its rotations.
Corollary 7. For 𝑏 = 2, the results from (20) to (24) coincide
with the results proved by Ramachandran et al. [8].
Corollary 8. For 𝑏 = 2, 𝑝 = 1, and 𝛽 = 1, the results from
(20) to (22) coincide with the results obtained by Ravichandran
et al. [4] for the class 𝐵𝛼 (𝜙).
Corollary 9. For 𝑏 = 2, 𝛼 = 0, and 𝛽 = 1, the results from
(20) to (24) coincide with the results obtained by Ali et al. [1]
for the class 𝑆𝑝∗ (𝜙).
},
2
(𝛼 − 1) (𝛼𝑝 + 2𝛽)
2
𝑔𝑝+1
Φ∗ (𝑝, 𝛼, 𝛽, 𝜇) =
2(𝛼𝑝 + 𝛽)
2
},
2
(𝛼𝑝 + 𝛽)
𝑔𝑝+2 𝑏𝑝𝐵12 (𝛼𝑝 + 2𝛽)
× {2𝐵2 − 𝑏𝑝𝐵12
(34)
2
(𝛼𝑝 + 𝛽)
𝑔𝑝+2 𝑏𝑝𝐵12 (𝛼𝑝 + 2𝛽)
− 𝑏𝑝𝐵12
𝑧𝐺󸀠 (𝑧) 𝐺𝜆 (𝑧) 𝛼
𝐺 (𝑧) 𝛼
2 2
( 𝑝 ) }
+ {(1 − 𝛽) ( 𝜆 𝑝 ) + 𝛽 𝜆
𝑏 𝑏
𝑧
𝑝𝐺𝜆 (𝑧)
𝑧
≺ 𝜙 (−
2(𝛼𝑝 + 𝛽)
2
𝑔𝑝+1
𝑧𝐹󸀠 (𝑧) 𝐹𝜆 (𝑧) 𝛼
𝐹 (𝑧) 𝛼
2 2
1 − + {(1 − 𝛽) ( 𝜆 𝑝 ) + 𝛽 𝜆
) }
(
𝑏 𝑏
𝑧
𝑝𝐹𝜆 (𝑧) 𝑧𝑝
𝑧 (𝑧 + 𝜆)
≺ 𝜙(
),
1 + 𝜆𝑧
(𝛼 − 1) (𝛼𝑝 + 2𝛽)
× {2 (𝐵2 + 𝐵1 )
≺ 𝜙 (𝑧𝑛−1 ) ,
1−
2
(𝛼𝑝 + 𝛽)
𝑔𝑝+2 𝑏𝑝𝐵12 (𝛼𝑝 + 2𝛽)
× {2 (𝐵2 − 𝐵1 )
𝐾𝜙𝑛 (0) = [𝐾𝜙𝑛 ] (0) − 1 = 0,
𝐹𝜆 (0) = 𝐹𝜆󸀠 (0) − 1 = 0,
2
𝑔𝑝+1
(𝛼 − 1) (𝛼𝑝 + 2𝛽)
2
2(𝛼𝑝 + 𝛽)
},
(𝛼𝑝 + 2𝛽) (2𝜇 (𝑔𝑝+2 /𝑔𝑝+1 ) + 𝛼 − 1)
2
2(𝛼𝑝 + 𝛽)
.
(35)
If 𝑓(𝑧) is of the form (1) and belongs to the class 𝑉𝑝,𝑏,𝛼,𝛽,𝑔 (𝜙),
then
󵄨󵄨
2 󵄨󵄨
󵄨󵄨
󵄨󵄨𝑎𝑝+2 − 𝜇𝑎𝑝+1
󵄨
󵄨
𝑏𝑝𝐵12 ∗
𝑏𝑝
{
{
−
(𝐵
Φ (𝑝, 𝛼, 𝛽, 𝜇)) ,
{
2
{
2
{
{ 2𝑔𝑝+2 (𝛼𝑝 + 2𝛽)
{
{
{
𝜇 < 𝜎1 ,
{
{
{
{
𝑏𝑝𝐵1
{
𝜎1 ≤ 𝜇 ≤ 𝜎2 ,
≤ { 2𝑔 (𝛼𝑝 + 2𝛽) ,
{
𝑝+2
{
{
{
{
𝑏𝑝𝐵12 ∗
𝑏𝑝
{
{
{
(
Φ (𝑝, 𝛼, 𝛽, 𝜇) − 𝐵2 ) ,
{
{
2
2𝑔𝑝+2 (𝛼𝑝 + 2𝛽)
{
{
{
𝜇 > 𝜎2 .
{
(36)
6
Journal of Applied Mathematics
Furthermore, for 𝜎1 ≤ 𝜇 ≤ 𝜎3 ,
Proof. Since 𝑓 ∈ 𝑉𝑝,𝑏,𝛼,𝛽,𝑔 (𝜙), therefore we have for a Schwarz
function 𝑤, such that
󵄨󵄨
2 󵄨󵄨
󵄨󵄨𝑎𝑝+2 − 𝜇𝑎𝑝+1
󵄨󵄨
󵄨
󵄨
+
𝛼
2
𝑔𝑝+1
2
𝑏𝑝𝐵1 𝑔𝑝+2
(2 (1 −
+
≤
1−
𝐵2 (𝛼𝑝 + 𝛽)
)
𝐵1 (𝛼𝑝 + 2𝛽)
(𝑓 ∗ 𝑔) (𝑧)
2 2
)
+ { (1 − 𝛽) (
𝑏 𝑏
𝑧𝑝
󸀠
+𝛽
𝑔𝑝+2
𝑏𝑝𝐵1
󵄨2
󵄨
+ 𝛼 − 1) ) 󵄨󵄨󵄨󵄨𝑎𝑝+1 󵄨󵄨󵄨󵄨
(2𝜇
2
𝑔𝑝+1
𝑏𝑝𝐵1
,
2𝑔𝑝+2 (𝛼𝑝 + 2𝛽)
Now,
𝛼
(37)
1−
(𝑓 ∗ 𝑔) (𝑧)
2 2
)
+ { (1 − 𝛽) (
𝑏 𝑏
𝑧𝑝
and for 𝜎3 ≤ 𝜇 ≤ 𝜎2 ,
󸀠
+𝛽
󵄨󵄨
2 󵄨󵄨
󵄨󵄨𝑎𝑝+2 − 𝜇𝑎𝑝+1
󵄨󵄨
󵄨
󵄨
+
2
𝑔𝑝+1
𝑏𝑝𝐵1 𝑔𝑝+2
=1+
2
𝐵 (𝛼𝑝 + 𝛽)
(2 (1 + 2 )
𝐵1 (𝛼𝑝 + 2𝛽)
−
≤
𝛼
𝑧(𝑓 ∗ 𝑔) (𝑧) (𝑓 ∗ 𝑔) (𝑧)
(
) } = 𝜙 (𝑤 (𝑧)) .
𝑧𝑝
𝑝 (𝑓 ∗ 𝑔) (𝑧)
(42)
𝑔𝑝+2
𝑏𝑝𝐵1
󵄨2
󵄨
+ 𝛼 − 1) ) 󵄨󵄨󵄨󵄨𝑎𝑝+1 󵄨󵄨󵄨󵄨
(2𝜇
2
𝑔𝑝+1
𝑏𝑝𝐵1
.
2𝑔𝑝+2 (𝛼𝑝 + 2𝛽)
𝛼
𝑧(𝑓 ∗ 𝑔) (𝑧) (𝑓 ∗ 𝑔) (𝑧)
) }
(
𝑧𝑝
𝑝 (𝑓 ∗ 𝑔) (𝑧)
2
1
(𝛼𝑝 + 𝛽) 𝑎𝑝+1 𝑔𝑝+1 𝑧 +
(𝛼𝑝 + 2𝛽)
𝑏𝑝
𝑏𝑝
2
2
𝑔𝑝+1
} 𝑧2 +
× {2𝑎𝑝+2 𝑔𝑝+2 + (𝛼 − 1) 𝑎𝑝+1
2
(𝛼𝑝 + 3𝛽)
𝑏𝑝
{𝑎𝑝+3 𝑔𝑝+3 + (𝛼 − 1) 𝑎𝑝+1 𝑎𝑝+2 𝑔𝑝+1 𝑔𝑝+2 } 3
×{
(𝛼 − 1) (𝛼 − 2) 3 3
}𝑧 + ⋅⋅⋅ .
+
𝑎𝑝+1 𝑔𝑝+1
6
}
{
(43)
(38)
Also, we obtain
For any complex number 𝜇,
𝜙 (𝑤 (𝑧)) = 1 + 𝐵1 𝑤1 𝑧 + (𝐵1 𝑤2 + 𝐵1 𝑤12 ) 𝑧2
󵄨󵄨
2 󵄨󵄨
󵄨󵄨𝑎𝑝+2 − 𝜇𝑎𝑝+1
󵄨󵄨
󵄨
󵄨
≤
󵄨󵄨 𝑏𝑝𝐵
𝑏𝑝𝐵1
𝐵 󵄨󵄨󵄨
󵄨
1 ∗
max {1, 󵄨󵄨󵄨
Φ (𝑝, 𝛼, 𝛽, 𝜇) − 2 󵄨󵄨󵄨} .
󵄨󵄨 2
𝐵1 󵄨󵄨
2𝑔𝑝+2 (𝛼𝑝 + 2𝛽)
(39)
Also,
𝑏𝑝𝐵1
󵄨󵄨
󵄨
󵄨󵄨𝑎𝑝+3 󵄨󵄨󵄨 ≤
󵄨
󵄨 2 (𝛼𝑝 + 3𝛽) 𝑔 𝐻 (𝑞1 , 𝑞2 ) ,
𝑝+3
(40)
+ (𝐵1 𝑤3 + 2𝐵2 𝑤1 𝑤2 + 𝐵3 𝑤13 ) 𝑧3 + ⋅ ⋅ ⋅ .
Comparing the coefficients of 𝑧, 𝑧2 , 𝑧3 and after simple
calculations, we obtain
𝑎𝑝+1 =
𝑏𝑝𝐵1 𝑤1
,
2𝑔𝑝+1 (𝛼𝑝 + 𝛽)
𝑎𝑝+2 =
𝑏𝑝𝐵1
2𝑔𝑝+2 (𝛼𝑝 + 2𝛽)
where 𝐻(𝑞1 , 𝑞2 ) is defined in Lemma 5 and
𝑞1 =
2𝐵2 𝑏𝑝𝐵1 (1 − 𝛼) (𝛼𝑝 + 3𝛽)
+
,
𝐵1
2 (𝛼𝑝 + 𝛽) (𝛼𝑝 + 2𝛽)
𝑞2 =
𝑏𝑝𝐵1 2 (𝛼 − 1) (2𝛼 − 1) (𝛼𝑝 + 3𝛽)
𝐵3
+(
)
3
𝐵1
2
6(𝛼𝑝 + 𝛽)
+
𝑏𝑝𝐵2 (1 − 𝛼) (𝛼𝑝 + 3𝛽)
.
2 (𝛼𝑝 + 𝛽) (𝛼𝑝 + 2𝛽)
These results are sharp.
(44)
× {𝑤2 − (
𝑎𝑝+3 =
𝑏𝑝𝐵1 (𝛼 − 1) (𝛼𝑝 + 2𝛽) 𝐵2
− ) 𝑤12 } ,
2
2
𝐵1
2(𝛼𝑝 + 𝛽)
𝑏𝑝𝐵1
{𝑤 + 𝑞1 𝑤1 𝑤2 + 𝑞2 𝑤13 } .
2𝑔𝑝+3 (𝛼𝑝 + 2𝛽) 3
(45)
(41)
The remaining proof of the theorem is similar to the proof of
Theorem 6.
Corollary 12. For 𝑏 = 2, the results from (36) to (40) coincide
with the results proved by Ramachandran et al. [8] for the class
𝑅𝑝,1,𝛼,𝛽,𝑔 (𝜙).
Journal of Applied Mathematics
7
Corollary 13. For 𝑏 = 2, 𝛼 = 0, and 𝛽 = 1, the results from
(36) to (40) coincide with the results obtained by Ali et al. [1]
∗
(𝜙).
for the class 𝑆𝑝,𝑔
Corollary 14. For 𝑏 = 2, 𝑝 = 1, 𝛼 = 0, and 𝛽 = 1,
𝑔2 =
2
Γ (3) Γ (2 − 𝜆)
=
,
Γ (3 − 𝜆)
2−𝜆
𝑔3 =
Γ (4) Γ (2 − 𝜆)
6
,
=
−
𝜆)
Γ (4 − 𝜆)
(2
(3 − 𝜆)
𝐵1 =
8
,
𝜋2
𝐵2 =
(46)
16
,
3𝜋2
the results from (36) to (38) coincide with the results obtained
by Srivastava and Mishra [9].
Acknowledgment
The work here is fully supported by LRGS/TD/2011/UKM/
ICT/03/02.
References
[1] R. M. Ali, V. Ravichandran, and N. Seenivasagan, “Coefficient
bounds for 𝑝-valent functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 35–46, 2007.
[2] W. C. Ma and D. Minda, “A unified treatment of some special
classes of univalent functions,” in Proceedings of the Conference
on Complex Analysis, pp. 157–169, International Press, 1994.
[3] S. Owa, “Properties of certain integral operators,” Southeast
Asian Bulletin of Mathematics, vol. 24, no. 3, pp. 411–419, 2000.
[4] V. Ravichandran, A. Gangadharan, and M. Darus, “FeketeSzegő inequality for certain class of Bazilevic functions,” Far
East Journal of Mathematical Sciences, vol. 15, no. 2, pp. 171–180,
2004.
[5] M. P. Chen, “On the regular functions satisfying 𝑅𝑒{𝑓(𝑧)/𝑧} >
𝜌,” Bulletin of the Institute of Mathematics. Academia Sinica, vol.
3, no. 1, pp. 65–70, 1975.
[6] F. R. Keogh and E. P. Merkes, “A coefficient inequality for certain
classes of analytic functions,” Proceedings of the American
Mathematical Society, vol. 20, pp. 8–12, 1969.
[7] D. V. Prokhorov and J. Szynal, “Inverse coefficients for
(𝛼, 𝛽)-convex functions,” Annales Universitatis Mariae CurieSkłodowska A, vol. 35, pp. 125–143, 1981.
[8] C. Ramachandran, S. Sivasubramanian, and H. Silverman,
“Certain coefficient bounds for 𝑝-valent functions,” International Journal of Mathematics and Mathematical Sciences, vol.
2007, Article ID 46576, 11 pages, 2007.
[9] H. M. Srivastava and A. K. Mishra, “Applications of fractional
calculus to parabolic starlike and uniformly convex functions,”
Computers & Mathematics with Applications, vol. 39, no. 3-4, pp.
57–69, 2000.
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