Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 847287, 6 pages
http://dx.doi.org/10.1155/2013/847287
Research Article
Certain Properties of a Class of Close-to-Convex Functions
Related to Conic Domains
Wasim Ul-Haq and Shahid Mahmood
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
Correspondence should be addressed to Wasim Ul-Haq; wasim474@hotmail.com
Received 30 September 2012; Revised 3 March 2013; Accepted 17 March 2013
Academic Editor: Nikolaos Papageorgiou
Copyright © 2013 W. Ul-Haq and S. Mahmood. 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.
We aim to define a new class of close-to-convex functions which is related to conic domains. Many interesting properties such as
sufficiency criteria, inclusion results, and integral preserving properties are investigated here. Some interesting consequences of our
results are also observed.
1. Introduction
Let A be the class of functions 𝑓
∞
𝑓 (𝑧) = 𝑧 + ∑ 𝑎𝑛 𝑧𝑛 ,
(1)
was considered by Acu [4]; for study details on these classes,
we refer to [5–7]. All these above mentioned classes were generalized to the classes SD(𝑘, 𝛿), CD(𝑘, 𝛿), and KD(𝑘, 𝛽, 𝛿)
by Shams et al. [8] and Srivastava et al. [9], respectively. The
classes SD(𝑘, 𝛿) and CD(𝑘, 𝛿) are defined as
𝑛=2
which are analytic in the open unit disc 𝐸 = {𝑧 ∈ C : |𝑧| <
1}. Let 𝑓 and 𝑔 be analytic in 𝐸, and we say that 𝑓 is subordinate to 𝑔, written as 𝑓(𝑧) ≺ 𝑔(𝑧) if there exists a Schwarz
function 𝑤, which is analytic in 𝐸 with 𝑤(0) = 0 and |𝑤(𝑧)| <
1 (𝑧 ∈ 𝐸), such that 𝑓(𝑧) = 𝑔(𝑤(𝑧)). In particular, when
𝑔 is univalent, then the above subordination is equivalent to
𝑓(0) = 𝑔(0) and 𝑓(𝐸) ⊆ 𝑔(𝐸); see [1].
Kanas and Wisniowska [2, 3] introduced and studied the
classes of 𝑘-uniformly convex functions denoted by 𝑘-UCV
and the corresponding class 𝑘-ST related by the Alexandertype relation. Later, the class 𝑘-uniformly close-to-convex
functions denoted by 𝑘-UK defined as
󵄨󵄨 󸀠
󵄨󵄨󵄨
𝑧𝑓󸀠 (𝑧)
󵄨 𝑧𝑓 (𝑧)
) > 𝑘 󵄨󵄨󵄨󵄨
− 1󵄨󵄨󵄨󵄨 ,
𝑔 (𝑧)
󵄨󵄨 𝑔 (𝑧)
󵄨󵄨
𝑔 (𝑧) ∈ 𝑘-ST, 𝑧 ∈ 𝐸}
Re (
󵄨󵄨 󸀠
󵄨󵄨
𝑧𝑓󸀠 (𝑧)
󵄨 𝑧𝑓 (𝑧)
󵄨
) > 𝑘 󵄨󵄨󵄨󵄨
− 1󵄨󵄨󵄨󵄨
𝑓 (𝑧)
󵄨󵄨 𝑓 (𝑧)
󵄨󵄨
+ 𝛿, 𝑧 ∈ 𝐸} ,
(3)
CD (𝑘, 𝛿) = {𝑓 ∈ A :
Re (1 +
𝑘-UK = {𝑓 ∈ A :
Re (
SD (𝑘, 𝛿) = {𝑓 ∈ A :
(2)
󵄨󵄨 󸀠󸀠
󵄨
𝑧𝑓󸀠󸀠 (𝑧)
󵄨󵄨 𝑧𝑓 (𝑧) 󵄨󵄨󵄨
󵄨
󵄨󵄨
)
>
𝑘
󵄨󵄨 󸀠
𝑓󸀠 (𝑧)
󵄨󵄨 𝑓 (𝑧) 󵄨󵄨󵄨
+ 𝛿, 𝑧 ∈ 𝐸} ,
where 𝑘 ≥ 0, 0 ≤ 𝛽, 𝛿 < 1. The class KD(𝑘, 𝛽, 𝛿) known
as 𝑘-uniformly close-to-convex functions of order 𝛽 type 𝛿
2
Abstract and Applied Analysis
is the class of all those functions 𝑓 ∈ A which satisfies the
following condition:
Re (
󵄨󵄨 󸀠
󵄨󵄨
𝑓󸀠 (𝑧)
󵄨󵄨 𝑓 (𝑧)
󵄨󵄨
󵄨
󵄨󵄨 + 𝛽,
)
≥
𝑘
−
1
󵄨󵄨 󸀠
󵄨󵄨
𝑔󸀠 (𝑧)
󵄨󵄨 𝑔 (𝑧)
󵄨
(4)
(𝑘 ≥ 0, 0 ≤ 𝛽, 𝛿 < 1; 𝑧 ∈ 𝐸) ,
for some 𝑔 ∈ CD(𝑘, 𝛿).
Motivated by the work of Noor et al. [10–13], we define
the following.
Definition 1. Let 𝑓 ∈ A. Then, 𝑓 is in the class TD(𝑘, 𝛼, 𝛽,
𝛾, 𝛿) if and only if, for 𝛼 ≥ 0, 0 ≤ 𝛽, 𝛾, 𝛿 < 1,
󵄨
󵄨
Re (𝐻 (𝛼, 𝛽, 𝛾; 𝑓, 𝑔)) ≥ 𝑘 󵄨󵄨󵄨𝐻 (𝛼, 𝛽, 𝛾; 𝑓, 𝑔) − 1󵄨󵄨󵄨 ,
(𝑘 ≥ 0; 𝑧 ∈ 𝐸) ,
(5)
1 − 𝛼 𝑓󸀠 (𝑧)
− 𝛽]
[
1 − 𝛽 𝑔󸀠 (𝑧)
+
𝛼 [
1−𝛾
𝑔󸀠 (𝑧)
[
(6)
󸀠
(𝑧𝑓󸀠 (𝑧))
1+𝑧
,
𝑘 = 0,
{
{
{
1−𝑧
{
2
{
{
1 + √𝑧
2
{
{
{
𝑘 = 1,
),
1 + 2 (log
{
{
𝜋
1
− √𝑧
{
{
{
{
{
2
2
{
{
1+
sinh2 [( arccos 𝑘) arctan ℎ√𝑧] ,
{
2
{
1
−
𝑘
𝜋
{
{
𝑝𝑘 (𝑧) = {
0 < 𝑘 < 1,
{
{
{
{
{
{
{
{
1
1
𝜋 𝑢(𝑧)/√𝑡
{
{
1+
𝑑𝑥
)
sin(
∫
{
{
2 −1
{
𝑘
2𝑅
(𝑡) 0
{
√1−𝑥2 √1−(𝑡𝑥)2
{
{
{
{
{
{
1
{
+
,
𝑘 > 1,
{ 𝑘2 − 1
(8)
where 𝑢(𝑧) = (𝑧 − √𝑡)/(1 − √𝑡𝑧), 𝑡 ∈ (0, 1), 𝑧 ∈ 𝐸, and
𝑧 is chosen such that 𝑘 = cosh(𝜋𝑅󸀠 (𝑡)/4𝑅(𝑡)), where 𝑅(𝑡) is
Legendre’s complete elliptic integral of the first kind and 𝑅󸀠 (𝑡)
is complementary integral of 𝑅(𝑡); see [2, 3].
for some 𝑔 ∈ CD(𝑘, 𝛿), where
𝐻 (𝛼, 𝛽, 𝛾; 𝑓, 𝑔) =
Extremal functions for these conic regions are denoted by
𝑝𝑘 (𝑧), which are analytic in 𝐸 and map 𝐸 onto Ω𝑘 such that
𝑝𝑘 (0) = 1 and 𝑝𝑘󸀠 (0) > 1. These functions are given as:
− 𝛾] .
]
We require the following results which are essential in our
investigations.
Lemma 2 (see [17, page 70]). Let ℎ be convex function in 𝐸
and 𝑞 : 𝐸 󳨃→ 𝐶 with Re 𝑞(𝑧) > 0, 𝑧 ∈ 𝐸. If 𝑝 is analytic in 𝐸
with 𝑝(0) = ℎ(0), then
Special Cases
(i) (𝑘, 0, 𝛽, 𝛾, 𝛿) = KD(𝑘, 𝛽, 𝛿); see [9].
(ii) TD(1, 0, 𝛽, 𝛾, 0) = UK(𝛽) and TD(1, 1, 𝛽, 𝛾, 0) =
UQ(𝛾), the classes of uniformly close-to-convex and
quasiconvex functions introduced and investigated in
[14].
(iii) TD(0, 𝛼, 0, 0 , 0) = Q𝛼 , the class of alpha quasiconvex
functions, introduced and studied in [11].
(iv) TD(0, 0 𝛽, 𝛾, 𝛿) = K(𝛽, 𝛿), the class of close-toconvex functions of order 𝛽 type 𝛿, [15].
(v) TD(0, 1, 𝛽, 𝛾, 𝛿) = 𝐶∗ (𝛾, 𝛿), the class of quasiconvex
functions of order 𝛾 type 𝛿, [16].
The conditions 𝑘 ≥ 0, 𝛼 ≥ 0, 0 ≤ 𝛽, 𝛾, 𝛿 < 1 on the parameters are assumed throughout the entire paper unless otherwise mentioned.
Geometric Interpretation. A function 𝑓 ∈ A is in the class
TD(𝑘, 𝛼, 𝛽, 𝛾, 𝛿) if and only if the functional 𝐻(𝛼, 𝛽, 𝛾, 𝛿; 𝑓,
𝑔) takes all the values in the conic domain Ω𝑘 defined as
follows:
Ω𝑘 = {𝑢 + 𝜄V; 𝑢 > 𝑘√(𝑢 − 1)2 + V2 } .
2. Preliminaries Result
(7)
𝑝 (𝑧) + 𝑞 (𝑧) 𝑧𝑝󸀠 (𝑧) ≺ ℎ (𝑧)
implies 𝑝 (𝑧) ≺ ℎ (𝑧) .
(9)
Lemma 3 (see [17, page 195]). Let ℎ be convex function in 𝐸
with ℎ(0) = 0 and 𝐴 ≥ 0. Suppose that 𝑗 ≥ 4/ℎ󸀠 (0) and that
𝐵(𝑧), 𝐶(𝑧), and 𝐷(𝑧) are analytic in 𝐸 and satisfy
Re 𝐵 (𝑧) ≥ 𝐴 + |𝐶 (𝑧) − 1| − Re (𝐶 (𝑧) − 1) + 𝑗𝐷 (𝑧) , (10)
for 𝑧 ∈ 𝐸. If 𝑝 is analytic in 𝐸 with 𝑝(𝑧) = 𝑎1 𝑧+𝑎2 𝑧2 +𝑎3 𝑧3 +⋅ ⋅ ⋅
and the following subordination relation holds:
𝐴𝑧2 𝑝󸀠󸀠 (𝑧) + 𝐵 (𝑧) 𝑧𝑝󸀠 (𝑧) + 𝐶 (𝑧) 𝑝 (𝑧) + 𝐷 (𝑧) ≺ ℎ (𝑧) ,
(11)
then
𝑝 (𝑧) ≺ ℎ (𝑧) .
(12)
Lemma 4 (see [12]). If 𝑓(𝑧) ≺ ℎ(𝑧) and 𝑔(𝑧) ≺ ℎ(𝑧), then for
𝛼 ∈ [0, 1],
(1 − 𝛼) 𝑓 (𝑧) + 𝛼𝑔 (𝑧) ≺ ℎ (𝑧) .
(13)
3. Main Results
First, we prove the following sufficiency criteria for the functions in the class TD(𝑘, 𝛼, 𝛽, 𝛾, 𝛿).
Abstract and Applied Analysis
3
Theorem 5. A function 𝑓 ∈ A is said to be in the class TD(𝑘,
𝛼, 𝛽, 𝛾, 𝛿), if
∞
∑ Φ𝑛 (𝑘; 𝛼, 𝛽, 𝛾, 𝛿) < (1 − 𝛽) (1 − 𝛾) ,
(14)
𝑛=2
𝑛
Using (1) and the series 𝑔(𝑧) = 𝑧 + ∑∞
𝑛=2 𝑏𝑛 𝑧 in (17), we have
󵄨
󵄨󵄨
󵄨󵄨𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) − 1󵄨󵄨󵄨
󵄨󵄨
𝑛−1
󵄨󵄨 (1 − 𝛼) (1 − 𝛾) (1 + ∑∞
)
𝑛=2 𝑛𝑎𝑛 𝑧
= 󵄨󵄨󵄨󵄨
∞
𝑛−1
󵄨󵄨 (1 − 𝛽) (1 − 𝛾) (1 + ∑𝑛=2 𝑛𝑏𝑛 𝑧 )
󵄨
+
where
󵄨 󵄨
= (𝑘 + 1) [(1 − 𝛼) (1 − 𝛾) 𝑛 +𝛼 (1 − 𝛽) 𝑛2 ] 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨
(15)
+
Proof. Let us assume that relation (6) holds. Now, it is sufficient to show that
󵄨
󵄨
𝑘 󵄨󵄨󵄨𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔)− 1󵄨󵄨󵄨 − Re [𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔)− 1] < 1.
(16)
First, we consider
󵄨
󵄨󵄨
󵄨󵄨𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) − 1󵄨󵄨󵄨
󵄨󵄨
󵄨 1 − 𝛼 𝑓󸀠 (𝑧)
= 󵄨󵄨󵄨󵄨
− 𝛽]
[
󵄨󵄨 1 − 𝛽 𝑔󸀠 (𝑧)
󵄨󵄨
󸀠
󸀠
󵄨󵄨󵄨
𝛼 [ (𝑧𝑓 (𝑧))
+
− 𝛾] − 1󵄨󵄨󵄨
󸀠
󵄨󵄨
1−𝛾
𝑔 (𝑧)
󵄨󵄨
]
[
󵄨󵄨
󸀠
󸀠
󵄨󵄨 1 − 𝛼 𝑓󸀠 (𝑧)
𝛼 (𝑧𝑓 (𝑧))
󵄨󵄨
= 󵄨󵄨
+
󵄨󵄨 1 − 𝛽 𝑔󸀠 (𝑧) 1 − 𝛾 𝑔󸀠 (𝑧)
󵄨󵄨
󵄨󵄨
𝛼𝛾
(1 − 𝛼) 𝛽
󵄨
−
−
− 1󵄨󵄨󵄨
󵄨󵄨
1−𝛽
1−𝛾
󵄨󵄨 (1 − 𝛼) (1 − 𝛾) 𝑓󸀠 (𝑧)
󵄨
= 󵄨󵄨󵄨󵄨
󵄨󵄨 (1 − 𝛽) (1 − 𝛾) 𝑔󸀠 (𝑧)
+
−
𝛼 (1 − 𝛽) (𝑧𝑓󸀠 (𝑧))
󸀠
(1 − 𝛽) (1 − 𝛾) 𝑔󸀠 (𝑧)
[(1 − 𝛾) + 𝛼 (𝛾 − 𝛽)] 𝑔󸀠 (𝑧) 󵄨󵄨󵄨󵄨
󵄨󵄨 .
(1 − 𝛽) (1 − 𝛾) 𝑔󸀠 (𝑧) 󵄨󵄨󵄨
𝑛−1 )
(1 − 𝛽) (1 − 𝛾) (1 + ∑∞
𝑛=2 𝑛𝑏𝑛 𝑧
𝑛−1 󵄨󵄨
[(1 − 𝛾) + 𝛼 (𝛾 − 𝛽)] (1 + ∑∞
) 󵄨󵄨󵄨
𝑛=2 𝑛𝑏𝑛 𝑧
󵄨󵄨
−
∞
(1 − 𝛽) (1 − 𝛾) (1 + ∑𝑛=2 𝑛𝑏𝑛 𝑧𝑛−1 ) 󵄨󵄨󵄨
󵄨
󵄨󵄨
∞
𝑛−1
󵄨󵄨 (1 − 𝛼) (1 − 𝛾) (∑𝑛=2 𝑛𝑎𝑛 𝑧 )
= 󵄨󵄨󵄨󵄨
𝑛−1 )
󵄨󵄨 (1 − 𝛽) (1 − 𝛾) (1 + ∑∞
𝑛=2 𝑛𝑏𝑛 𝑧
󵄨
Φ𝑛 (𝑘; 𝛼, 𝛽, 𝛾, 𝛿)
󵄨 󵄨
+ [(𝑘 + 2 − 𝛽) (1 − 𝛾) +𝛼 (𝑘 + 1) (𝛾 − 𝛽) 𝑛 󵄨󵄨󵄨𝑏𝑛 󵄨󵄨󵄨] .
2
𝑛−1
𝛼 (1 − 𝛽) (1 + ∑∞
)
𝑛=2 𝑛 𝑎𝑛 𝑧
(17)
2
𝑛−1
𝛼 (1 − 𝛽) (∑∞
)
𝑛=2 𝑛 𝑎𝑛 𝑧
𝑛−1 )
(1 − 𝛽) (1 − 𝛾) (1 + ∑∞
𝑛=2 𝑛𝑏𝑛 𝑧
𝑛−1 󵄨󵄨
[(1 − 𝛾) + 𝛼 (𝛾 − 𝛽)] (∑∞
) 󵄨󵄨
𝑛=2 𝑛𝑏𝑛 𝑧
󵄨󵄨
−
󵄨
∞
(1 − 𝛽) (1 − 𝛾) (1 + ∑𝑛=2 𝑛𝑏𝑛 𝑧𝑛−1 ) 󵄨󵄨󵄨
󵄨
󵄨󵄨
󵄨
󵄨
∞
𝑛−1
󵄨󵄨 (1 − 𝛼) (1 − 𝛾) (∑𝑛=2 𝑛 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨 |𝑧| )
≤ 󵄨󵄨󵄨󵄨
󵄨󵄨 󵄨󵄨 𝑛−1 )
󵄨󵄨 (1 − 𝛽) (1 − 𝛾) (1 − ∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨 |𝑧|
󵄨
𝑛−1
2 󵄨󵄨 󵄨󵄨
𝛼 (1 − 𝛽) (∑∞
)
𝑛=2 𝑛 󵄨󵄨𝑎𝑛 󵄨󵄨 |𝑧|
+
󵄨 󵄨󵄨 𝑛−1
∞
󵄨
(1 − 𝛽) (1 − 𝛾) (1 − ∑𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨 |𝑧| )
󵄨󵄨 󵄨󵄨 𝑛−1 󵄨󵄨󵄨
[(1 − 𝛾) + 𝛼 (𝛾 − 𝛽)] (∑∞
) 󵄨󵄨
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨 |𝑧|
+
󵄨 󵄨󵄨 𝑛−1 󵄨󵄨󵄨󵄨
∞
󵄨
(1 − 𝛽) (1 − 𝛾) (1 − ∑𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨 |𝑧| ) 󵄨󵄨
󵄨󵄨 (1 − 𝛼) (1 − 𝛾) (∑∞ 𝑛 󵄨󵄨𝑎 󵄨󵄨)
󵄨
𝑛=2 󵄨󵄨 𝑛 󵄨󵄨
≤ 󵄨󵄨󵄨󵄨
󵄨󵄨 󵄨󵄨
󵄨󵄨 (1 − 𝛽) (1 − 𝛾) (1 − ∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨)
2 󵄨󵄨 󵄨󵄨
𝛼 (1 − 𝛽) (∑∞
𝑛=2 𝑛 󵄨󵄨𝑎𝑛 󵄨󵄨)
+
󵄨󵄨 󵄨󵄨
(1 − 𝛽) (1 − 𝛾) (1 − ∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨)
󵄨󵄨 󵄨󵄨 󵄨󵄨
[(1 − 𝛾) + 𝛼 (𝛾 − 𝛽)] (∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨) 󵄨󵄨󵄨
+
󵄨󵄨 󵄨󵄨 󵄨󵄨 .
(1 − 𝛽) (1 − 𝛾) (1 − ∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨) 󵄨󵄨
(18)
Now,
󵄨
󵄨
𝑘 󵄨󵄨󵄨𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) − 1󵄨󵄨󵄨 − Re [𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) − 1]
󵄨
󵄨
≤ (𝑘 + 1) 󵄨󵄨󵄨𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) − 1󵄨󵄨󵄨
󵄨 󵄨
∞
(1 − 𝛼) (1 − 𝛾) (∑𝑛=2 𝑛 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨)
≤ (𝑘 + 1) [
󵄨󵄨 󵄨󵄨
(1 − 𝛽) (1 − 𝛾) (1 − ∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨)
2 󵄨󵄨 󵄨󵄨
𝛼 (1 − 𝛽) (∑∞
𝑛=2 𝑛 󵄨󵄨𝑎𝑛 󵄨󵄨)
+
󵄨󵄨 󵄨󵄨
(1 − 𝛽) (1 − 𝛾) (1 − ∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨)
󵄨󵄨 󵄨󵄨
[(1 − 𝛾) + 𝛼 (𝛾 − 𝛽)] (∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨)
+
󵄨󵄨 󵄨󵄨 ] .
(1 − 𝛽) (1 − 𝛾) (1 − ∑∞
𝑛=2 𝑛 󵄨󵄨𝑏𝑛 󵄨󵄨)
(19)
4
Abstract and Applied Analysis
󸀠
where 𝜓(𝑧) = (𝑧𝑔󸀠 (𝑧)) /𝑔󸀠 (𝑧). Using (24) and (25) in relation
(6), we obtain
The last inequality is bounded above by 1, if
∞
∞
𝑛=2
𝑛=2
󵄨 󵄨
󵄨 󵄨
(𝑘 + 1) [(1 − 𝛼) (1 − 𝛾) ( ∑ 𝑛 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨) + 𝛼 (1 − 𝛽) ( ∑ 𝑛2 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨)
𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓 (𝑧))
∞
󵄨 󵄨
+ [(1 − 𝛾) + 𝛼 (𝛾 − 𝛽)] ( ∑ 𝑛 󵄨󵄨󵄨𝑏𝑛 󵄨󵄨󵄨)]
𝑛=2
∞
󵄨 󵄨
≤ (1 − 𝛽) (1 − 𝛾) (1 − ∑ 𝑛 󵄨󵄨󵄨𝑏𝑛 󵄨󵄨󵄨) .
𝑛=2
=
𝛼
1−𝛼
[𝑝 (𝑧) − 𝛽] +
[𝑧𝑝󸀠 (𝑧) + 𝑝 (𝑧) 𝜓 (𝑧) − 𝛾]
1−𝛽
1−𝛾
=
(1−𝛼) (1−𝛾)+𝛼 (1−𝛽) 𝜓 (𝑧)
𝛼
] 𝑝 (𝑧)
𝑧𝑝󸀠 (𝑧)+[
1−𝛾
(1 − 𝛽) (1 − 𝛾)
−
(20)
𝛽 (1 − 𝛼) (1 − 𝛾) + 𝛼𝛾 (1 − 𝛽)
(1 − 𝛽) (1 − 𝛾)
= 𝐵 (𝑧) 𝑧𝑝󸀠 (𝑧) + 𝐶 (𝑧) 𝑝 (𝑧) + 𝐷 (𝑧) ,
Hence,
(26)
∞
∑ Φ𝑛 (𝑘; 𝛼, 𝛽, 𝛾, 𝛿) ≤ (1 − 𝛽) (1 − 𝛾) ,
(21)
𝑛=2
where Φ𝑛 (𝑘; 𝛼, 𝛽, 𝛾, 𝛿) is given by (15). This completes the
proof.
When we take 𝛼 = 0, 𝑘 = 1, and 𝑔(𝑧) = 𝑧 in the above
theorem, we obtain the following sufficient condition for the
functions to be in the class UK(𝛽) which is proved in [14].
Corollary 6 (see [14]). A function 𝑓 ∈ 𝐴 is said to be in the
class UK(𝛽) if
∞
󵄨 󵄨 (1 − 𝛽)
∑ 𝑛 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨 ≤
.
2
𝑛=2
(22)
Corollary 7 (see [14]). A function 𝑓 ∈ 𝐴 is said to be in the
class UQ(𝛾) if
where
𝐵 (𝑧) =
𝐶 (𝑧) =
𝐷 (𝑧) = −
𝛽 (1 − 𝛼) (1 − 𝛾) + 𝛼𝛾 (1 − 𝛽)
.
(1 − 𝛽) (1 − 𝛾)
(27)
Now, since 𝑓 ∈ TD(𝑘, 𝛼, 𝛽, 𝛾, 𝛿), we have
𝐵 (𝑧) 𝑧𝑝󸀠 (𝑧) + 𝐶 (𝑧) 𝑝 (𝑧) + 𝐷 (𝑧) ≺ 𝑝𝑘 (𝑧) .
(23)
𝐵 (𝑧) 𝑧𝑝∗󸀠 (𝑧) + 𝐶 (𝑧) 𝑝∗ (𝑧) + 𝐷∗ (𝑧) ≺ 𝑝𝑘∗ (𝑧) ,
Proof. Let
󸀠
𝑓 (𝑧)
= 𝑝 (𝑧) ,
𝑔󸀠 (𝑧)
(24)
where 𝑝(𝑧) is analytic and 𝑝(0) = 1. Now differentiating (24),
we have
󸀠
(𝑧𝑓󸀠 (𝑧))
𝑔󸀠 (𝑧)
= 𝑧𝑝󸀠 (𝑧) + 𝑝 (𝑧) 𝜓 (𝑧) ,
(25)
(29)
where 𝐷∗ (𝑧) = 𝐶(𝑧) + 𝐷(𝑧) − 1. Using Lemma 3 with 𝐴 = 0,
we obtain
𝑝∗ (𝑧) ≺ 𝑝𝑘∗ (𝑧) .
(30)
𝑓󸀠 (𝑧)
= 𝑝 (𝑧) ≺ 𝑝𝑘 (𝑧) .
𝑔󸀠 (𝑧)
(31)
This implies that
The above corollary is obtained when we take 𝛼 = 1, 𝑘 =
1, and 𝑔(𝑧) = 𝑧 in Theorem 5.
Theorem 8. Let 𝑓 ∈ TD(𝑘, 𝛼, 𝛽, 𝛾, 𝛿) and 𝛼 ≥ 4(𝛽/(1 + 3𝛽)).
Then, 𝑓 ∈ KD(𝑘, 0, 𝛿).
(28)
Replacing 𝑝(𝑧) by 𝑝∗ (𝑧) = 𝑝(𝑧) − 1 and 𝑝𝑘 (𝑧) by 𝑝𝑘∗ (𝑧) =
𝑝𝑘 (𝑧) − 1, the above subordination is equivalent to
∞
󵄨 󵄨 (1 − 𝛾)
∑ 𝑛2 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨 ≤
.
2
𝑛=2
(1−𝛼) (1−𝛾)+𝛼 (1−𝛽) 𝜓 (𝑧)
,
(1−𝛽) (1−𝛾)
𝛼
,
1−𝛾
Hence, 𝑓 ∈ KD(𝑘, 0, 𝛿). This completes the proof.
Corollary 9. Let 𝑓 ∈ TD(𝑘, 𝛼, 0, 0, 0) = Q𝛼 . Then, 𝑓 ∈
KD(0, 0, 0) = K. That is, Q𝛼 ⊂ K, 𝛼 ≥ 0.
The above result is well-known inclusion proved in [11].
For 𝑓 ∈ 𝐴, consider the following integral operator defined by
𝐹 (𝑧) = 𝐼𝑚 [𝑓] =
𝑚 + 1 𝑧 𝑚−1
∫ 𝑡 𝑓 (𝑡) 𝑑𝑡,
𝑧𝑚 0
𝑚 = 1, 2, 3 . . . .
(32)
This operator was given by Bernardi [18] in 1969. In particular,
the operator 𝐼1 was considered by Libera [19]. Now let us
prove the following.
Abstract and Applied Analysis
5
Theorem 10. Let 𝑓 ∈ TD(𝑘, 𝛼, 𝛽, 𝛾, 𝛿). Then, 𝐼𝑚 [𝑓] ∈
KD(𝑘, 0, 𝛿).
Now proceeding in the similar manner as in the proof of
Theorem 5 and using Lemma 3 with 𝐴 = 0, we obtain
Proof. Let the function 𝑔 be such that (6) is satisfied. It can
easily be seen that according to [4], the function 𝐺 = 𝐼𝑚 [𝑓] ∈
CD(𝑘, 𝛿), and from (32), we deduce
𝑓󸀠 (𝑧)
= ℎ (𝑧) ≺ 𝑝𝑘 (𝑧) .
𝑔󸀠 (𝑧)
(1 + 𝑚) 𝑓󸀠 (𝑧) = (1 + 𝑚) 𝐹󸀠 (𝑧) + 𝑧𝐹󸀠󸀠 (𝑧) ,
󸀠
󸀠
󸀠󸀠
(1 + 𝑚) 𝑔 (𝑧) = (1 + 𝑚) 𝐺 (𝑧) + 𝑧𝐺 (𝑧) .
󸀠
󸀠
If we let 𝑝(𝑧) = 𝐹 (𝑧)/𝐺 (𝑧) and 𝑞(𝑧) = 1/(𝑚 + 1 +
(𝑧𝐺󸀠󸀠 (𝑧)/𝐺󸀠 )), then simple computations yield us
󸀠
󸀠
From (36), it implies that
(33)
(34)
𝑝 (𝑧) + 𝑧𝑝󸀠 (𝑧) 𝑞 (𝑧) ≺ 𝑝𝑘 (𝑧) .
Theorem 11. For 𝛼 > 𝛼1 ≥ 0,
TD (𝑘, 𝛼, 𝛽, 𝛾, 𝛿) ⊆ TD (𝑘, 𝛼1 , 𝛽, 𝛾, 𝛿) .
𝑧𝑝 (𝑧) 𝑔󸀠 (𝑧) + 𝑧𝑝 (𝑧) 𝑔󸀠󸀠 (𝑧) + 𝑧𝑔󸀠 (𝑧) 𝑝󸀠 (𝑧) (35)
=
(1 + 𝑚) 𝐺󸀠 (𝑧) + 𝑧𝐺󸀠󸀠 (𝑧)
𝐻 (𝛼1 , 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) =
󸀠
(36)
󸀠
(37)
󸀠
where 𝜓(𝑧) = (𝑧𝑔 (𝑧)) /𝑔 (𝑧). Using (36) and (37) in (6), we
have
𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓 (𝑧))
=
𝛼
1−𝛼
[ℎ (𝑧) − 𝛽] +
[𝑧ℎ󸀠 (𝑧) + ℎ (𝑧) 𝜓 (𝑧) − 𝛾]
1−𝛽
1−𝛾
=
𝛼
𝑧ℎ󸀠 (𝑧)
1−𝛾
+[
−
(1 − 𝛼) (1 − 𝛾) + 𝛼 (1 − 𝛽) 𝜓 (𝑧)
] ℎ (𝑧)
(1 − 𝛽) (1 − 𝛾)
= 𝐵 (𝑧) 𝑧ℎ (𝑧) + 𝐶 (𝑧) ℎ (𝑧) + 𝐷 (𝑧) ,
(38)
where
𝛼
,
1−𝛾
=
𝑓󸀠 (𝑧)
𝛼
1
− 𝛽]
(1 − 1 ) [ 󸀠
1−𝛽
𝛼
𝑔 (𝑧)
𝐶 (𝑧) =
(𝑧𝑓󸀠 (𝑧))
𝛼1 [1 − 𝛼 𝑓󸀠 (𝑧)
𝛼
+
− 𝛽) +
− 𝛾)]
(
( 󸀠
𝛼 1 − 𝛽 𝑔󸀠 (𝑧)
1−𝛾
𝑔 (𝑧)
]
[
𝛼
𝛼
= (1 − 1) 𝐻 (0, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔)+ 1 𝐻 (𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) .
𝛼
𝛼
(44)
Now, since 𝑓 ∈ TD(𝑘, 𝛼, 𝛽, 𝛾, 𝛿), we have 𝐻(𝛼, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) ≺
𝑝𝑘 (𝑧). Also, Theorem 8 gives us that 𝐻(0, 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) ≺
𝑝𝑘 (𝑧). The use of Lemma 4 leads us to the required relation;
that is, 𝐻(𝛼1 , 𝛽, 𝛾, 𝛿; 𝑓, 𝑔) ≺ 𝑝𝑘 (𝑧). This completes the
proof.
Acknowledgment
𝛽 (1 − 𝛼) (1 − 𝛾) + 𝛼𝛾 (1 − 𝛽)
(1 − 𝛽) (1 − 𝛾)
󸀠
𝐵 (𝑧) =
After some simple computations, we have
󸀠
󸀠
󸀠
= 𝜓 (𝑧) ℎ (𝑧) + 𝑧ℎ󸀠 (𝑧) ,
(43)
𝐻 (𝛼1 , 𝛽, 𝛾, 𝛿; 𝑓, 𝑔)
where ℎ(𝑧) is analytic and 𝑝(0) = 1. From (36), we have
𝑔󸀠 (𝑧)
1 − 𝛼1 𝑓󸀠 (𝑧)
− 𝛽]
[
1 − 𝛽 𝑔󸀠 (𝑧)
(𝑧𝑓󸀠 (𝑧))
𝛼
+ 1 [
− 𝛾] .
1−𝛾
𝑔󸀠 (𝑧)
]
[
Let
(𝑧𝑓󸀠 (𝑧))
(42)
Proof. Let 𝑓 ∈ TD(𝑘, 𝛼, 𝛽, 𝛾, 𝛿). Then, consider
= 𝑝 (𝑧) + 𝑞 (𝑧) 𝑧𝑝󸀠 (𝑧) .
𝑓󸀠 (𝑧)
= 𝑝 (𝑧) + 𝑧𝑝󸀠 (𝑧) 𝑞 (𝑧) = ℎ (𝑧) ,
𝑔󸀠 (𝑧)
(41)
By employing Lemma 2, we immediately obtain the desired
result.
󸀠󸀠
𝑓 (𝑧) (1 + 𝑚) 𝐹 (𝑧) + 𝑧𝐹 (𝑧)
=
𝑔󸀠 (𝑧) (1 + 𝑚) 𝐺󸀠 (𝑧) + 𝑧𝐺󸀠󸀠 (𝑧)
(40)
(1 − 𝛼) (1 − 𝛾) + 𝛼 (1 − 𝛽) 𝜓 (𝑧)
,
(1 − 𝛽) (1 − 𝛾)
𝛽 (1 − 𝛼) (1 − 𝛾) + 𝛼𝛾 (1 − 𝛽)
𝐷 (𝑧) = −
.
(1 − 𝛽) (1 − 𝛾)
(39)
The authors would like to thank the reviewer of this paper
for his/her valuable comments on the earlier version of this
paper. They would also like to acknowledge Prof. Dr. Ehsan
Ali, VC AWKUM, for the financial support on the publication
of this article.
References
[1] A. W. Goodman, Univalent Functions. Vol. I, Polygonal Publishing House, Washington, DC, USA, 1983.
[2] S. Kanas and A. Wisniowska, “Conic regions and 𝑘-uniform
convexity,” Journal of Computational and Applied Mathematics,
vol. 105, no. 1-2, pp. 327–336, 1999.
6
[3] S. Kanas and A. Wisniowska, “Conic domains and starlike functions,” Revue Roumaine de Mathématiques Pures et Appliquées,
vol. 45, no. 4, pp. 647–657, 2000.
[4] M. Acu, “On a subclass of 𝑛-uniformly close to convex functions,” General Mathematics, vol. 14, no. 1, pp. 55–64, 2006.
[5] E. Aqlan, J. M. Jahangiri, and S. R. Kulkarni, “New classes of 𝑘uniformly convex and starlike functions,” Tamkang Journal of
Mathematics, vol. 35, no. 3, pp. 1–7, 2004.
[6] S. Kanas, “Alternative characterization of the class 𝑘-𝑈𝐶𝑉 and
related classes of univalent functions,” Serdica. Mathematical
Journal, vol. 25, no. 4, pp. 341–350, 1999.
[7] S. Kanas and H. M. Srivastava, “Linear operators associated with
𝑘-uniformly convex functions,” Integral Transforms and Special
Functions, vol. 9, no. 2, pp. 121–132, 2000.
[8] S. Shams, S. R. Kulkarni, and J. M. Jahangiri, “Classes of uniformly starlike and convex functions,” International Journal of
Mathematics and Mathematical Sciences, vol. 2004, no. 55, pp.
2959–2961, 2004.
[9] H. M. Srivastava, S.-H. Li, and H. Tang, “Certain classes of 𝑘uniformly close-to-convex functions and other related functions defined by using the Dziok-Srivastava operator,” Bulletin
of Mathematical Analysis and Applications, vol. 1, no. 3, pp. 49–
63, 2009.
[10] K. I. Noor, “On a generalization of uniformly convex and related
functions,” Computers & Mathematics with Applications, vol. 61,
no. 1, pp. 117–125, 2011.
[11] K. I. Noor and F. M. Al-Oboudi, “Alpha-quasi-convex univalent
functions,” Caribbean Journal of Mathematics, vol. 3, pp. 1–8,
1984.
[12] K. I. Noor and S. N. Malik, “On generalized bounded Mocanu
variation associated with conic domain,” Mathematical and
Computer Modelling, vol. 55, no. 3-4, pp. 844–852, 2012.
[13] K. I. Noor, M. Arif, and W. Ul-Haq, “On 𝑘-uniformly close-toconvex functions of complex order,” Applied Mathematics and
Computation, vol. 215, no. 2, pp. 629–635, 2009.
[14] K. G. Subramanian, T. V. Sudharsan, and H. Silverman, “On uniformly close-to-convex functions and uniformly quasiconvex
functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 48, pp. 3053–3058, 2003.
[15] R. J. Libera, “Some radius of convexity problems,” Duke Mathematical Journal, vol. 31, pp. 143–158, 1964.
[16] K. I. Noor, “On quasi-convex functions and related topics,”
International Journal of Mathematics and Mathematical Sciences, vol. 2, pp. 241–258, 1987.
[17] S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory
and Applications, vol. 225 of Monographs and Textbooks in Pure
and Applied Mathematics, Marcel Dekker, New York, NY, USA,
2000.
[18] S. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol. 135, pp. 429–
446, 1969.
[19] R. J. Libera, “Some classes of regular univalent functions,” Proceedings of the American Mathematical Society, vol. 16, pp. 755–
758, 1965.
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http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014