# Coefficient Inequalities for a Subclass ofp-Valent Analytic Functions

Authors: Muhammad Arif; Janusz Sokół; Muhammad Ayaz
Journal: The Scientific World Journal
 (2014)

---

## Abstract

The aim of this paper is to study the problem of coefficient bounds for a newly defined subclass ofp-valent analytic functions. Many known results appear as special consequences of our work.


---

## Body

## 1. Introduction

Let𝒜
(
p
) denote the class of functions f
(
z
) of the form
(1)
f
(
z
)
=
z
p
+
∑
n
=
p
+
1
∞
‍
a
n
z
n
,
(
p
∈
ℕ
)
,
which are analytic and multivalent in the open unit disk 𝕌
=
{
z
:
|
z
|
<
1
}. Also let 𝒮
p
* and 𝒦
p denote the well-known classes of p-valent starlike functions and p-valent convex functions, respectively.Forf
(
z
)
∈
𝒜
(
p
) given by (1) and g
(
z
)
∈
𝒜
(
p
) given by
(2)
g
(
z
)
=
z
p
+
∑
n
=
p
+
1
∞
‍
b
n
z
n
,
(
p
∈
ℕ
)
,
the Hadamard product (or convolution) of f
(
z
) and g
(
z
) is given by
(3)
(
f
*
g
)
(
z
)
=
z
p
+
∑
n
=
p
+
1
∞
‍
a
n
b
n
z
n
,
(
z
∈
𝕌
)
.
Motivated by Ruscheweyh operator [1], Goel and Sohi [2] introduced a differential operator D
δ
+
p
-
1 for p-valent analytic functions given by
(4)
D
δ
+
p
-
1
f
(
z
)
=
z
(
1
-
z
)
δ
+
p
*
f
(
z
)
=
z
p
+
∑
n
=
p
+
1
∞
‍
φ
n
(
δ
)
a
n
z
n
,
with δ
>
-
p,
(5)
φ
n
(
δ
)
=
(
δ
+
p
)
n
-
p
(
n
-
p
)
!
,
and (
x
)
n is a Pochhammer symbol given by
(6)
(
x
)
n
=
{
1
,
n
=
0
,
x
(
x
+
1
)
(
x
+
2
)
⋯
(
x
+
n
-
1
)
,
n
∈
ℕ
.
It is obvious that when δ is any integer greater than -
p,
(7)
D
δ
+
p
-
1
f
(
z
)
=
z
p
(
z
δ
-
1
f
(
z
)
)
(
δ
+
p
-
1
)
(
δ
+
p
-
1
)
!
.
The following identity can be easily established:
(8)
(
δ
+
p
)
D
δ
+
p
f
(
z
)
=
δ
D
δ
+
p
-
1
f
(
z
)
+
z
(
D
δ
+
p
-
1
f
(
z
)
)
′
.
Using the generalized Ruscheweyh operator, we define a subclass of 𝒱
𝒟
p
λ
(
δ
,
b
,
α
,
β
) of p-valent analytic functions as follows.Definition 1.
An analyticp-valent function f
(
z
) of the form (1) belongs to the class 𝒱
𝒟
p
λ
(
δ
,
b
,
α
,
β
), if and only if
(9)
Re
{
e
i
λ
(
1
-
2
b
+
2
b
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
)
}
>
α
|
2
b
(
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
-
1
)
|
+
β
cos
λ
,
where α
≥
0,  b
∈
ℂ
∖
{
0
}, δ
>
-
p, λ is real with |
λ
|
<
(
π
/
2
), and 0
≤
β
<
1.
By giving specific values toα, β, λ, p, b, and δ in 𝒱
𝒟
p
λ
(
δ
,
b
,
α
,
β
), we obtain many important subclasses studied by various authors in earlier papers; see for details [3–6]; we list some of them as follows:(i)
𝒱
𝒟
1
λ
(
0,2
,
0,0
)
≡
𝒮
λ
* and 𝒱
𝒟
1
λ
(
1,1
,
0,0
)
≡
𝒦
λ, studied by Spacek [7] and Robertson [8], respectively; for the advancement work see [9–11];
(ii)
𝒱
𝒟
1
0
(
0,2
,
α
,
β
)
≡
𝒮
𝒟
(
α
,
β
) and 𝒱
𝒟
1
0
(
1,1
,
α
,
β
)
≡
𝒦
𝒟
(
α
,
β
), studied by both Owa et al. and Shams et al. [12, 13];
(iii)
𝒱
𝒟
1
λ
(
0,2
,
1,0
)
≡
𝒰
𝒮
𝒫
(
λ
) and 𝒱
𝒟
1
λ
(
1,1
,
1,0
)
≡
𝒰
𝒞
𝒮
𝒫
(
λ
), introduced by Ravichandran et al. [14];
(iv)
𝒱
𝒟
1
0
(
δ
,
b
,
α
,
β
)
≡
𝒱
𝒟
(
δ
,
b
,
α
,
β
), considered by Latha [15];
(v)
𝒱
𝒟
1
0
(
0,2
,
0
,
β
)
≡
𝒮
*
(
β
) and 𝒱
𝒟
1
0
(
1,1
,
0
,
β
)
≡
𝒦
(
β
), the well-known classes of starlike and convex functions of order β.From the above special cases we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.We will assume throughout our discussion, unless otherwise stated, thatα
≥
0, 0
≤
β
<
1, δ
>
-
1, λ is real with |
λ
|
<
(
π
/
2
), and b
∈
ℂ
∖
{
0
}.


## 2. Main Results

Theorem 2.
Letf
(
z
)
∈
𝒱
𝒟
p
λ
(
δ
,
b
,
α
,
β
) with 0
≤
α
≤
β. Then f
(
z
)
∈
𝒱
𝒟
p
λ
(
δ
,
b
,
0
,
ζ
), where ζ
=
(
β
-
α
)
/
(
1
-
α
).Proof.
Letf
(
z
)
∈
𝒱
𝒟
p
λ
(
δ
,
b
,
α
,
β
). Then we obtain
(10)
Re
e
i
λ
{
1
-
2
b
+
2
b
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
}
>
α
|
2
b
(
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
-
1
)
|
+
β
cos
λ
>
α
Re
e
i
λ
(
1
-
2
b
+
2
b
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
)
-
α
Re
e
i
λ
+
β
cos
λ
,
and this implies
(11)
Re
[
e
i
λ
(
1
-
2
b
+
2
b
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
)
]
>
ζ
cos
λ
.
Also if 0
≤
α
≤
β, then we can easily obtain
(12)
0
≤
ζ
<
1
,
and this completes the proof.Theorem 3.
Iff
(
z
)
∈
𝒱
𝒟
p
λ
(
δ
,
b
,
α
,
β
), then
(13)
|
a
2
|
≤
|
b
|
|
η
|
|
1
-
α
|
,
and
(14)
|
a
n
+
p
-
1
|
≤
(
δ
+
p
)
|
b
|
|
η
|
(
n
-
1
)
|
1
-
α
|
φ
n
+
p
-
1
(
δ
)
×
∏
j
=
1
n
-
2
‍
(
1
+
(
δ
+
p
)
|
b
|
|
η
|
j
|
1
-
α
|
)
,
n
≥
3
,
where φ
n
+
p
-
1
(
δ
) is given by (5) and
(15)
η
=
(
1
-
β
)
cos
λ
+
i
(
1
-
α
)
sin
λ
.Proof.
Letf
(
z
)
∈
𝒱
𝒟
p
λ
(
δ
,
b
,
α
,
β
). Then by Theorem 2, we have
(16)
Re
[
e
i
λ
(
1
-
2
b
+
2
b
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
)
]
>
(
β
-
α
1
-
α
)
cos
λ
,
(
z
∈
E
)
.
Let us define p
(
z
) by
(17)
e
i
λ
(
1
-
2
b
+
2
b
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
)
=
[
(
1
-
β
1
-
α
)
p
(
z
)
+
(
β
-
α
1
-
α
)
]
cos
λ
+
i
sin
λ
.
Then p
(
z
) is analytic in E with p
(
0
)
=
1 and Re
p
(
z
)
>
0, z
∈
E. Let
(18)
p
(
z
)
=
1
+
∑
n
=
1
∞
‍
p
n
z
n
,
z
∈
E
.
Then (17) becomes
(19)
1
-
2
b
+
2
b
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
=
1
+
(
1
-
β
)
cos
λ
+
i
(
1
-
α
)
sin
λ
e
i
λ
(
1
-
α
)
∑
n
=
1
∞
‍
p
n
z
n
.
That is,
(20)
2
e
i
λ
(
1
-
α
)
[
D
δ
+
p
f
(
z
)
-
D
δ
+
p
-
1
f
(
z
)
]
=
b
η
D
δ
+
p
-
1
f
(
z
)
∑
n
=
1
∞
‍
p
n
z
n
,
where η is given by (15). Using (8) in (20), we obtain
(21)
2
e
i
λ
(
1
-
α
)
[
z
(
D
δ
+
p
-
1
f
(
z
)
)
′
-
p
D
δ
+
p
-
1
f
(
z
)
]
=
b
(
δ
+
p
)
η
D
δ
+
p
-
1
f
(
z
)
∑
n
=
1
∞
‍
p
n
z
n
,
or, equivalently,
(22)
2
e
i
λ
(
1
-
α
)
[
∑
k
=
p
+
1
∞
‍
(
k
-
p
)
φ
k
(
δ
)
a
k
z
k
]
=
b
(
δ
+
p
)
η
[
z
p
+
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
a
k
z
k
]
(
∑
n
=
1
∞
‍
p
n
z
n
)
.
Comparing the coefficients of z
n
+
p
-
1 on both sides,
(23)
2
e
i
λ
(
1
-
α
)
(
n
-
1
)
φ
n
+
p
-
1
(
δ
)
a
n
+
p
-
1
=
b
(
δ
+
p
)
η
{
p
1
a
n
+
p
-
2
φ
n
+
p
-
2
(
δ
)
+
⋯
+
p
n
-
1
}
.
Taking absolute on both sides and then applying the coefficient estimates |
p
n
|
≤
2 for Caratheodory functions [3], we have
(24)
|
a
n
+
p
-
1
|
≤
|
b
|
(
δ
+
p
)
|
η
|
(
n
-
1
)
|
1
-
α
|
φ
n
+
p
-
1
(
δ
)
×
{
1
+
φ
p
+
1
(
δ
)
|
a
p
+
1
|
+
⋯
+
φ
n
+
p
-
2
(
δ
)
|
a
n
+
p
-
2
|
}
.
We apply mathematical induction on (24). So for n
=
2,
(25)
|
a
p
+
1
|
≤
|
b
|
|
η
|
|
1
-
α
|
,
which shows that (13) is true. For n
=
3,
(26)
|
a
p
+
2
|
≤
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
(
2
)
φ
p
+
2
(
δ
)
{
1
+
φ
p
+
1
(
δ
)
|
a
p
+
1
|
}
,
and using (13), we have
(27)
|
a
p
+
2
|
≤
|
b
|
|
η
|
(
δ
+
p
)
(
1
-
α
)
(
2
)
φ
p
+
2
(
δ
)
{
1
+
|
b
|
|
η
|
(
δ
+
p
)
(
1
-
α
)
}
.
Therefore, (14) holds for n
=
3.
Assume that (14) is true for n
=
k; that is,
(28)
|
a
k
+
p
-
1
|
≤
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
(
k
-
1
)
φ
k
+
p
-
1
(
δ
)
∏
j
=
1
k
-
2
(
1
+
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
j
)
.
Consider
(29)
|
a
k
+
p
|
≤
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
(
k
)
φ
k
+
p
(
δ
)
×
{
∏
j
=
1
k
-
2
(
1
+
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
j
)
(
1
+
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
)
+
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
(
2
)
×
(
1
+
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
)
+
⋯
+
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
(
k
-
1
)
×
∏
j
=
1
k
-
2
(
1
+
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
j
)
}
=
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
(
k
)
φ
k
+
p
(
δ
)
∏
j
=
1
k
-
1
(
1
+
|
b
|
|
η
|
(
δ
+
p
)
|
1
-
α
|
j
)
.
Therefore, the result is true for n
=
k
+
1, and hence by using mathematical induction, (14) holds true for all n
≥
3.
If we putλ
=
0, p
=
1, b
=
2, and δ
=
0 in Theorem 3, we obtain the result proved in [12].Corollary 4.
Iff
(
z
)
∈
𝒮
𝒟
(
α
,
β
), then
(30)
|
a
2
|
≤
2
(
1
-
β
)
|
1
-
α
|
,
|
a
n
|
≤
2
(
1
-
β
)
(
n
-
1
)
|
1
-
α
|
∏
j
=
1
n
-
2
‍
(
1
+
2
(
1
-
β
)
j
|
1
-
α
|
)
,
(
n
≥
3
)
.
If one takes α
=
0 in Corollary 4, one obtains the following inequality:
(31)
|
a
n
|
≤
1
(
n
-
1
)
!
∏
j
=
2
n
‍
(
j
-
2
β
)
,
(
n
≥
2
)
,
which was proved by Robertson [16].
By settingλ
=
0, p
=
1, b
=
1, and δ
=
1 in Theorem 3, one obtains the result proved in [12].Corollary 5.
Iff
(
z
)
∈
𝒦
𝒟
(
α
,
β
), then
(32)
|
a
2
|
≤
(
1
-
β
)
|
1
-
α
|
,
|
a
n
|
≤
2
(
1
-
β
)
n
(
n
-
1
)
|
1
-
α
|
∏
j
=
1
n
-
2
(
1
+
2
(
1
-
β
)
j
|
1
-
α
|
)
,
(
n
≥
3
)
.
Letting α
=
0 in Corollary 5, one gets the following inequality proved by Robertson [16]:
(33)
|
a
n
|
≤
1
n
!
∏
j
=
2
n
‍
(
j
-
2
β
)
,
(
n
≥
2
)
.Theorem 6.
Iff
(
z
)
∈
𝒜
(
p
) and satisfies
(34)
2
(
α
+
1
)
(
δ
+
p
)
|
b
|
sin
λ
2
+
β
(
δ
+
p
)
|
b
|
cos
λ
+
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
|
a
k
|
×
[
(
α
+
1
)
{
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
2
(
k
-
p
)
}
-
β
(
δ
+
p
)
|
b
|
cos
λ
+
(
δ
+
p
)
|
b
|
(
α
+
1
)
{
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
2
(
k
-
p
)
}
]
<
(
δ
+
p
)
|
b
|
,
where φ
k
(
δ
) is given by (5), then f
(
z
)
∈
𝒱
𝒟
p
λ
(
δ
,
b
,
α
,
β
).Proof.
Suppose (34) holds. Also let us suppose
(35)
Q
(
z
)
=
e
i
λ
(
1
-
2
b
+
2
b
D
δ
+
p
f
(
z
)
D
δ
+
p
-
1
f
(
z
)
)
.
Then
(36)
|
Q
(
z
)
-
1
|
=
|
(
b
e
i
λ
-
2
e
i
λ
-
b
)
D
δ
+
p
-
1
f
(
z
)
+
2
e
i
λ
D
δ
+
p
f
(
z
)
b
D
δ
+
p
-
1
f
(
z
)
|
.
Using (8) and then simplifications gives
(37)
|
Q
(
z
)
-
1
|
=
|
(
b
(
δ
+
p
)
D
δ
+
p
-
1
f
(
z
)
)
-
1
(
D
δ
+
p
-
1
f
(
z
)
+
2
e
i
λ
z
(
D
δ
+
p
-
1
f
(
z
)
)
′
{
b
δ
(
e
i
λ
-
1
)
+
b
p
(
e
i
λ
-
1
)
-
2
p
e
i
λ
}
l
×
D
δ
+
p
-
1
f
(
z
)
+
2
e
i
λ
z
(
D
δ
+
p
-
1
f
(
z
)
)
′
)
×
(
b
(
δ
+
p
)
D
δ
+
p
-
1
f
(
z
)
)
-
1
(
D
δ
+
p
-
1
f
(
z
)
+
2
e
i
λ
z
(
D
δ
+
p
-
1
f
(
z
)
)
′
{
b
δ
(
e
i
λ
-
1
)
+
b
p
(
e
i
λ
-
1
)
-
2
p
e
i
λ
}
|
≤
(
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
|
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
a
k
z
k
|
×
[
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
(
k
-
p
)
]
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
|
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
a
k
z
k
|
)
×
(
(
δ
+
p
)
|
b
|
(
1
-
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
|
a
k
|
)
)
-
1
.
Now consider
(38)
α
|
Q
(
z
)
-
1
|
-
Re
|
Q
(
z
)
-
1
|
+
β
cos
λ
≤
(
α
+
1
)
|
Q
(
z
)
-
1
|
+
β
cos
λ
≤
(
(
(
δ
+
p
)
|
b
|
(
1
-
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
|
a
k
|
)
)
-
1
(
[
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
(
k
-
p
)
]
}
(
α
+
1
)
{
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
|
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
a
k
z
k
|
lllllll
×
|
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
a
k
z
k
|
[
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
(
k
-
p
)
]
}
)
×
(
(
δ
+
p
)
|
b
|
(
1
-
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
|
a
k
|
)
)
-
1
)
+
β
cos
λ
.
The last expression is bounded by 1 if
(39)
2
(
α
+
1
)
(
δ
+
p
)
|
b
|
sin
λ
2
+
β
(
δ
+
p
)
|
b
|
cos
λ
+
∑
k
=
p
+
1
∞
‍
φ
k
(
δ
)
|
a
k
|
×
[
(
α
+
1
)
{
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
2
(
k
-
p
)
}
l
-
β
(
δ
+
p
)
|
b
|
cos
λ
+
(
δ
+
p
)
|
b
|
{
2
(
δ
+
p
)
|
b
|
sin
λ
2
+
2
(
k
-
p
)
}
]
<
(
δ
+
p
)
|
b
|
,
and this completes the proof.Forλ
=
0, δ
=
-
p
+
1, b
=
2, and α
=
0 in Theorem 6, we obtain the following.Corollary 7.
Iff
(
z
)
∈
𝒜
(
p
) and satisfies
(40)
∑
k
=
p
+
1
∞
(
k
+
1
-
p
-
β
)
|
a
k
|
<
(
1
-
β
)
,
then f
(
z
)
∈
𝒮
p
*
(
β
), the class of p-valent starlike functions of order β.
Forλ
=
0, δ
=
-
p
+
2, b
=
1, and α
=
0 in Theorem 6, one has the following.Corollary 8.
Iff
(
z
)
∈
𝒜
(
p
) and satisfies
(41)
∑
k
=
p
+
1
∞
(
k
-
p
+
1
)
(
k
+
1
-
p
-
β
)
|
a
k
|
<
(
1
-
β
)
,
then f
(
z
)
∈
𝒦
p
(
β
), the class of p-valent convex functions of order β.
Further forp
=
1 in both the last two corollaries, one obtains the results for the classes 𝒮
*
(
β
) and 𝒦
(
β
) which was proved by Merkes et al. [17] and Silverman [18], respectively.



---

*Source: 801751-2014-02-04.xml*