# On a New Class ofp-Valent Meromorphic Functions Defined in Conic Domains

Authors: Mohammed Ali Alamri; Maslina Darus
Journal: The Scientific World Journal
 (2016)

---

## Abstract

We define a new class of multivalent meromorphic functions using the generalised hypergeometric function. We derived this class related to conic domain. It is also shown that this new class of functions, under certain conditions, becomes a class of starlike functions. Some results on inclusion and closure properties are also derived.


---

## Body

## 1. Introduction

LetM
p denote the class of functions of the form (1)
f
z
=
1
z
p
+
∑
n
=
1
∞
a
n
z
n
-
p
,
p
∈
N
=
1,2
,
3
,
…
,which are analytic and p-valent in the punctured unit disc centred at origin E
∗
=
{
z
:
0
<
z
<
1
}
=
E
∖
{
0
}. Also by f
(
z
)
≺
g
(
z
) we mean f
(
z
) is subordinate to g
(
z
) which implies the existence of an analytic function, called Schwartz function w
(
z
) with |
w
(
z
)
|
<
1, for z
∈
E
∗ such that f
(
z
)
=
g
(
w
(
z
)
), where f
(
z
) and g
(
z
) are multivalent meromorphic functions. Note that if g is univalent in E then the above subordination is equivalent to f
(
0
)
=
g
(
0
) and f
(
E
)
⊂
g
(
E
).The set of points, for0
<
γ
<
1 and k
∈
0
,
∞, (2)
Ω
k
,
γ
=
γ
Ω
k
+
1
-
γ
,where (3)
Ω
k
=
u
+
i
v
:
u
>
k
u
-
1
2
+
v
2
,
u
>
0
,[1] showed that the extremal functions q
k
,
γ
(
z
) for conic regions are convex univalent and given by (4)
q
k
,
γ
z
=
1
+
1
-
2
γ
z
1
-
z
,
k
=
0
,
1
+
2
γ
1
-
k
2
2
π
arccos
⁡
k
arctanh
⁡
z
,
0
<
k
<
1
,
1
+
2
γ
π
2
log
⁡
1
+
z
1
-
z
2
,
k
=
1
,
1
+
γ
k
2
-
1
sin
⁡
π
2
R
t
∫
0
u
z
/
t
1
1
-
x
2
1
-
t
x
2
d
x
-
1
,
k
>
1
,where R
(
t
) is Legendre’s complete elliptic integral of the first kind with R
′
(
t
)
=
1
-
t
2 as its complementary integral, u
(
z
)
=
(
z
-
t
)
/
(
1
-
t
z
), t
∈
(
0,1
), and z
∈
E is chosen in such a way that k
=
cosh
⁡
π
R
′
(
t
)
/
R
(
t
).The generalised hypergeometric functionF
s
q
(
α
1
,
…
,
α
q
;
β
1
,
…
,
β
s
;
z
) for complex parameters α
1
,
…
,
α
q and β
1
,
…
,
β
s with β
j
≠
0
,
-
1
,
-
2
,
-
3
,
…, for j
=
1,2
,
3
,
…
,
s, is defined as (5)
F
s
q
α
1
,
…
,
α
q
;
β
1
,
…
,
β
s
;
z
=
∑
n
=
0
∞
α
1
n
⋯
α
q
n
β
1
n
⋯
β
s
n
n
!
z
n
=
1
+
∑
n
=
1
∞
α
1
n
⋯
α
q
n
β
1
n
⋯
β
s
n
z
n
n
!
,with q
≤
s
+
1, q
,
s
∈
N
0
=
N
∪
{
0
}, and α
n is the well-known Pochhammer symbol related to the factorial and the Gamma function by the relation (6)
α
n
=
α
+
n
-
1
!
α
-
1
!
=
Γ
α
+
n
Γ
α
.Also (5) implies (7)
h
p
α
1
,
…
,
α
q
;
β
1
,
…
,
β
s
;
z
=
z
-
p
F
s
q
α
1
,
…
,
α
q
;
β
1
,
…
,
β
s
;
z
∈
M
p
.Liu and Srivastava [2] defined a linear operator for functions belonging to the class of multivalent meromorphic function H
p
(
α
1
,
…
,
α
q
;
β
1
,
…
,
β
s
)
:
M
p
→
M
p as follows: (8)
H
p
α
1
,
…
,
α
q
;
β
1
,
…
,
β
s
=
h
p
α
1
,
…
,
α
q
;
β
1
,
…
,
β
s
;
z
⋆
f
z
.If we assume for brevity that H
p
,
q
,
s
(
α
1
)
=
H
p
(
α
1
,
…
,
α
q
;
β
1
,
…
,
β
s
) then the following identity holds for this operator: (9)
z
H
p
,
q
,
s
α
1
f
z
′
=
α
1
H
p
,
q
,
s
α
1
+
1
f
z
-
α
1
+
p
H
p
,
q
,
s
α
1
f
z
.Shareef [3] defined and studied subclass M
Q
p
(
k
,
λ
,
α
1
) of meromorphic function associated with conic domain, for k
≥
0, 0
≤
λ
<
1, and p
≥
1, as follows: (10)
-
1
p
z
H
p
,
q
,
s
α
1
f
z
′
+
λ
z
2
H
p
,
q
,
s
α
1
f
z
′
′
1
-
λ
H
p
,
q
,
s
α
1
f
z
+
λ
z
H
p
,
q
,
s
α
1
f
z
′
≺
q
k
,
γ
z
.We now define a new subclassM
Q
p
(
b
,
k
,
λ
,
α
1
) of meromorphic function associated with conic domain, for k
≥
0, 0
≤
λ
<
1, p
≥
1, and b
≥
1, as follows: (11)
-
1
p
1
+
1
b
z
H
p
,
q
,
s
α
1
f
z
′
+
λ
z
2
H
p
,
q
,
s
α
1
f
z
′
′
1
-
λ
H
p
,
q
,
s
α
1
f
z
+
λ
z
H
p
,
q
,
s
α
1
f
z
′
-
b
≺
q
k
,
γ
z
.Since q
k
,
γ is a convex and univalent function, for h
(
z
)
≺
q
k
,
γ
(
z
) it means h
(
E
∗
) is contained in q
k
,
γ
(
E
∗
), where (12)
h
z
=
-
1
p
1
+
1
b
z
H
p
,
q
,
s
α
1
f
z
′
+
λ
z
2
H
p
,
q
,
s
α
1
f
z
′
′
1
-
λ
H
p
,
q
,
s
α
1
f
z
+
λ
z
H
p
,
q
,
s
α
1
f
z
′
-
b
.In the next two sections, for brevity, we drop the subscripts of the operator H
p
,
q
,
s
(
α
1
).


## 2. Preliminary Results

Lemma 1 (see [4]).
Leth
2
(
z
) be convex in E and R
(
λ
h
2
(
z
)
+
μ
)
>
0, where μ
∈
C, λ
∈
C
∖
{
0
}, and z
∈
E. If h
1
(
z
) is analytic in E, with h
1
(
0
)
=
h
2
(
0
), then (13)
h
1
z
+
z
h
1
′
z
λ
h
1
z
+
μ
≺
h
2
z
i
m
p
l
i
e
s
h
1
z
≺
h
2
z
.Lemma 2 (see [5]).
Leth
(
z
)
=
1
+
∑
n
=
1
∞
c
n
z
n and H
z
=
1
+
∑
n
=
1
∞
d
n
z
n and h
≺
H. If H
(
z
) is univalent and convex in E, then (14)
c
n
≤
d
1
,for n
≥
1.Lemma 3 (see [6]).
Ifq
k
,
γ
(
z
)
=
1
+
q
1
z
+
q
2
z
2
+
⋯ then (15)
q
1
=
8
1
-
γ
arccos
⁡
k
2
π
2
1
-
k
2
,
0
<
k
<
1
,
8
1
-
γ
π
2
,
k
=
1
,
π
2
1
-
γ
4
t
k
2
-
1
k
2
t
1
+
t
,
k
>
1
.One now states and proves the main results.


## 3. Main Results

In this section we explore some of the geometric properties exhibited by the classM
Q
p
(
b
,
k
,
λ
,
α
1
).We begin by discussing an inclusion property for the classM
Q
p
(
b
,
k
,
λ
,
α
1
).Theorem 4.
IfR
(
α
1
)
>
p
R
(
b
q
k
,
λ
(
z
)
-
1
) then (16)
M
Q
p
b
,
k
,
λ
,
α
1
+
1
⊂
M
Q
p
b
,
k
,
λ
,
α
1
.Proof.
Letf
∈
M
Q
p
(
b
,
k
,
λ
,
α
1
+
1
) and set (17)
-
1
p
1
+
1
b
z
H
α
1
f
z
′
+
λ
z
2
H
α
1
f
z
′
′
1
-
λ
H
α
1
f
z
+
λ
z
H
α
1
f
z
′
-
b
=
h
z
.But differentiating (9) with respect to z we get (18)
z
H
α
1
f
z
′
=
α
1
H
α
1
+
1
f
z
′
-
α
1
+
p
H
α
1
f
z
′
,
z
H
α
1
f
z
′
′
+
H
α
1
f
z
′
=
α
1
H
α
1
+
1
f
z
′
-
α
1
+
p
H
α
1
f
z
′
,
z
2
H
α
1
f
z
′
′
=
α
1
z
H
α
1
+
1
f
z
′
-
α
1
+
p
+
1
z
H
α
1
f
z
′
.Putting (18) in (17) we have (19)
λ
z
α
1
H
α
+
1
f
z
′
+
1
-
λ
α
1
+
p
+
1
z
H
α
1
f
z
′
1
-
λ
H
α
1
f
z
+
λ
z
H
α
1
f
z
′
=
-
b
p
h
z
,
λ
α
1
z
H
α
1
+
1
f
z
′
+
1
-
λ
α
1
H
α
1
+
1
f
z
1
-
λ
H
α
1
f
z
+
λ
z
H
α
1
f
z
′
=
-
b
p
h
z
+
α
1
+
p
.Taking logarithmic derivative of (19) we have(20)
z
H
α
1
+
1
f
z
′
+
λ
z
2
H
α
1
+
1
f
z
′
′
1
-
λ
H
α
1
+
1
f
z
+
λ
z
H
α
1
+
1
f
z
′
+
b
p
h
z
+
b
b
-
1
=
-
b
p
z
h
′
z
-
b
p
h
z
+
α
1
+
p
,
z
H
α
1
+
1
f
z
′
+
λ
z
2
H
α
1
+
1
f
z
′
′
1
-
λ
H
α
1
+
1
f
z
+
λ
z
H
α
1
+
1
f
z
′
=
-
b
p
h
z
+
z
h
′
z
-
b
p
h
z
+
α
1
+
p
-
b
b
-
1
,
-
1
p
1
+
1
b
z
H
α
1
+
1
f
z
′
+
λ
z
2
H
α
1
+
1
f
z
′
′
1
-
λ
H
α
1
+
1
f
z
+
λ
z
H
α
1
+
1
f
z
′
-
b
=
h
z
+
z
h
′
z
-
b
p
h
z
+
α
1
+
p
.Since f
∈
M
Q
q
(
b
,
k
,
λ
,
α
1
+
1
), therefore (21)
h
z
+
z
h
′
z
-
b
p
h
z
+
α
1
+
p
≺
q
k
,
γ
z
.Using Lemma 2 we have (22)
h
z
≺
q
k
,
γ
z
,provided R
(
-
b
p
q
k
,
λ
(
z
)
+
α
1
+
p
)
>
0 or equivalently R
α
1
>
p
R
(
b
q
k
,
γ
(
z
)
-
1
). Hence, f
∈
M
Q
p
(
b
,
k
,
λ
,
α
1
).We now show that the classM
Q
p
(
b
,
k
,
λ
,
α
1
) is closed under a certain integral.Theorem 5.
Iff
(
z
)
∈
M
Q
p
(
b
,
k
,
λ
,
α
1
), then the integral (23)
B
η
H
α
f
z
=
η
-
p
z
η
∫
0
z
t
η
-
1
H
α
f
z
d
tmaps f
(
z
) into M
Q
p
(
b
,
k
,
λ
,
α
1
).Proof.
From (23) we have (24)
z
η
B
η
H
α
f
z
=
η
-
p
∫
0
z
t
η
-
1
H
α
f
z
d
t
.Note that (25)
B
η
H
α
f
z
=
H
α
B
η
f
z
.Differentiating (24) above we get (26)
η
z
η
-
1
B
η
H
α
f
z
+
z
B
η
H
α
f
z
′
=
η
-
p
z
η
-
1
H
α
f
z
,
η
B
η
H
α
f
z
+
z
B
η
H
α
f
z
′
=
η
-
p
H
α
f
z
.Differentiate again (27)
η
B
η
H
α
f
z
′
+
z
B
η
H
α
f
z
′
′
+
B
η
H
α
f
z
′
=
η
-
p
H
α
f
z
′
,
z
B
η
f
z
′
′
=
η
-
p
B
η
H
α
f
z
′
-
η
+
1
B
η
f
z
′
.Now let (28)
-
1
p
1
+
1
b
z
B
η
H
α
1
f
′
+
λ
z
2
B
η
H
α
1
f
′
′
1
-
λ
B
η
H
α
1
f
+
λ
z
B
η
H
α
1
f
′
-
b
=
g
z
.Using (26) and (27) in (28) we get (29)
z
B
η
H
α
1
f
′
+
λ
z
η
-
p
H
α
1
f
′
-
z
1
+
η
B
η
H
α
1
f
′
1
-
λ
B
η
H
α
1
f
+
λ
z
B
η
H
α
1
f
′
=
-
b
p
g
z
+
b
2
-
b
,
λ
z
η
-
p
H
α
1
f
′
+
1
-
λ
1
+
η
η
-
p
H
α
1
f
-
η
B
η
H
α
1
f
1
-
λ
B
η
H
α
1
f
+
λ
z
B
η
H
α
1
f
′
=
-
b
p
g
z
+
b
2
-
b
,
η
-
p
λ
z
H
α
1
f
′
-
1
-
λ
H
α
1
f
-
λ
η
z
B
η
H
α
1
f
′
-
1
-
λ
η
B
η
H
α
1
f
1
-
λ
B
η
H
α
1
f
+
λ
z
B
η
H
α
1
f
′
=
-
b
p
g
z
+
b
2
-
b
,
η
-
p
λ
z
H
α
1
f
′
-
1
-
λ
H
α
1
f
1
-
λ
B
η
H
α
1
f
+
λ
z
B
η
H
α
1
f
′
=
-
b
p
g
z
+
b
2
-
b
+
η
.Now taking logarithmic derivative we have (30)
η
-
p
λ
z
H
α
1
f
′
′
+
λ
H
α
1
f
′
+
1
-
λ
H
α
1
f
′
η
-
p
λ
z
H
α
1
f
′
-
1
-
λ
H
α
1
f
-
1
-
λ
B
η
H
α
1
f
′
+
λ
z
B
η
H
α
1
f
′
′
+
λ
B
η
H
α
1
f
′
1
-
λ
B
η
H
α
1
f
+
λ
z
B
η
H
α
1
f
′
=
-
b
p
g
′
z
-
b
p
g
z
+
b
2
-
b
,
-
1
p
1
+
1
b
z
H
α
1
f
′
+
λ
z
2
H
α
1
f
′
′
1
-
λ
H
α
1
f
+
λ
z
H
α
1
f
′
-
b
=
g
z
+
z
g
′
z
-
b
p
g
z
+
b
2
-
b
.Using Lemma 2 we get g
(
z
)
≺
q
k
,
γ
(
z
) which implies (31)
B
η
f
z
∈
M
Q
p
b
,
k
,
λ
,
α
1
.This proves the assertion.Now we get coefficient estimates of the classM
Q
p
(
b
,
k
,
λ
,
α
1
).Theorem 6.
Iff
(
z
)
∈
M
Q
p
(
b
,
k
,
λ
,
α
1
) and f
(
z
) is given by (1) then (32)
a
n
≤
b
p
q
1
b
1
-
b
+
b
p
-
p
+
n
∑
k
=
0
n
-
1
a
k
,
a
0
=
1provided (33)
2
λ
p
-
1
+
3
-
n
λ
<
k
λ
f
o
r
1
-
λ
p
+
n
-
1
λ
>
0
,
2
λ
p
-
1
+
3
-
n
λ
>
k
λ
f
o
r
1
-
λ
p
+
n
-
1
λ
<
0
,for all k
≤
n.Proof.
Letf
(
z
)
∈
M
Q
p
(
b
,
k
,
λ
,
α
1
); then, by definition, we have (34)
-
1
p
1
+
1
b
z
H
α
1
f
z
′
+
λ
z
2
H
α
1
f
z
′
′
1
-
λ
H
α
1
f
z
+
λ
z
H
α
1
f
z
′
-
b
=
h
z
≺
q
k
,
γ
z
,
-
z
H
α
1
f
z
′
+
λ
z
2
H
α
1
f
z
′
′
p
1
-
λ
H
α
1
f
z
+
λ
z
H
α
1
f
z
′
=
b
h
z
+
b
p
-
b
2
p
,which gives (35)
-
z
H
α
1
f
z
′
+
λ
z
2
H
α
1
f
z
′
′
p
=
b
h
z
+
b
1
-
b
p
1
-
λ
H
α
1
f
z
+
λ
z
H
α
1
f
z
′
.
Let us writeH
α
1
f
z
=
f
z; then (36)
f
z
=
z
-
p
+
∑
n
=
0
∞
a
n
z
n
-
p
,
f
′
z
=
-
p
z
-
p
-
1
+
∑
n
=
0
∞
a
n
n
-
p
z
n
-
p
-
1
,
z
f
′
z
=
-
p
z
-
p
+
∑
n
=
0
∞
a
n
n
-
p
z
n
-
p
,
f
′
′
z
=
-
p
-
p
-
1
z
-
p
-
2
+
∑
n
=
0
∞
a
n
n
-
p
n
-
p
-
1
z
n
-
p
-
2
,
z
2
f
′
′
z
=
-
p
-
p
-
1
z
-
p
+
∑
n
=
0
∞
a
n
n
-
p
n
-
p
-
1
z
n
-
p
.Assuming h
(
z
)
=
1
+
∑
n
=
1
∞
c
n
z
n, then (35) becomes (37)
z
-
p
-
∑
n
=
0
∞
a
n
n
-
p
p
z
n
-
p
+
λ
-
p
+
1
z
-
p
+
∑
n
=
0
∞
a
n
n
-
p
n
-
p
-
1
p
z
n
-
p
,
1
-
λ
p
-
λ
z
-
p
-
∑
n
=
1
∞
a
n
n
-
p
p
1
+
λ
n
-
p
-
1
z
n
-
p
=
b
+
∑
n
=
1
∞
b
c
n
z
n
+
b
1
-
b
p
1
-
λ
p
-
λ
z
-
p
+
∑
n
=
1
∞
a
n
1
+
λ
n
-
p
-
1
z
n
-
p
.From (37) we have (38)
1
-
λ
p
-
λ
z
-
p
-
∑
n
=
1
∞
a
n
n
-
p
p
1
+
λ
n
-
p
-
1
z
n
-
p
=
b
+
b
c
1
z
+
b
c
2
z
2
+
b
c
3
z
3
+
⋯
+
b
c
n
z
n
+
⋯
+
b
1
-
b
p
1
-
λ
p
-
λ
z
-
p
+
∑
n
=
1
∞
a
n
1
+
λ
n
-
p
-
1
z
n
-
p
.
Now comparing coefficients ofz
1
-
p we have (39)
-
a
1
1
-
p
p
1
-
λ
p
=
a
1
b
1
-
λ
p
+
a
1
b
1
-
b
p
1
-
λ
p
+
b
c
1
1
-
λ
p
-
λ
,
-
a
1
1
-
λ
p
b
1
-
b
+
b
p
-
p
+
1
p
=
b
c
1
1
-
λ
p
-
λ
,
a
1
=
-
p
1
-
λ
p
b
1
-
b
+
b
p
-
p
+
1
1
-
λ
p
-
λ
b
c
1
,and comparing the coefficients of z
2
-
p gives (40)
-
a
2
2
-
p
p
1
+
λ
1
-
p
=
a
2
1
+
λ
1
-
p
b
+
b
1
-
b
p
+
a
1
1
-
λ
p
b
c
1
+
b
c
2
1
-
λ
p
-
λ
,
-
a
2
1
+
λ
1
-
p
2
-
p
p
+
b
p
+
b
1
-
b
p
=
a
1
1
-
λ
p
b
c
1
+
b
c
2
1
-
λ
p
-
λ
,
-
a
2
1
+
λ
1
-
p
b
1
-
b
+
b
p
-
p
+
2
p
=
a
1
1
-
λ
p
b
c
1
+
b
c
2
1
-
λ
p
-
λ
,
a
2
=
-
p
1
+
λ
1
-
p
b
1
-
b
+
b
p
-
p
+
2
1
-
λ
p
-
λ
b
c
2
+
1
-
λ
p
b
a
1
c
1
,and for the coefficient of z
3
-
p we have(41)
-
a
3
1
+
λ
2
-
p
b
1
-
b
+
b
p
-
p
+
3
p
=
b
c
1
a
2
1
+
λ
1
-
p
+
b
c
2
a
1
1
-
λ
p
+
b
c
3
1
-
λ
p
-
λ
,
a
3
=
-
p
1
+
λ
2
-
p
b
1
-
b
+
b
p
-
p
+
3
1
-
λ
p
-
λ
b
c
3
+
1
-
λ
p
b
c
2
a
1
+
1
+
λ
1
-
p
b
c
1
a
2
,which generalise to (42)
a
n
=
-
p
1
-
λ
p
+
n
-
1
λ
b
1
-
b
+
b
p
-
p
+
n
1
-
λ
p
-
λ
b
c
n
+
1
-
λ
p
b
c
n
-
1
a
1
+
1
-
λ
p
+
λ
b
c
n
-
2
a
2
+
⋯
+
1
-
λ
p
+
n
-
2
λ
b
c
1
a
n
-
1
.The above expression can also be written as (43)
a
n
=
-
p
1
-
λ
p
+
n
-
1
λ
b
1
-
b
+
b
p
-
p
+
n
∑
k
=
1
n
1
-
λ
p
+
k
-
2
λ
c
n
-
k
-
1
a
k
-
1
,
with
a
0
=
1
.
Now taking(44)
A
n
,
k
=
-
1
-
λ
p
+
k
-
2
λ
1
-
λ
p
+
n
-
1
λ
,
n
≥
k
,we have (45)
A
n
,
k
<
1
,
if
2
λ
p
-
1
+
3
-
n
λ
<
k
λ
for
1
-
λ
p
+
n
-
1
λ
>
0
,
2
λ
p
-
1
+
3
-
n
λ
>
k
λ
for
1
-
λ
p
+
n
-
1
λ
<
0
,for all n
≥
k. Since q
k
,
γ
(
z
) is univalent and q
k
,
γ
(
E
) is convex, applying Rogosinski’s theorem we have (46)
c
n
≤
q
1
,where q
1 is given in (15). Under the conditions given in (45), expressions (39)–(42) give (47)
a
1
≤
b
p
q
1
b
1
-
b
+
b
p
-
p
+
1
,
a
2
≤
b
p
q
1
b
1
-
b
+
b
p
-
p
+
2
1
+
a
1
,
a
3
≤
b
p
q
1
b
1
-
b
+
b
p
-
p
+
3
1
+
a
1
+
a
2
,
⋮
a
n
≤
b
p
q
1
b
1
-
b
+
b
p
-
p
+
n
1
+
a
1
+
a
2
+
⋯
+
a
n
-
1
.This can also be written as (48)
a
n
≤
b
p
q
1
b
1
-
b
+
b
p
-
p
+
n
∑
k
=
0
n
-
1
a
k
.This concludes the proof.



---

*Source: 6360250-2016-07-26.xml*