# A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized Hypergeometric Functions

Authors: F. Ghanim; M. Darus
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

We introduce a new class of meromorphically analytic functions, which is defined by means of a Hadamard product (or convolution) involving some suitably normalized meromorphically functions related to Cho-Kwon-Srivastava operator. A characterization property giving the coefficient bounds is obtained for this class of functions. The other related properties, which are investigated in this paper, include distortion and the radii of starlikeness and convexity. We also consider several applications of our main results to generalized hypergeometric functions.


---

## Body

## 1. Introduction

A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic functionf
(
z
) is a function of the form
(1)
f
(
z
)
=
g
(
z
)
h
(
z
)
,
where g
(
z
) and h
(
z
) are entire functions with h
(
z
)
≠
0 (see [1, page 64]). A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain. An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere. For example, the gamma function is meromorphic in the whole complex plane C.In the present paper, we initiate the study of functions which are meromorphic in the punctured diskU
*
=
{
z
:
0
<
|
z
|
<
1
} with a Laurent expansion about the origin; see [2].LetA be the class of analytic functions h
(
z
) with h
(
0
)
=
1, which are convex and univalent in the open unit disk U
=
U
*
∪
{
0
} and for which
(2)
R
{
h
(
z
)
}
>
0
,
(
z
∈
U
*
)
.
For functions f and g analytic in U, we say that f is subordinate to g and write
(3)
f
≺
g
in
U
or
f
(
z
)
≺
g
(
z
)
,
(
z
∈
U
*
)
if there exists an analytic function w
(
z
) in U such that
(4)
|
w
(
z
)
|
≤
|
z
|
,
f
(
z
)
=
g
(
w
(
z
)
)
,
(
z
∈
U
*
)
.
Furthermore, if the function g is univalent in U, then
(5)
f
(
z
)
≺
g
(
z
)
⟺
f
(
0
)
=
g
(
0
)
,
f
(
U
)
⊆
g
(
U
)
,
(
z
∈
U
*
)
.This paper is divided into two sections; the first introduces a new class of meromorphically analytic functions, which is defined by means of a Hadamard product (or convolution) involving linear operator. The second section highlights some applications of the main results involving generalized hypergeometric functions.


## 2. Preliminaries

LetΣ denote the class of meromorphic functions f
(
z
) normalized by
(6)
f
(
z
)
=
1
z
+
∑
n
=
1
∞
a
n
z
n
,
which are analytic in the punctured unit disk U
*
=
{
z
:
0
<
|
z
|
<
1
}. For 0
≤
β, we denote by S
*
(
β
) and k
(
β
) the subclasses of Σ consisting of all meromorphic functions which are, respectively, starlike of order β and convex of order β in U.For functionsf
j
(
z
)  (
j
=
1
;
2
) defined by
(7)
f
j
(
z
)
=
1
z
+
∑
n
=
1
∞
a
n
,
j
z
n
,
we denote the Hadamard product (or convolution) of f
1
(
z
) and f
2
(
z
) by
(8)
(
f
1
*
f
2
)
=
1
z
+
∑
n
=
1
∞
‍
a
n
,
1
a
n
,
2
z
n
.
Cho et al. [3] and Ghanim and Darus [4] studied the following function:
(9)
q
λ
,
μ
(
z
)
=
1
z
+
∑
n
=
1
∞
(
λ
n
+
1
+
λ
)
μ
z
n
,
(
λ
>
0
,
μ
≥
0
)
.
Corresponding to the function q
λ
,
μ
(
z
) and using the Hadamard product for f
(
z
)
∈
Σ, we define a new linear operator L
(
λ
,
μ
) on Σ by
(10)
L
λ
,
μ
f
(
z
)
=
(
f
(
z
)
*
q
λ
,
μ
(
z
)
)
=
1
z
+
∑
n
=
1
∞
(
λ
n
+
λ
+
1
)
μ
|
a
n
|
z
n
.The Hadamard product or convolution of the functionsf given by (10) with the functions L
t
,
a
g and L
t
,
a
h given, respectively, by
(11)
L
λ
,
μ
g
(
z
)
=
1
z
+
∑
n
=
1
∞
(
λ
n
+
λ
+
1
)
μ
|
b
n
|
z
n
,
h
h
h
h
h
h
h
h
(
z
∈
U
*
,
g
(
z
)
∈
Σ
)
,
L
λ
,
μ
h
(
z
)
=
1
z
+
∑
n
=
1
∞
(
λ
n
+
λ
+
1
)
μ
|
c
n
|
z
n
,
h
h
h
h
h
h
h
h
i
i
(
z
∈
U
*
,
h
(
z
)
∈
Σ
)
,
can be expressed as follows:
(12)
L
λ
,
μ
(
f
*
g
)
(
z
)
=
1
z
+
∑
n
=
1
∞
(
λ
n
+
λ
+
1
)
μ
|
a
n
b
n
|
z
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
z
∈
U
*
)
,
L
λ
,
μ
(
f
*
h
)
(
z
)
=
1
z
+
∑
n
=
1
∞
(
λ
n
+
λ
+
1
)
μ
|
a
n
c
n
|
z
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
z
∈
U
*
)
.By applying the subordination definition, we introduce here a new classΣ
λ
μ
(
ρ
,
A
,
B
) of meromorphic functions, which is defined as follows.Definition 1.
A functionf
∈
Σ of the form (6) is said to be in the class Σ
λ
μ
(
ρ
,
A
,
B
) if it satisfies the following subordination property:
(13)
ρ
L
λ
,
μ
(
f
*
g
)
(
z
)
L
λ
,
μ
(
f
*
h
)
(
z
)
≺
ρ
-
(
A
-
B
)
z
1
+
B
z
,
(
z
∈
U
*
)
,
where -
1
≤
B
<
A
≤
1, ρ
>
0, with condition 0
≤
|
c
n
|
≤
|
b
n
| and L
(
λ
,
μ
)
(
f
*
h
)
(
z
)
≠
0.As for the second result of this paper on applications involving generalized hypergeometric functions, we need to utilize the well-known Gaussian hypergeometric function. One denotesϕ
(
α
,
β
;
z
) the class of the function given by
(14)
ϕ
~
(
α
,
β
;
z
)
=
1
z
+
∑
n
=
0
∞
(
α
)
n
+
1
(
β
)
n
+
1
z
n
,
for β
≠
0
,
-
1
,
-
2
,
…, and α
∈
C
∖
{
0
}, where (
λ
)
n
=
λ
(
λ
+
1
)
n
+
1 is the Pochhammer symbol. We note that
(15)
ϕ
~
(
α
,
β
;
z
)
=
1
z
F
1
2
(
1
,
α
,
β
;
z
)
,
where
(16)
F
1
2
(
b
,
α
,
β
;
z
)
=
∑
n
=
0
∞
(
b
)
n
(
α
)
n
(
β
)
n
z
n
n
!
is the well-known Gaussian hypergeometric function.Corresponding to the functionsϕ
~
(
α
,
β
;
z
) and q
λ
,
μ
(
z
) given in (9) and using the Hadamard product for f
(
z
)
∈
Σ, we define a new linear operator L
(
α
,
β
,
λ
,
μ
) on Σ by
(17)
L
(
α
,
β
,
λ
,
μ
)
f
(
z
)
=
(
f
(
z
)
*
ϕ
(
α
,
β
;
z
)
*
q
λ
,
μ
(
z
)
)
=
1
z
+
∑
n
=
1
∞
‍
(
α
)
n
+
1
(
β
)
n
+
1
(
λ
n
+
λ
+
1
)
μ
|
a
n
|
z
n
.
The meromorphic functions with the generalized hypergeometric functions were considered recently by Cho and Kim [5], Dziok and Srivastava [6, 7], Ghanim [8], Ghanim et al. [9, 10], and Liu and Srivastava [11, 12].Now, it follows from (17) that
(18)
z
(
L
(
α
,
β
,
λ
,
μ
)
f
(
z
)
)
′
=
α
L
(
α
+
1
,
β
,
λ
,
μ
)
f
(
z
)
-
(
α
+
1
)
L
(
α
,
β
,
λ
,
μ
)
f
(
z
)
.The subordination relation (13) in conjunction with (17) takes the following form:
(19)
ρ
L
(
α
+
1
,
β
,
λ
,
μ
)
f
(
z
)
L
(
α
,
β
,
λ
,
μ
)
f
(
z
)
≺
ρ
-
(
A
-
B
)
z
1
+
B
z
,
h
h
h
h
h
h
h
h
h
h
h
h
h
(
0
≤
B
<
A
≤
1
,
ρ
>
0
)
.Definition 2.
A functionf
∈
Σ of the form (6) is said to be in the class Σ
λ
μ
(
ρ
,
α
,
β
,
A
,
B
) if it satisfies the subordination relation (19) above.


## 3. Characterization and Other Related Properties

In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a functionf
∈
Σ of the form (6) to belong to the class Σ
λ
μ
(
ρ
,
A
,
B
) of meromorphically analytic functions.Theorem 3.
The functionf
∈
Σ is said to be a member of the class Σ
λ
μ
(
ρ
,
A
,
B
) if and only if it satisfies
(20)
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
(
ρ
|
b
n
|
(
1
+
B
)
h
h
h
h
h
h
h
h
h
h
h
h
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
|
a
n
|
≤
A
-
B
.
The equality is attained for the function f
n
(
z
) given by
(21)
f
n
(
z
)
=
1
z
+
∑
n
=
1
∞
(
A
-
B
)
(
n
+
λ
+
1
)
μ
λ
μ
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
z
n
.Proof.
Letf of the form (6) belong to the class Σ
λ
μ
(
ρ
,
A
,
B
). Then, in view of (12), we find that
(22)
|
ρ
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
a
n
(
b
n
-
c
n
)
z
n
+
1
×
(
(
A
-
B
)
-
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
hhhh
-
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
h
h
h
h
h
h
h
h
h
×
(
ρ
B
b
n
+
c
n
{
(
A
-
B
)
-
ρ
B
}
)
a
n
z
n
+
1
(
A
-
B
)
-
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
)
-
1
|
≤
ρ
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
a
n
(
b
n
-
c
n
)
|
z
n
+
1
|
h
h
×
(
(
A
-
B
)
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
hhh
h
h
-
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
h
h
h
h
h
h
h
h
h
h
h
h
h
×
(
A
-
B
)
-
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
(
ρ
B
b
n
+
c
n
{
(
A
-
B
)
-
ρ
B
}
)
a
n
|
z
n
+
1
|
)
-
1
≤
1
.
Putting |
z
|
=
r  (
0
≤
r
<
1
) and noting the fact that the denominator in the above inequality remains positive by virtue of the constraints stated in (13) for all r
∈
[
0,1
), we easily arrive at the desired inequality (20) by letting z
→
1.
Conversely, if we assume that the inequality (20) holds true in the simplified form (22), it can readily be shown that
(23)
|
ρ
{
(
(
f
*
g
)
(
z
)
)
-
(
(
f
*
h
)
(
z
)
)
}
ρ
B
(
(
f
*
g
)
(
z
)
)
+
{
ρ
(
A
-
B
)
-
ρ
B
}
(
(
f
*
h
)
(
z
)
)
|
<
1
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
z
∈
U
*
)
,
which is equivalent to our condition of theorem, so that f
∈
Σ
λ
μ
(
ρ
,
A
,
B
), hence the theorem.Theorem3 immediately yields the following result.Corollary 4.
If the functionf
∈
Σ belongs to the class Σ
λ
μ
(
ρ
,
A
,
B
), then
(24)
|
a
n
|
≤
(
A
-
B
)
(
n
+
λ
+
1
)
μ
λ
μ
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
,
n
≥
1
,
where the equality holds true for the functions f
n
(
z
) given by (21).We now state the following growth and distortion properties for the classΣ
λ
μ
(
ρ
,
A
,
B
).Theorem 5.
If the functionf defined by (6) is in the class Σ
λ
μ
(
ρ
,
A
,
B
), then, for 0
<
|
z
|
=
r
<
1, one has
(25)
1
r
-
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
r
≤
|
f
(
z
)
|
≤
1
r
+
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
r
,
1
r
2
-
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
≤
|
f
′
(
z
)
|
≤
1
r
2
+
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
.Proof.
Sincef
∈
Σ
λ
μ
(
ρ
,
A
,
B
), Theorem 3 readily yields the inequality
(26)
∑
n
=
1
∞
‍
|
a
n
|
≤
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
.
Thus, for 0
<
|
z
|
=
r
<
1 and utilizing (26), we have
(27)
|
f
(
z
)
|
=
1
|
z
|
+
∑
n
=
1
∞
‍
|
a
n
|
|
z
|
n
≤
1
r
+
r
∑
n
=
1
∞
‍
|
a
n
|
≤
1
r
+
r
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
,
|
f
(
z
)
|
=
1
|
z
|
-
∑
n
=
1
∞
‍
|
a
n
|
|
z
|
n
≥
1
r
-
r
∑
n
=
1
∞
‍
|
a
n
|
≥
1
r
-
r
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
.
Also from Theorem 3, we get
(28)
∑
n
=
1
∞
‍
n
|
a
n
|
≤
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
.
Hence
(29)
|
f
′
(
z
)
|
=
1
|
z
|
2
+
∑
n
=
1
∞
n
|
a
n
|
|
z
|
n
-
1
≤
1
r
2
+
∑
n
=
1
∞
|
a
n
|
≤
1
r
2
+
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
,
|
f
′
(
z
)
|
=
1
|
z
|
2
-
∑
n
=
1
∞
‍
n
|
a
n
|
|
z
|
n
-
1
≥
1
r
2
-
∑
n
=
1
∞
n
|
a
n
|
≥
1
r
2
-
(
A
-
B
)
(
2
+
λ
)
μ
λ
μ
(
ρ
|
b
1
|
(
1
+
B
)
-
|
c
1
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
.
This completes the proof of Theorem 5.We next determine the radii of meromorphic starlikeness and meromorphic convexity of the classΣ
λ
μ
(
ρ
,
A
,
B
), which are given by Theorems 6 and 7 below.Theorem 6.
If the functionf defined by (6) is in the class Σ
λ
μ
(
ρ
,
A
,
B
), then f is meromorphic starlike of order δ in the disk |
z
|
<
r
1, where
(30)
r
1
=
inf
⁡
n
≥
1
{
(
1
-
δ
)
(
(
A
-
B
)
(
n
+
2
-
δ
)
)
-
1
h
h
h
×
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
h
h
h
×
(
(
A
-
B
)
(
n
+
2
-
δ
)
)
-
1
}
1
/
(
n
+
1
)
.
The equality is attained for the function f
n
(
z
) given by (21).Proof.
It suffices to prove that(31)
|
z
(
f
(
z
)
)
′
f
(
z
)
+
1
|
≤
1
-
δ
.
For |
z
|
<
r
1, we have
(32)
|
z
(
f
(
z
)
)
′
f
(
z
)
+
1
|
=
|
∑
n
=
1
∞
(
n
+
1
)
(
λ
/
(
n
+
1
+
λ
)
)
μ
a
n
z
n
1
/
z
+
∑
n
=
1
∞
‍
(
λ
/
(
n
+
1
+
λ
)
)
μ
a
n
z
n
|
=
|
∑
n
=
1
∞
‍
(
n
+
1
)
(
λ
/
(
n
+
λ
+
1
)
)
μ
a
n
z
n
+
1
1
+
∑
n
=
1
∞
‍
(
λ
/
(
n
+
λ
+
1
)
)
μ
a
n
z
n
+
1
|
≤
∑
n
=
1
∞
‍
(
n
+
1
)
(
λ
/
(
n
+
λ
+
1
)
)
μ
|
a
n
|
|
z
|
n
+
1
1
+
∑
n
=
1
∞
‍
(
λ
/
(
n
+
λ
+
1
)
)
μ
|
a
n
|
|
z
|
n
+
1
.
Hence (32) holds true for
(33)
∑
n
=
1
∞
(
n
+
1
)
(
λ
n
+
λ
+
1
)
μ
|
a
n
|
|
z
|
n
+
1
≤
(
1
-
δ
)
(
1
-
∑
n
=
1
∞
‍
(
λ
n
+
λ
+
1
)
μ
|
a
n
|
|
z
|
n
+
1
)
or
(34)
∑
n
=
1
∞
(
n
+
2
-
δ
)
(
λ
/
(
n
+
λ
+
1
)
)
μ
|
a
n
|
|
z
|
n
+
1
(
1
-
δ
)
≤
1
.
With the aid of (20) and (34), it is true to say that for fixed n
(35)
(
n
+
2
-
δ
)
(
λ
/
(
n
+
1
+
λ
)
)
μ
|
z
|
n
+
1
(
1
-
δ
)
≤
(
λ
n
+
λ
+
1
)
μ
h
h
h
×
(
λ
n
+
λ
+
1
)
μ
(
ρ
|
b
n
|
(
1
+
B
)
+
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
h
h
h
×
(
A
-
B
)
-
1
,
n
≥
1
.
Solving (35) for |
z
|, we obtain
(36)
|
z
|
≤
{
(
1
-
δ
)
(
(
n
+
2
-
δ
)
(
A
-
B
)
)
-
1
h
h
×
(
ρ
|
b
n
|
(
1
+
B
)
+
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
h
h
×
(
(
n
+
2
-
δ
)
(
A
-
B
)
)
-
1
}
n
+
1
.
This completes the proof of Theorem 6.Theorem 7.
If the functionf defined by (6) is in the class Σ
λ
μ
(
ρ
,
A
,
B
), then f is meromorphic convex of order δ in the disk |
z
|
<
r
2, where
(37)
r
2
=
inf
⁡
n
≥
1
{
(
1
-
δ
)
(
n
(
n
+
2
-
δ
)
(
A
-
B
)
)
-
1
h
h
h
h
×
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
h
h
h
h
×
(
n
(
n
+
2
-
δ
)
(
A
-
B
)
)
-
1
}
1
/
(
n
+
1
)
.
The equality is attained for the function f
n
(
z
) given by (21).Proof.
By using the same technique employed in the proof of Theorem6, we can show that
(38)
|
z
(
f
(
z
)
)
′′
(
f
(
z
)
)
′
+
2
|
≤
1
-
δ
.
For |
z
|
<
r
1 and with the aid of Theorem 3, we have the assertion of Theorem 7.


## 4. Applications Involving Generalized Hypergeometric Functions

Theorem 8.
The functionf
∈
Σ is said to be a member of the class Σ
λ
μ
(
ρ
,
α
,
β
,
A
,
B
) if and only if it satisfies
(39)
∑
n
=
1
∞
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
h
h
×
(
α
)
n
+
1
(
β
)
n
+
1
(
λ
n
+
λ
+
1
)
μ
|
a
n
|
≤
A
-
B
.
The equality is attained for the function f
n
(
z
) given by
(40)
f
n
(
z
)
=
1
z
+
∑
n
=
1
∞
(
A
-
B
)
(
n
+
λ
+
1
)
μ
λ
μ
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
z
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
n
≥
1
.Proof.
By using the same technique employed in the proof of Theorem3 along with Definition 2, we can prove Theorem 8.The following consequences of Theorem8 can be deduced by applying (39) and (40) along with Definition 2.Corollary 9.
If the functionf
∈
Σ belongs to the class Σ
λ
μ
(
ρ
,
α
,
β
,
A
,
B
), then
(41)
|
a
n
|
≤
(
A
-
B
)
(
n
+
λ
+
1
)
μ
(
β
)
n
+
1
λ
μ
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
(
α
)
n
+
1
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
≥
1
,
where the equality holds true for the functions f
n
(
z
) given by (40).Corollary 10.
If the functionf defined by (6) is in the class Σ
λ
μ
(
ρ
,
α
,
β
,
A
,
B
), then f is meromorphic starlike of order δ in the disk |
z
|
<
r
3, where
(42)
r
3
=
inf
⁡
n
≥
1
{
(
1
-
δ
)
(
(
A
-
B
)
(
n
+
2
-
δ
)
)
-
1
h
h
h
h
×
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
h
h
h
h
i
×
(
(
A
-
B
)
(
n
+
2
-
δ
)
)
-
1
}
1
/
(
n
+
1
)
.
The equality is attained for the function f
n
(
z
) given by (40).Corollary 11.
If the functionf defined by (6) is in the class Σ
λ
μ
(
ρ
,
α
,
β
,
A
,
B
), then f is meromorphic convex of order δ in the disk |
z
|
<
r
4, where
(43)
r
4
=
inf
⁡
n
≥
1
{
(
1
-
δ
)
×
(
n
(
n
+
2
-
δ
)
(
A
-
B
)
)
-
1
h
h
h
h
×
(
ρ
|
b
n
|
(
1
+
B
)
-
|
c
n
|
(
ρ
(
1
+
B
)
+
A
-
B
)
)
h
h
h
h
×
(
n
(
n
+
2
-
δ
)
(
A
-
B
)
)
-
1
}
1
/
(
n
+
1
)
.
The equality is attained for the function f
n
(
z
) given by (40).



---

*Source: 374821-2014-07-09.xml*