1. Introduction
Let
Mp denote the class of functions of the form
(1)
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, for 0 <
γ < 1 and
k ∈ [0,
∞,
(2)
where
(3)
[
1] showed that the extremal functions
qk,γ(
z) for conic regions are convex univalent and given by
(4)
where
R(
t) is Legendre’s complete elliptic integral of the first kind with
as its complementary integral,
,
t ∈ (0,1), and
z ∈
E is chosen in such a way that
k = cosh(
πR′(
t)/
R(
t)).
The generalised hypergeometric function
for complex parameters
α1, …,
αq and
β1, …,
βs with
βj ≠ 0, −1, −2, −3, …, for
j = 1,2, 3, …,
s, is defined as
(5)
with
q ≤
s + 1,
, and (
α)
n is the well-known Pochhammer symbol related to the factorial and the Gamma function by the relation
(6)
Also (
5) implies
(7)
Liu and Srivastava [
2] defined a linear operator for functions belonging to the class of multivalent meromorphic function
Hp(
α1, …,
αq;
β1, …,
βs) :
Mp →
Mp as follows:
(8)
If we assume for brevity that
Hp,q,s(
α1) =
Hp(
α1, …,
αq;
β1, …,
βs) then the following identity holds for this operator:
(9)
Shareef [
3] defined and studied subclass
MQp(
k,
λ,
α1) of meromorphic function associated with conic domain, for
k ≥ 0, 0 ≤
λ < 1, and
p ≥ 1, as follows:
(10)
We now define a new subclass
MQp(
b,
k,
λ,
α1) of meromorphic function associated with conic domain, for
k ≥ 0, 0 ≤
λ < 1,
p ≥ 1, and
b ≥ 1, as follows:
(11)
Since
qk,γ is a convex and univalent function, for
h(
z)≺
qk,γ(
z) it means
h(
E∗) is contained in
qk,γ(
E∗), where
(12)
In the next two sections, for brevity, we drop the subscripts of the operator
Hp,q,s(
α1).
2. Preliminary Results
Lemma 1 (see [4].)Let h2(z) be convex in E and , where , , and z ∈ E. If h1(z) is analytic in E, with h1(0) = h2(0), then
(13)
Lemma 2 (see [5].)Let and and h≺H. If H(z) is univalent and convex in E, then
(14)
for
n ≥ 1.
Lemma 3 (see [6].)If qk,γ(z) = 1 + q1z + q2z2 + ⋯ then
(15)
One now states and proves the main results.
3. Main Results
In this section we explore some of the geometric properties exhibited by the class MQp(b, k, λ, α1).
We begin by discussing an inclusion property for the class MQp(b, k, λ, α1).
Proof. Let f ∈ MQp(b, k, λ, α1 + 1) and set
(17)
But differentiating (
9) with respect to
z we get
(18)
Putting (
18) in (
17) we have
(19)
Taking logarithmic derivative of (
19) we have
(20)
Since
f ∈
MQq(
b,
k,
λ,
α1 + 1), therefore
(21)
Using Lemma
2 we have
(22)
provided
or equivalently
. Hence,
f ∈
MQp(
b,
k,
λ,
α1).
We now show that the class MQp(b, k, λ, α1) is closed under a certain integral.
Theorem 5. If f(z) ∈ MQp(b, k, λ, α1), then the integral
(23)
maps
f(
z) into
MQp(
b,
k,
λ,
α1).
Proof. From (23) we have
(24)
Note that
(25)
Differentiating (
24) above we get
(26)
Differentiate again
(27)
Now let
(28)
Using (
26) and (
27) in (
28) we get
(29)
Now taking logarithmic derivative we have
(30)
Using Lemma
2 we get
g(
z)≺
qk,γ(
z) which implies
(31)
This proves the assertion.
Now we get coefficient estimates of the class MQp(b, k, λ, α1).
Theorem 6. If f(z) ∈ MQp(b, k, λ, α1) and f(z) is given by (1) then
(32)
provided
(33)
for all
k ≤
n.
Proof. Let f(z) ∈ MQp(b, k, λ, α1); then, by definition, we have
(34)
which gives
(35)
Let us write H(α1)f(z) = f(z); then
(36)
Assuming
, then (
35) becomes
(37)
From (
37) we have
(38)
Now comparing coefficients of z1−p we have
(39)
and comparing the coefficients of
z2−p gives
(40)
and for the coefficient of
z3−p we have
(41)
which generalise to
(42)
The above expression can also be written as
(43)
Now taking
(44)
we have
(45)
for all
n ≥
k. Since
qk,γ(
z) is univalent and
qk,γ(
E) is convex, applying Rogosinski’s theorem we have
(46)
where
q1 is given in (
15). Under the conditions given in (
45), expressions (
39)–(
42) give
(47)
This can also be written as
(48)
This concludes the proof.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work here is supported by AP-2013-009.
