International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 1 (2014), 68-77
http://www.etamaths.com

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE

FUNCTIONS OF COMPLEX ORDER

T. JANANI AND G. MURUGUSUNDARAMOORTHY∗

Abstract. In the present investigation, we define a new subclass of meromor-

phic bi-univalent functions class Σ′ of complex order γ ∈ C\{0}, and obtain
the estimates for the coefficients |b0| and |b1|. Further we pointed out several
new or known consequences of our result.

1. Introduction and Definitions

Denote by A the class of analytic functions of the form

(1.1) f(z) = z +

∞∑
n=2

anz
n

which are univalent in the open unit disc

∆ = {z : |z| < 1}.

Also denote by S the class of all functions in A which are univalent and normalized
by the conditions

f(0) = 0 = f′(0) − 1
in ∆. Some of the important and well-investigated subclasses of the univalent func-
tion class S includes the class S∗(α)(0 ≤ α < 1) of starlike functions of order α in
∆ and the class K(α)(0 ≤ α < 1) of convex functions of order α

<
(
z f′(z)

f(z)

)
> α or <

(
1 +

z f′′(z)

f′(z)

)
> α, (z ∈ ∆)

respectively.Further a function f(z) ∈A is said to be in the class S(γ) of univalent
function of complex order γ(γ ∈ C\{0}) if and only if

f(z)

z
6= 0 and <

(
1 +

1

γ

[
zf′(z)

f(z)
− 1
])

> 0,z ∈ ∆.

By taking γ = (1 − α)cosβ e−iβ, |β| < π
2

and 0 ≤ α < 1, the class S((1 −
α)cosβ e−iβ) ≡ S(α,β) called the generalized class of β-spiral-like functions of
order α(0 ≤ α < 1).

An analytic function ϕ is subordinate to an analytic function ψ, written by

ϕ(z) ≺ ψ(z),

2000 Mathematics Subject Classification. 30C45 , 30C50.
Key words and phrases. Analytic functions; Univalent functions; Meromorphic functions; Bi-

univalent functions; Bi-starlike and bi-convex functions of complex order; Coefficient bounds.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

68



COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS 69

provided there is an analytic function ω defined on ∆ with

ω(0) = 0 and |ω(z)| < 1

satisfying

ϕ(z) = ψ(ω(z)).

Ma and Minda [9] unified various subclasses of starlike and convex functions for
which either of the quantity

z f′(z)

f(z)
or 1 +

z f′′(z)

f′(z)

is subordinate to a more general superordinate function. For this purpose, they
considered an analytic function φ with positive real part in the unit disk ∆,φ(0) =
1,φ′(0) > 0 and φ maps ∆ onto a region starlike with respect to 1 and symmetric
with respect to the real axis.

The class of Ma-Minda starlike functions consists of functions f ∈ A satisfying
the subordination

z f′(z)

f(z)
≺ φ(z).

Similarly, the class of Ma-Minda convex functions consists of functions f ∈ A
satisfying the subordination

1 +
z f′′(z)

f′(z)
≺ φ(z).

It is well known that every function f ∈S has an inverse f−1, defined by

f−1(f(z)) = z, (z ∈ ∆)
and f(f−1(w)) = w, (|w| < r0(f); r0(f) ≥ 1/4)

where

(1.2) f−1(w) = w −a2w2 + (2a22 −a3)w
3 − (5a32 − 5a2a3 + a4)w

4 + · · · .

A function f ∈A given by (1.1), is said to be bi-univalent in ∆ if both f(z) and
f−1(z) are univalent in ∆, these classes are denoted by Σ. Earlier, Brannan and
Taha [2] introduced certain subclasses of bi-univalent function class Σ, namely bi-
starlike functions S∗Σ(α) and bi-convex function KΣ(α) of order α corresponding
to the function classes S∗(α) and K(α) respectively. For each of the function
classes S∗Σ(α) and KΣ(α), non-sharp estimates on the first two Taylor-Maclaurin
coefficients |a2| and |a3| were found [2, 17]. But the coefficient problem for each of
the following Taylor-Maclaurin coefficients:

|an| (n ∈ N\{1, 2}; N := {1, 2, 3, · · ·})

is still an open problem(see[1, 2, 8, 10, 17]). Recently several interesting subclass-
es of the bi-univalent function class Σ have been introduced and studied in the
literature(see[15, 18, 19]).

A function f is bi-starlike of Ma-Minda type or bi-convex of Ma-Minda type
if both f and f−1 are respectively Ma-Minda starlike or convex. These classes
are denoted respectively by S∗Σ(φ) and KΣ(φ).In the sequel, it is assumed that
φ is an analytic function with positive real part in the unit disk ∆, satisfying



70 JANANI AND MURUGUSUNDARAMOORTHY

φ(0) = 1,φ′(0) > 0 and φ(∆) is symmetric with respect to the real axis. Such a
function has a series expansion of the form

(1.3) φ(z) = 1 + B1z + B2z
2 + B3z

3 + · · · , (B1 > 0).
Let Σ′ denote the class of meromorphic univalent functions g of the form

(1.4) g(z) = z + b0 +

∞∑
n=1

bn
zn

defined on the domain ∆∗ = {z : 1 < |z| < ∞}. Since g ∈ Σ′ is univalent, it has an
inverse g−1 = h that satisfy

g−1(g(z)) = z, (z ∈ ∆∗)
and

g(g−1(w)) = w, (M < |w| < ∞,M > 0)
where

(1.5) g−1(w) = h(w) = w +

∞∑
n=0

Cn
wn

, (M < |w| < ∞).

Analogous to the bi-univalent analytic functions, a function g ∈ Σ′ is said to
be meromorphic bi-univalent if g−1 ∈ Σ′. We denote the class of all meromorphic
bi-univalent functions by MΣ′. Estimates on the coefficients of meromorphic uni-
valent functions were widely investigated in the literature, for example, Schiffer[13]
obtained the estimate |b2| ≤ 23 for meromorphic univalent functions g ∈ Σ

′ with

b0 = 0 and Duren [3] gave an elementary proof of the inequality |bn| ≤ 2(n+1) on the
coefficient of meromorphic univalent functions g ∈ Σ′ with bk = 0 for 1 ≤ k < n2 .
For the coefficient of the inverse of meromorphic univalent functions h ∈ MΣ′ ,
Springer [14] proved that |C3| ≤ 1 and |C3 + 12C

2
1| ≤

1
2

and conjectured that

|C2n−1| ≤
(2n−1)!
n!(n−1)!, (n = 1, 2, ...).

In 1977, Kubota [7] has proved that the Springer conjecture is true for n =
3, 4, 5 and subsequently Schober [12] obtained a sharp bounds for the coefficients
C2n−1, 1 ≤ n ≤ 7 of the inverse of meromorphic univalent functions in ∆∗. Recently,
Kapoor and Mishra [6] (see [16]) found the coefficient estimates for a class consisting
of inverses of meromorphic starlike univalent functions of order α in ∆∗.

Motivated by the earlier work of [4, 5, 6, 20], in the present investigation, a new
subclass of meromorphic bi-univalent functions class Σ′ of complex order γ ∈ C\{0},
is introduced and estimates for the coefficients |b0| and |b1| of functions in the newly
introduced subclass are obtained. Several new consequences of the results are also
pointed out.

Definition 1.1. For 0 ≤ λ ≤ 1,µ ≥ 0,µ > λ a function g(z) ∈ Σ′ given by (1.4)
is said to be in the class MγΣ′ (λ,µ,φ) if the following conditions are satisfied:

(1.6) 1 +
1

γ

[
(1 −λ)

(
g(z)

z

)µ
+ λg′(z)

(
g(z)

z

)µ−1
− 1

]
≺ φ(z)

and

(1.7) 1 +
1

γ

[
(1 −λ)

(
h(w)

w

)µ
+ λh′(w)

(
h(w)

w

)µ−1
− 1

]
≺ φ(w)

where z,w ∈ ∆∗, γ ∈ C\{0} and the function h is given by (1.5).



COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS 71

By suitably specializing the parameters λ and µ, we state the new subclasses
of the class meromorphic bi-univalent functions of complex order MγΣ′ (λ,µ,φ) as
illustrated in the following Examples.

Example 1.1. For 0 ≤ λ < 1,µ = 1 a function g ∈ Σ′ given by (1.4) is said
to be in the class MγΣ′ (λ, 1,φ) ≡ F

γ
Σ′ (λ,φ) if it satisfies the following conditions

respectively:

1 +
1

γ

[
(1 −λ)

(
g(z)

z

)
+ λg′(z) − 1

]
≺ φ(z)

and

1 +
1

γ

[
(1 −λ)

(
h(w)

w

)
+ λh′(w) − 1

]
≺ φ(w)

where z,w ∈ ∆∗, γ ∈ C\{0} and the function h is given by (1.5).

Example 1.2. For λ = 1, 0 ≤ µ < 1 a function g ∈ Σ′ given by (1.4) is said
to be in the class MγΣ′ (1,µ,φ) ≡ B

γ
Σ′ (µ,φ) if it satisfies the following conditions

respectively:

1 +
1

γ

[
g′(z)

(
g(z)

z

)µ−1
− 1

]
≺ φ(z)

and

1 +
1

γ

[
h′(w)

(
h(w)

w

)µ−1
− 1

]
≺ φ(w)

where z,w ∈ ∆∗, γ ∈ C\{0} and the function h is given by (1.5).

Example 1.3. For λ = 1,µ = 0, a function g ∈ Σ′ given by (1.4) is said to be in
the class MγΣ′ (1, 0,φ) ≡S

γ
Σ′ (φ) if it satisfies the following conditions respectively:

1 +
1

γ

(
zg′(z)

g(z)
− 1
)
≺ φ(z)

and

1 +
1

γ

(
wh′(w)

h(w)
− 1
)
≺ φ(w)

where z,w ∈ ∆∗, γ ∈ C\{0} and the function h is given by (1.5).

2. Coefficient estimates for the function class MγΣ′ (λ,µ,φ)

In this section we obtain the coefficients |b0| and |b1| for g ∈M
γ
Σ′ (λ,µ,φ) asso-

ciating the given functions with the functions having positive real part. In order to
prove our result we recall the following lemma.

Lemma 2.1. [11] If Φ ∈ P, the class of all functions with <(Φ(z)) > 0, (z ∈ ∆)
then

|ck| ≤ 2, for each k,

where

Φ(z) = 1 + c1z + c2z
2 + · · · for z ∈ ∆.



72 JANANI AND MURUGUSUNDARAMOORTHY

Define the functions p and q in P given by

p(z) =
1 + u(z)

1 −u(z)
= 1 +

p1
z

+
p2
z2

+ · · ·

and

q(z) =
1 + v(z)

1 −v(z)
= 1 +

q1
z

+
q2
z2

+ · · · .

It follows that

u(z) =
p(z) − 1
p(z) + 1

=
1

2

[
p1
z

+

(
p2 −

p21
2

)
1

z2
+ · · ·

]
and

v(z) =
q(z) − 1
q(z) + 1

=
1

2

[
q1
z

+

(
q2 −

q21
2

)
1

z2
+ · · ·

]
.

Note that for the functions p(z),q(z) ∈P, we have

|pi| ≤ 2 and |qi| ≤ 2 for each i.

Theorem 2.1. Let g is given by (1.4) be in the class MγΣ′ (λ,µ,φ). Then

(2.1) |b0| ≤
∣∣∣∣ γB1µ−λ

∣∣∣∣
and

(2.2) |b1| ≤

∣∣∣∣∣∣ γ
√(

(µ− 1)γB21
2(µ−λ)2

)2
+

(
B2

µ− 2λ

)2 ∣∣∣∣∣∣
where γ ∈ C\{0}, 0 ≤ λ ≤ 1,µ ≥ 0,µ > λ and z,w ∈ ∆∗.

Proof. It follows from (1.6) and (1.7) that

(2.3) 1 +
1

γ

[
(1 −λ)

(
g(z)

z

)µ
+ λg′(z)

(
g(z)

z

)µ−1
− 1

]
= φ(u(z))

and

(2.4) 1 +
1

γ

[
(1 −λ)

(
h(w)

w

)µ
+ λh′(w)

(
h(w)

w

)µ−1
− 1

]
= φ(v(w)).

In light of (1.4), (1.5), (1.6) and (1.7), we have

(2.5) 1 +
1

γ

[
(1 −λ)

(
g(z)

z

)µ
+ λg′(z)

(
g(z)

z

)µ−1
− 1

]

= 1 +
1

γ

[
(µ−λ)

b0
z

+ (µ− 2λ)[
(µ− 1)

2
b20 + b1]

1

z2
+ ...

]
= 1 + B1p1

1

2z
+

[
1

2
B1(p2 −

p21
2

) +
1

4
B2p

2
1

]
1

z2
+ ...



COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS 73

and

(2.6) 1 +
1

γ

[
(1 −λ)

(
h(w)

w

)µ
+ λh′(w)

(
h(w)

w

)µ−1
− 1

]

= 1 +
1

γ

[
−(µ−λ)

b0
z

+ (µ− 2λ)[
(µ− 1)

2
b20 − b1]

1

z2
+ ...

]
= 1 + B1q1

1

2w
+

[
1

2
B1(q2 −

q21
2

) +
1

4
B2q

2
1

]
1

w2
+ ...

Now, equating the coefficients in (2.5) and (2.6), we get

(2.7)
1

γ
(µ−λ)b0 =

1

2
B1p1,

(2.8)
1

γ
(µ− 2λ)

[
(µ− 1)

b20
2

+ b1

]
=

1

2
B1(p2 −

p21
2

) +
1

4
B2p

2
1,

(2.9) −
1

γ
(µ−λ)b0 =

1

2
B1q1,

and

(2.10)
1

γ
(µ− 2λ)

[
(µ− 1)

b20
2
− b1

]
=

1

2
B1(q2 −

q21
2

) +
1

4
B2q

2
1.

From (2.7) and (2.9), we get

(2.11) p1 = −q1
and

8(µ−λ)2b20 = γ
2B21 (p

2
1 + q

2
1 ).

Hence,

(2.12) b20 =
γ2B21 (p

2
1 + q

2
1 )

8(µ−λ)2
.

Applying Lemma (2.1) for the coefficients p1 and q1, we have

|b0| ≤
∣∣∣∣ γB1µ−λ

∣∣∣∣ .
Next, in order to find the bound on |b1| from (2.8), (2.10) and (2.11), we obtain

(2.13) (µ− 2λ)2 b21 = (µ− 2λ)
2(µ− 1)2

b40
4

−γ2
(
B21
4
p2q2 + (B2 −B1)B1(p2 + q2)

p21
8

+ (B1 −B2)2
p41
16

)
.

Using (2.12) and applying Lemma (2.1) once again for the coefficients p1,p2 and
q2, we get

|b1| ≤

∣∣∣∣∣∣ γ
√(

(µ− 1)γB21
2(µ−λ)2

)2
+

(
B2

µ− 2λ

)2 ∣∣∣∣∣∣ .
�



74 JANANI AND MURUGUSUNDARAMOORTHY

Corollary 2.1. Let g(z) is given by (1.4) be in the class FγΣ′ (λ,φ). Then

(2.14) |b0| ≤
∣∣∣∣ γB11 −λ

∣∣∣∣
and

(2.15) |b1| ≤
∣∣∣∣ γB22λ− 1

∣∣∣∣
where γ ∈ C\{0}, 0 ≤ λ < 1 and z,w ∈ ∆∗.

Corollary 2.2. Let g(z) is given by (1.4) be in the class BγΣ′ (µ,φ). Then

(2.16) |b0| ≤
∣∣∣∣ γB1µ− 1

∣∣∣∣
and

(2.17) |b1| ≤

∣∣∣∣∣∣ γ
√(

γB21
2(µ− 1)

)2
+

(
B2
µ− 2

)2 ∣∣∣∣∣∣
where γ ∈ C\{0}, 0 ≤ µ < 1 and z,w ∈ ∆∗.

Corollary 2.3. Let g(z) is given by (1.4) be in the class SγΣ′ (φ). Then

(2.18) |b0| ≤ |γ B1|

and

(2.19) |b1| ≤
∣∣∣∣γ2
√
γ2B41 + B

2
2

∣∣∣∣
where γ ∈ C\{0} and z,w ∈ ∆∗.

3. Corollaries and concluding Remarks

Analogous to (1.3), by setting φ(z) as given below:

(3.1) φ(z) =

(
1 + z

1 −z

)α
= 1 + 2αz + 2α2z2 + · · · (0 < α ≤ 1),

we have

B1 = 2α, B2 = 2α
2.

For γ = 1 and φ(z)is given by (3.1) we state the following corollaries:

Corollary 3.1. Let g is given by (1.4) be in the class M1Σ′ (λ,µ,
(

1+z
1−z

)α
) ≡

MΣ′ (λ,α). Then

|b0| ≤
2α

|µ−λ|
and

|b1| ≤

∣∣∣∣∣ 2α2
√

(µ− 1)2
(µ−λ)4

+
1

(µ− 2λ)2

∣∣∣∣∣
where 0 < λ ≤ 1,µ ≥ 0,µ > λ and z,w ∈ ∆∗.



COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS 75

Corollary 3.2. Let g(z) is given by (1.4) be in the class F1Σ′ (λ,
(

1+z
1−z

)α
) ≡FΣ′ (λ,α),

then

|b0| ≤
2α

|1 −λ|
and

|b1| ≤
2α2

|1 − 2λ|
where 0 ≤ λ < 1 and z,w ∈ ∆∗.

Corollary 3.3. Let g(z) is given by (1.4) be in the class B1Σ′ (λ,
(

1+z
1−z

)α
) ≡BΣ′ (µ,α),

then

|b0| ≤
2α

|µ− 1|
and

|b1| ≤

∣∣∣∣∣ 2α2
√

1

(µ− 1)2
+

1

(µ− 2)2

∣∣∣∣∣
where 0 ≤ µ < 1 and z,w ∈ ∆∗.

Corollary 3.4. Let g(z) is given by (1.4) be in the class S1Σ′
([

1+z
1−z

]α)
≡ SΣ′ (α)

then

|b0| ≤ 2α
and

|b1| ≤ α2
√

5

where z,w ∈ ∆∗.

On the other hand if we take

(3.2) φ(z) =
1 + (1 − 2β)z

1 −z
= 1 + 2(1 −β)z + 2(1 −β)z2 + · · · (0 ≤ β < 1),

then

B1 = B2 = 2(1 −β).
For γ = 1 and φ(z)is given by (3.2) we state the following corollaries:

Corollary 3.5. Let g is given by (1.4) be in the class M1Σ′
(
λ,µ,

1+(1−2β)z
1−z

)
≡

MΣ′ (λ,µ,β). Then

|b0| ≤
2(1 −β)
|µ−λ|

and

|b1| ≤

∣∣∣∣∣ 2(1 −β)
√

(µ− 1)2(1 −β)2
(µ−λ)4

+
1

(µ− 2λ)2

∣∣∣∣∣
where 0 ≤ λ ≤ 1,µ ≥ 0,µ > λ and z,w ∈ ∆∗.

Remark 3.1. We obtain the estimates |b0| and |b1| as obtained in the Corollaries
3.2 to 3.4 for function g given by (1.4) are in the subclasses defined in Examples
1.1 to 1.3.



76 JANANI AND MURUGUSUNDARAMOORTHY

Concluding Remarks: Let a function g ∈ Σ′ given by (1.4). By taking γ =
(1 − α)cosβ e−iβ, |β| < π

2
, 0 ≤ α < 1 the class MγΣ′ (λ,µ,φ) ≡ M

β
Σ′ (α,λ,µ,φ)

called the generalized class of β− bi spiral-like functions of order α(0 ≤ α < 1)
satisfying the following conditions.

eiβ

[
(1 −λ)

(
g(z)

z

)µ
+ λg′(z)

(
g(z)

z

)µ−1]
≺ [φ(z)(1 −α) + α]cos β + isinβ

and

eiβ

[
(1 −λ)

(
h(w)

w

)µ
+ λh′(w)

(
h(w)

w

)µ−1]
≺ [φ(w)(1 −α) + α]cos β + isinβ

where 0 ≤ λ ≤ 1,µ ≥ 0 and z,w ∈ ∆∗ and the function h is given by (1.5).
For function g ∈MβΣ′ (α,λ,µ,φ) given by (1.4),by choosing φ(z) = (

1+z
1−z ), (or φ(z) =

1+Az
1+Bz

,−1 ≤ B < A ≤ 1), we obtain the estimates |b0| and |b1| by routine procedure
(as in Theorem2.1) and so we omit the details.

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School of Advanced Sciences, VIT University, Vellore - 632 014, India

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