COEFFICIENT ESTIMATES OF MEROMORPHICBI- STARLIKE FUNCTIONS OF COMPLEX ORDER

T. Janani and G. Murugusundaramoorthy∗

∗ Corresponding Author,School of Advanced Sciences,

VIT University, Vellore - 632 014, India.Email : janani.t2013@vit.ac.in;gmsmoorthy@yahoo.com

Abstract. In the present investigation, we define a new subclass of meromorphic bi-univalentfunctions 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.

2010 Mathematics Subject Classification: 30C45 , 30C50.

Keywords: Analytic functions; Univalent functions; Meromorphic functions; Bi-univalentfunctions; Bi-starlike and bi-convex functions of complex order; Coefficient bounds.

1. Introduction and Definitions

Denote by A the class of analytic functions of the form

f(z) = z +∞∑n=2

anzn (1.1)

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 theconditions

f(0) = 0 = f ′(0)− 1

in ∆. Some of the important and well-investigated subclasses of the univalent function classS includes the class S∗(α)(0 ≤ α < 1) of starlike functions of order α in ∆ and the classK(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 ofcomplex order γ(γ ∈ C \ {0}) if and only if

f(z)

z6= 0 and <

(1 +

1

γ

[zf ′(z)

f(z)− 1

])> 0, z ∈ ∆.

By taking γ = (1 − α)cosβ e−iβ, |β| < π2

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),

1

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS... 2

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 eitherof 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 ananalytic 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 subordi-nation

z f ′(z)

f(z)≺ φ(z).

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

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

f−1(w) = w − a2w2 + (2a2

2 − a3)w3 − (5a32 − 5a2a3 + a4)w4 + · · · . (1.2)

A function f ∈ A given by (1.1), is said to be bi-univalent in ∆ if both f(z) and f−1(z) areunivalent in ∆, these classes are denoted by Σ. Earlier, Brannan and Taha [2] introduced certainsubclasses of bi-univalent function class Σ, namely bi-starlike functions S∗Σ(α) and bi-convexfunction 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 ofthe 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 subclasses 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 andf−1 are respectively Ma-Minda starlike or convex. These classes are denoted respectively byS∗Σ(φ) and KΣ(φ).In the sequel, it is assumed that φ is an analytic function with positive realpart in the unit disk ∆, satisfying φ(0) = 1, φ′(0) > 0 and φ(∆) is symmetric with respect tothe real axis. Such a function has a series expansion of the form

φ(z) = 1 +B1z +B2z2 +B3z

3 + · · · , (B1 > 0). (1.3)

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS... 3

Let Σ′ denote the class of meromorphic univalent functions g of the form

g(z) = z + b0 +∞∑n=1

bnzn

(1.4)

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

g−1(g(z)) = z, (z ∈ ∆∗)

and

g(g−1(w)) = w, (M < |w| <∞,M > 0)

where

g−1(w) = h(w) = w +∞∑n=0

Cnwn

, (M < |w| <∞). (1.5)

Analogous to the bi-univalent analytic functions, a function g ∈ Σ′ is said to be meromorphicbi-univalent if g−1 ∈ Σ′. We denote the class of all meromorphic bi-univalent functions byMΣ′ .Estimates on the coefficients of meromorphic univalent functions were widely investigated in theliterature, for example, Schiffer[13] obtained the estimate |b2| ≤ 2

3for 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 < n

2.

For the coefficient of the inverse of meromorphic univalent functions h ∈ MΣ′ , Springer [14]

proved that |C3| ≤ 1 and |C3 + 12C2

1 | ≤ 12

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 andsubsequently Schober [12] obtained a sharp bounds for the coefficients C2n−1, 1 ≤ n ≤ 7 ofthe 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 starlikeunivalent functions of order α in ∆∗.

Motivated by the earlier work of [4, 5, 6, 20], in the present investigation, a new subclassof meromorphic bi-univalent functions class Σ′ of complex order γ ∈ C\{0}, is introduced andestimates for the coefficients |b0| and |b1| of functions in the newly introduced subclass areobtained. 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 bein the class Mγ

Σ′(λ, µ, φ) if the following conditions are satisfied:

1 +1

γ

[(1− λ)

(g(z)

z

)µ+ λg′(z)

(g(z)

z

)µ−1

− 1

]≺ φ(z) (1.6)

and

1 +1

γ

[(1− λ)

(h(w)

w

)µ+ λh′(w)

(h(w)

w

)µ−1

− 1

]≺ φ(w) (1.7)

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

By suitably specializing the parameters λ and µ, we state the new subclasses of the classmeromorphic bi-univalent functions of complex orderMγ

Σ′(λ, µ, φ) as illustrated in the followingExamples.

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS... 4

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

Σ′(λ, 1, φ) ≡ FγΣ′(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 classMγ

Σ′(1, µ, φ) ≡ BγΣ′(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 classMγ

Σ′(1, 0, φ) ≡ SγΣ′(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 ∈ MγΣ′(λ, µ, φ) associating the

given functions with the functions having positive real part. In order to prove our result werecall 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 + c2z2 + · · · for z ∈ ∆.

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+ · · · .

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS... 5

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

|b0| ≤∣∣∣∣ γB1

µ− λ

∣∣∣∣ (2.1)

and

|b1| ≤

∣∣∣∣∣∣ γ√(

(µ− 1)γB21

2(µ− λ)2

)2

+

(B2

µ− 2λ

)2

∣∣∣∣∣∣ (2.2)

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

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

1 +1

γ

[(1− λ)

(g(z)

z

)µ+ λg′(z)

(g(z)

z

)µ−1

− 1

]= φ(u(z)) (2.3)

and

1 +1

γ

[(1− λ)

(h(w)

w

)µ+ λh′(w)

(h(w)

w

)µ−1

− 1

]= φ(v(w)). (2.4)

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

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)

2b2

0 + b1]1

z2+ ...

]= 1 +B1p1

1

2z+

[1

2B1(p2 −

p21

2) +

1

4B2p

21

]1

z2+ ... (2.5)

and

1 +1

γ

[(1− λ)

(h(w)

w

)µ+ λh′(w)

(h(w)

w

)µ−1

− 1

]

= 1 +1

γ

[−(µ− λ)

b0

w+ (µ− 2λ)[

(µ− 1)

2b2

0 − b1]1

w2+ ...

]= 1 +B1q1

1

2w+

[1

2B1(q2 −

q21

2) +

1

4B2q

21

]1

w2+ ... (2.6)

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

1

γ(µ− λ)b0 =

1

2B1p1, (2.7)

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS... 6

1

γ(µ− 2λ)

[(µ− 1)

b20

2+ b1

]=

1

2B1(p2 −

p21

2) +

1

4B2p

21, (2.8)

− 1

γ(µ− λ)b0 =

1

2B1q1, (2.9)

and1

γ(µ− 2λ)

[(µ− 1)

b20

2− b1

]=

1

2B1(q2 −

q21

2) +

1

4B2q

21. (2.10)

From (2.7) and (2.9), we getp1 = −q1 (2.11)

and8(µ− λ)2b2

0 = γ2B21(p2

1 + q21).

Hence,

b20 =

γ2B21(p2

1 + q21)

8(µ− λ)2. (2.12)

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

|b0| ≤∣∣∣∣ γB1

µ− λ

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

(µ− 2λ)2 b21 = (µ− 2λ)2(µ− 1)2 b

40

4

− γ2

(B2

1

4p2q2 + (B2 −B1)B1(p2 + q2)

p21

8+ (B1 −B2)2 p4

1

16

). (2.13)

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

∣∣∣∣∣∣ .�

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

|b0| ≤∣∣∣∣ γB1

1− λ

∣∣∣∣ (2.14)

and

|b1| ≤∣∣∣∣ γB2

2λ− 1

∣∣∣∣ (2.15)

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

|b0| ≤∣∣∣∣ γB1

µ− 1

∣∣∣∣ (2.16)

and

|b1| ≤

∣∣∣∣∣∣ γ√(

γB21

2(µ− 1)

)2

+

(B2

µ− 2

)2

∣∣∣∣∣∣ (2.17)

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

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS... 7

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

|b0| ≤ |γ B1| (2.18)

and

|b1| ≤∣∣∣∣γ2√γ2B4

1 +B22

∣∣∣∣ (2.19)

where γ ∈ C\{0} and z, w ∈ ∆∗.

3. Corollaries and concluding Remarks

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

φ(z) =

(1 + z

1− z

)α= 1 + 2αz + 2α2z2 + · · · (0 < α ≤ 1), (3.1)

we have

B1 = 2α, B2 = 2α2.

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+z1−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 ∈ ∆∗.

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

(1+z1−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+z1−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 ∈ ∆∗.

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS... 8

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

([1+z1−z

]α) ≡ SΣ′(α) then

|b0| ≤ 2α

and

|b1| ≤ α2√

5

where z, w ∈ ∆∗.

On the other hand if we take

φ(z) =1 + (1− 2β)z

1− z= 1 + 2(1− β)z + 2(1− β)z2 + · · · (0 ≤ β < 1), (3.2)

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 3.4for function g given by (1.4) are in the subclasses defined in Examples 1.1 to 1.3.

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 Mγ

Σ′(λ, µ, φ) ≡ MβΣ′(α, λ, µ, φ) called the gen-

eralized class of β− bi spiral-like functions of order α(0 ≤ α < 1) satisfying the followingconditions.

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+Az1+Bz

,−1 ≤ B < A ≤ 1), we obtain the estimates |b0| and |b1| by routine procedure (as inTheorem2.1) and so we omit the details.

COEFFICIENT ESTIMATES OF MEROMORPHIC BI- STARLIKE FUNCTIONS... 9

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