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 Σ0 of complex orderγ∈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 byAthe class of analytic functions of the form
(1.1) f(z) =z+
∞ X
n=2
anzn
which are univalent in the open unit disc
∆ ={z:|z|<1}.
Also denote bySthe class of all functions inAwhich are univalent and normalized by the conditions
f(0) = 0 =f0(0)−1
in ∆.Some of the important and well-investigated subclasses of the univalent func-tion classS includes the class S∗(α)(0≤α <1) of starlike functions of orderαin
∆ and the classK(α)(0≤α <1) of convex functions of orderα
<
z f0(z)
f(z)
> α or <
1 + z f
00(z)
f0(z)
> α,(z∈∆)
respectively.Further a functionf(z)∈ Ais said to be in the classS(γ) of univalent function of complex orderγ(γ∈C\ {0}) if and only if
f(z)
z 6= 0 and <
1 + 1 γ
zf0(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
orderα(0≤α <1).
An analytic functionϕis subordinate to an analytic functionψ,written by
ϕ(z)≺ψ(z),
2000Mathematics 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.
provided there is an analytic functionω 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 f0(z)
f(z) or 1 +
z f00(z) f0(z)
is subordinate to a more general superordinate function. For this purpose, they considered an analytic functionφwith positive real part in the unit disk ∆, φ(0) = 1, φ0(0)>0 andφmaps ∆ 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 ∈ Asatisfying the subordination
z f0(z)
f(z) ≺φ(z).
Similarly, the class of Ma-Minda convex functions consists of functions f ∈ A satisfying the subordination
1 + z f
00(z)
f0(z) ≺φ(z).
It is well known that every functionf ∈ S has an inversef−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)w4+· · ·.
A functionf ∈ Agiven by (1.1), is said to be bi-univalent in ∆ if bothf(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 SΣ∗(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∗
φ(0) = 1, φ0(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+B2z2+B3z3+· · ·, (B1>0).
Let Σ0 denote the class of meromorphic univalent functionsg of the form
(1.4) g(z) =z+b0+
∞ X
n=1
bn
zn
defined on the domain ∆∗={z: 1<|z|<∞}.Sinceg∈Σ0 is univalent, it has an inverseg−1=hthat 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+
∞ X
n=0
Cn
wn, (M <|w|<∞).
Analogous to the bi-univalent analytic functions, a function g ∈ Σ0 is said to be meromorphic bi-univalent ifg−1 ∈Σ0. We denote the class of all meromorphic
bi-univalent functions by MΣ0. Estimates on the coefficients of meromorphic
uni-valent functions were widely investigated in the literature, for example, Schiffer[13] obtained the estimate |b2| ≤ 2
3 for meromorphic univalent functions g ∈ Σ
0 with
b0= 0 and Duren [3] gave an elementary proof of the inequality|bn| ≤ (n+1)2 on the
coefficient of meromorphic univalent functionsg ∈Σ0 with bk = 0 for 1≤k < n2.
For the coefficient of the inverse of meromorphic univalent functions h ∈ MΣ0,
Springer [14] proved that |C3| ≤ 1 and |C3 + 12C12| ≤ 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 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 orderαin ∆∗.
Motivated by the earlier work of [4, 5, 6, 20], in the present investigation, a new subclass of meromorphic bi-univalent functions class Σ0of complex orderγ∈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 > λa function g(z)∈Σ0 given by (1.4) is said to be in the class MγΣ0(λ, µ, φ)if the following conditions are satisfied:
(1.6) 1 + 1
γ
"
(1−λ)
g(z)
z
µ
+λg0(z)
g(z)
z
µ−1
−1
#
≺φ(z)
and
(1.7) 1 + 1
γ
"
(1−λ)
h(w) w
µ
+λh0(w)
h(w) w
µ−1
−1
#
≺φ(w)
wherez, w∈∆∗, γ∈
By suitably specializing the parameters λ and µ, we state the new subclasses of the class meromorphic bi-univalent functions of complex order MγΣ0(λ, µ, φ) as
illustrated in the following Examples.
Example 1.1. For 0 ≤ λ < 1, µ = 1 a function g ∈ Σ0 given by (1.4) is said to be in the class MγΣ0(λ,1, φ) ≡ F
γ
Σ0(λ, φ) if it satisfies the following conditions
respectively:
1 + 1 γ
(1−λ)
g(z)
z
+λg0(z)−1
≺φ(z)
and
1 + 1 γ
(1−λ)
h(w)
w
+λh0(w)−1
≺φ(w)
wherez, w∈∆∗, γ∈C\{0} and the function his given by (1.5).
Example 1.2. For λ = 1,0 ≤ µ < 1 a function g ∈ Σ0 given by (1.4) is said to be in the class MγΣ0(1, µ, φ) ≡ B
γ
Σ0(µ, φ) if it satisfies the following conditions
respectively:
1 + 1 γ
"
g0(z)
g(z)
z
µ−1
−1
#
≺φ(z)
and
1 + 1 γ
"
h0(w)
h(w)
w
µ−1
−1
#
≺φ(w)
wherez, w∈∆∗, γ∈C\{0} and the function his given by (1.5).
Example 1.3. Forλ= 1, µ= 0,a function g∈Σ0 given by (1.4) is said to be in the class MγΣ0(1,0, φ)≡ S
γ
Σ0(φ)if it satisfies the following conditions respectively:
1 + 1 γ
zg0(z)
g(z) −1
≺φ(z)
and
1 + 1 γ
wh0(w)
h(w) −1
≺φ(w)
wherez, w∈∆∗, γ∈C\{0} and the function his given by (1.5).
2. Coefficient estimates for the function classMγΣ0(λ, µ, φ)
In this section we obtain the coefficients|b0|and|b1|forg∈ MγΣ0(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 eachk,
where
Define the functionspandqin 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 p 1 z +
p2− p2
1 2
1
z2 +· · ·
and
v(z) = q(z)−1 q(z) + 1=
1 2 q 1 z +
q2− q21
2
1
z2+· · ·
.
Note that for the functionsp(z), q(z)∈ P,we have
|pi| ≤2 and |qi| ≤2 for eachi.
Theorem 2.1. Let g is given by (1.4) be in the classMγΣ0(λ, µ, φ). Then
(2.1) |b0| ≤
γB1 µ−λ and
(2.2) |b1| ≤
γ s (µ
−1)γB2 1 2(µ−λ)2
2
+
B
2 µ−2λ
2
whereγ∈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
+λg0(z)
g(z)
z
µ−1
−1
#
=φ(u(z))
and
(2.4) 1 + 1
γ
"
(1−λ)
h(w)
w
µ
+λh0(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
+λg0(z)
g(z)
z
µ−1
−1
#
= 1 +1 γ
(µ−λ)b0
z + (µ−2λ)[ (µ−1)
2 b
2 0+b1]
1 z2 +...
= 1 +B1p1 1 2z +
1
2B1(p2− p2
1 2 ) +
1 4B2p
2 1
1
and
(2.6) 1 + 1 γ
"
(1−λ)
h(w)
w
µ
+λh0(w)
h(w)
w
µ−1
−1
#
= 1 + 1 γ
−(µ−λ)b0
z + (µ−2λ)[ (µ−1)
2 b
2 0−b1]
1 z2 +...
= 1 +B1q1 1 2w+
1
2B1(q2− q2
1 2 ) +
1 4B2q
2 1
1
w2 +...
Now, equating the coefficients in (2.5) and (2.6), we get
(2.7) 1
γ(µ−λ)b0= 1 2B1p1,
(2.8) 1
γ(µ−2λ)
(µ−1)b 2 0 2 +b1
=1
2B1(p2− p2
1 2 ) +
1 4B2p
2 1,
(2.9) −1
γ(µ−λ)b0= 1 2B1q1,
and
(2.10) 1
γ(µ−2λ)
(µ−1)b 2 0 2 −b1
=1
2B1(q2− q2
1 2) +
1 4B2q
2 1.
From (2.7) and (2.9), we get
(2.11) p1=−q1
and
8(µ−λ)2b20=γ2B12(p21+q12).
Hence,
(2.12) b20=γ
2B2
1(p21+q12) 8(µ−λ)2 .
Applying Lemma (2.1) for the coefficientsp1andq1, 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
(2.13) (µ−2λ)2 b21= (µ−2λ)2(µ−1)2 b4
0 4
−γ2
B2 1
4 p2q2+ (B2−B1)B1(p2+q2) p2
1
8 + (B1−B2) 2 p41
16
.
Using (2.12) and applying Lemma (2.1) once again for the coefficientsp1, p2and q2, we get
|b1| ≤
γ s
(µ−1)γB2 1 2(µ−λ)2
2
+
B
2 µ−2λ
Corollary 2.1. Let g(z)is given by (1.4) be in the classFΣγ0(λ, φ). Then
(2.14) |b0| ≤
γB1 1−λ
and
(2.15) |b1| ≤
γB2 2λ−1
whereγ∈C\{0},0≤λ <1 and z, w∈∆∗.
Corollary 2.2. Let g(z)is given by (1.4) be in the classBγΣ0(µ, φ).Then
(2.16) |b0| ≤
γB1 µ−1
and
(2.17) |b1| ≤
γ
s
γB2 1 2(µ−1)
2
+
B
2 µ−2
2
whereγ∈C\{0},0≤µ <1 and z, w∈∆∗.
Corollary 2.3. Let g(z)is given by (1.4) be in the classSΣγ0(φ). Then
(2.18) |b0| ≤ |γ B1|
and
(2.19) |b1| ≤
γ 2
q
γ2B4 1+B22
whereγ∈C\{0} and z, w∈∆∗.
3. Corollaries and concluding Remarks
Analogous to (1.3), by settingφ(z) as given below:
(3.1) φ(z) =
1 +z
1−z
α
= 1 + 2αz+ 2α2z2+· · · (0< α≤1),
we have
B1= 2α, B2= 2α2.
Forγ= 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 M1 Σ0(λ, µ,
1+z
1−z
α
) ≡
MΣ0(λ, α).Then
|b0| ≤ 2α |µ−λ| and
|b1| ≤
2α2
s
(µ−1)2 (µ−λ)4 +
1 (µ−2λ)2
Corollary 3.2. Letg(z)is given by (1.4) be in the classF1 Σ0(λ,
1+z
1−z
α
)≡ FΣ0(λ, α),
then
|b0| ≤ 2α |1−λ| and
|b1| ≤ 2α 2
|1−2λ| where0≤λ <1 and z, w∈∆∗.
Corollary 3.3. Letg(z)is given by (1.4) be in the classB1 Σ0(λ,
1+z
1−z
α
)≡ BΣ0(µ, α),
then
|b0| ≤ 2α |µ−1| and
|b1| ≤
2α2
s
1 (µ−1)2 +
1 (µ−2)2
where0≤µ <1and z, w∈∆∗.
Corollary 3.4. Let g(z)is given by (1.4) be in the class S1 Σ0
h
1+z
1−z
iα
≡ SΣ0(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)z
2+· · · (0≤β <1),
then
B1=B2= 2(1−β).
Forγ= 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 M1 Σ0
λ, µ,1+(11−−z2β)z ≡ MΣ0(λ, µ, β). Then
|b0| ≤ 2(1−β) |µ−λ| and
|b1| ≤
2(1−β)
s
(µ−1)2(1−β)2 (µ−λ)4 +
1 (µ−2λ)2
where0≤λ≤1, µ≥0, µ > λand z, w∈∆∗.
Concluding Remarks: Let a functiong ∈Σ0 given by (1.4). By taking γ= (1−α)cosβ e−iβ, |β| < π
2, 0 ≤ α < 1 the class M
γ
Σ0(λ, µ, φ) ≡ M β
Σ0(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
+λg0(z)
g(z)
z
µ−1#
≺[φ(z)(1−α) +α]cos β+isinβ
and
eiβ
"
(1−λ)
h(w)
w
µ
+λh0(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 functionhis given by (1.5).
For functiong∈ MβΣ0(α, λ, µ, φ) given by (1.4),by choosingφ(z) = (1+1−zz),(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
∗
