APPLYING RUSCHEWEYH DERIVATIVE ON TWO SUB-CLASSES OF BI-UNIVALENT FUNCTIONS
ABDUL RAHMAN S. JUMA1, FATEH S. AZIZ2
DEPARTMENT OF MATHEMATICS, ALANBAR UNIVERSITY, RAMADI-IRAQ1,
DEPARTMENT OF MATHEMATICS, SALAHADDIN UNIVERSITY,ERBIL-IRAQ2,
Abstract. The Ruscheweyh derivative has been applied in this paper to in-vestigate two subclasses of the function class Σ of bi-univalent functions defined in the open unit disc. We find estimates on the coefficients|a2|and|a3|for functions in these subclasses.
Keywords : Analytic and univalent functions; Bi-univalent functions;λ- convex functions; Ruscheweyh derivative; Coefficient bounds.
AMS Subject Classifications : 30C45
1. Introduction and definitions
Let Ω denote the class of all functions of the form:
(1.1) f(z) =z+
∞ X
n=2 anzn,
which are analytic in the open unit disc U ={z :|z|<1}. Let M(λ) denote the
class ofλ-convex functions inU defined as follows see [5]:
M(λ) ={f ∈Ω :Re[(1−λ)zff(0(zz))+λ(1 + zff000((zz)))]>0, λ≥0}.
Further, byS we shall denote the class of all functions in Ω which are univalent
in U ( for details, see [3],[4],[10]).
It is well known that every functionf ∈S has an inversef−1, defined by
f−1(f(z)) =z (z∈U),
and
f(f−1(w)) =w (|w|< r
o(f) :ro(f)≥ 14),
where
f−1(w) =w−a
2w2+ (2a22−a3)w3−(5a32−5a2a3+a4)w4+... .
A functionf(z)∈Ω is said to be bi-univalent inU if both f(z) andf−1(z) are
univalent inU see [10].
1E-mail:
2E-mail: [email protected]
Let Σ denote the class of bi-univalent functions in U given by (1.1). Brannan
and Taha [2] (see also [6]) introduced certain subclasses of the bi-univalent function
class Σ similar to the familiar subclasses δ∗(α) and K(α) of starlike and convex
functions of order α(0 < α ≤1), respectively (see [7]). Thus, following Brannan
and Taha [2] (see also [6]), a function f(z) ∈ Ω is in the class δ∗Σ(α) of strongly
bi-starlike functions of order α(o < α ≤ 1) if each of the following conditions is
satisfied:
f ∈Σ, |arg{zff0((zz)))|<απ2 (0< α≤1, z∈U),
and
|arg{wgg0((ww)))|<απ2 (0< α≤1, w∈U)
where g is the extension off−1to U. The classesδ∗
Σ(α) and KΣ(α), of bi-starlike
functions of orderαand bi-convex functions of orderα, corresponding (respectively)
to the function classesδ∗(α) andK(α), were also introduced analogously. For each
of the function classes δΣ∗(α) and KΣ∗(α), they found non-sharp estimates on the
first two Taylor-Maclaurin coefficients|a2|and|a3|.
The object of the present paper is to introduce two subclasses of the function
class Σ applying the Ruscheweyh derivative, where Ruscheweyh [9] observed that
(1.2) Dnf(z) = z(z
n−1f(z))(n)
n! ,
for n∈No ={0,1,2, ...}. This symbol Dnf(z), n∈No is called by Al- Amiri [1],
thenthorder Ruscheweyh derivative off(z).
We note that Dof(z) =f(z), D1f(z) =zf0(z) and
(1.3) Dnf(z) =z+
∞ X
k=2
σ(n, k)akzk,
where
(1.4) σ(n, k) =
n+k−1
n
,
and find estimates on the coefficients|a2|and|a3|for functions in these subclasses
of the function class Σ employing the techniques used by Xiao-FeiLi et al.[11].
For deriving our main results, the following lemma needed to be mentioned [8].
Lemma 1.1. If h ∈ P then |ck| ≤ 2 for each k, where P is the family of all
functionshanalytic inU for whichRe(h(z))>0, h(z) = 1 +c1z+c2z2+c3z3+...
forz∈U.
2. Coefficient bounds for the function class FΣ(α, λ)
Definition 2.1. A functionf(z) given by (1.1) is said to be in the class FΣ(α, λ)
if the following conditions are satisfied:
f ∈Σ, |arg{(1−λ)z(D nf(z))0
Dnf(z) +λ[1 +
z(Dnf(z))00
(Dnf(z))0 ]}|< απ
2
(0< α≤1, λ≥0, z∈U),
(2.1)
and
|arg{(1−λ)w(D ng(w))0
Dng(w) +λ[1 +
w(Dng(w))00 (Dng(w))0 ]}|<
απ
2
(0< α≤1, λ≥0, w∈U),
(2.2)
where the functiongis the extension off−1given by
(2.3) g(w) =w−a2w2+ (2a22−a3)w3−(5a32−5a2a3+a4)w4+....
We note that forn= 0 the classFΣ(α, λ) reduces to the classBΣ(α, λ) introduced
and studied by Xiao-FeiLi et al.[11].
Theorem 2.2. Let f(z) given by (1.1) be in the class FΣ(α, λ),0 < α ≤ 1 and
λ≥0. Then
(2.4) |a2| ≤
2α p
α(1−4nλ−λ2)−αn2(λ2+ 4λ+ 1) + (n+ 1)2(λ+ 1)2,
and
(2.5) |a3| ≤
2α
(n+ 1)(n+ 2)(1 + 2λ)+
4α2
(n+ 1)2(λ+ 1)2.
Proof:
We can write the argument inequalities in (2.1) and (2.2) equivalently as follows:
(2.6) (1−λ)z(D nf(z))0
Dnf(z) +λ[1 +
z(Dnf(z))00
(Dnf(z))0 ] = [p(z)]
α,
and
(2.7) (1−λ)w(D ng(w))0
Dng(w) +λ[1 +
w(Dng(w))00
(Dng(w))0 ] = [q(w)]
α,
wherep(z) andq(w) satisfy the following inequalities
Re(p(z))>0 (z∈U) and Re(q(w))>0 (w∈U).
Furthermore, the functionsp(z) andq(w) have the forms
(2.8) p(z) = 1 +p1z+p2z2+p3z3+...,
and
(2.9) q(w) = 1 +q1w+q2w2+q3w3+... .
And
g(w) is given as in (2.3).
Now equating the coefficients in equations (2.6) and (2.7), we get
(2.10) (n+ 1)(1 +λ)a2=p1α,
(2.11) (n+ 1)(n+ 2)(1 + 2λ)a3=p2α+
α(α−1)
2 p
2 1+
1 + 3λ
(1 +λ)2p 2 1α
2.
and
(2.12) −(n+ 1)(1 +λ)a2=q1α,
(2.13) (n+ 1)(n+ 2)(1 + 2λ)(2a22−a3) =q2α+
α(α−1)
2 q
2 1+
1 + 3λ
(1 +λ)2q 2 1α
2.
From equations (2.10) and (2.12), we get
(2.14) p1=−q1,
also we get
(2.15) 2(n+ 1)2(λ+ 1)2a22=α2(p21+q12).
From (2.11),(2.13) and (2.15) we obtain
(2.16) a22=
α2(p2+q2)
α(1−4nλ−λ2)−αn2(λ2+ 4λ+ 1) + (n+ 1)2(λ+ 1)2.
Applying lemma (1.1) for the coefficientsp2andq2,we get
(2.17) |a2| ≤
2α p
α(1−4nλ−λ2)−αn2(λ2+ 4λ+ 1) + (n+ 1)2(λ+ 1)2.
Next, in order to find the bound on|a3|, by subtracting (2.13) from (2.11), we get
2(n+ 1)(n+ 2)(1 + 2λ)a3−2(n+ 1)(n+ 2)(1 + 2λ)a22=
α(p2−q2) +
α(α−1) 2 (p
2 1−q
2 1) +
1 + 3λ
(1 +λ)2α 2(p2
1−q 2 1).
(2.18)
Upon substituting the value ofa2
2from (2.15) and observing thatp21=q21 it follows
that
(2.19) a3=
α(p2−q2)
2(n+ 1)(n+ 2)(1 + 2λ)+
α2(p2 1+q12)
2(n+ 1)2(λ+ 1)2,
Applying lemma (1.1) once again for the coefficients p1, p2, q1 and q2, we readily
get
(2.20) |a3| ≤
2α
(n+ 1)(n+ 2)(1 + 2λ)+
4α2
(n+ 1)2(λ+ 1)2,
This completes the proof of Theorem 2.2.
Puttingn= 0 in Theorem 2.2 we have
Corollary 2.3. Let f(z) given by (1.1) be in the class MΣ(α, λ),0 < α ≤1, λ ≥
0, z∈U. Then
(2.21) |a2| ≤
2α p
α(1−λ2) + (1 +λ)2,
and
(2.22) |a3| ≤
α
1 + 2λ+
4α2
(1 +λ)2.
.
3. Coefficient bounds for the function classFΣ(β, λ)
Definition 3.1. A functionf(z) given by (1.1) is said to be in the classFΣ(β, λ)
if the following conditions are satisfied:
f ∈Σ, Re{(1−λ)z(D nf(z))0
Dnf(z) +λ[1 +
z(Dnf(z))00 (Dnf(z))0 ]}> β
(0≤β <1, λ≥0, z∈U),
(3.1)
and
Re{(1−λ)w(D ng(w))0
Dng(w) +λ[1 +
w(Dng(w))00
(Dng(w))0 ]}> β
(0≤β <1, λ≥0, w∈U),
(3.2)
where the functiong(w) is given as in (2.3).
We note that forn= 0 the classFΣ(β, λ) reduces to the classBΣ(β, λ) introduced
and studied by Xiao-FeiLi et al.[11].
Theorem 3.2. Let f(z) given by (1.1) be in the class FΣ(β, λ),0 ≤ β < 1 and
λ≥0. Then
(3.3) |a2| ≤
s
2(1−β) (n+ 1)((1−n)λ+ 1),
and
(3.4) |a3| ≤
2(1−β)
(n+ 1)(n+ 2)(1 + 2λ)+
4(1−β)2
(1 +n)2(1 +λ)2.
Proof:
The argument inequalities in (3.1) and (3.2) equivalently can be written as
fol-lows:
(3.5) (1−λ)z(D nf(z))0
Dnf(z) +λ[1 +
z(Dnf(z))00
(Dnf(z))0 ] =β+ (1−β)p(z),
and
(3.6) (1−λ)w(D ng(w))0
Dng(w) +λ[1 +
w(Dng(w))00
(Dng(w))0 ] =β+ (1−β)q(w),
whereg(w), p(z), andq(w) have the forms (2.3),(2.8) and (2.9) respectively.
Equating coefficients in equations (3.5) and (3.6) yields
(3.7) (n+ 1)(1 +λ)a2=p1(1−β),
(3.8) (n+ 1)(n+ 2)(1 + 2λ)a3=p2(1−β) +
1 + 3λ
(1 +λ)2p 2 1(1−β)
2.
and
(3.9) −(n+ 1)(1 +λ)a2=q1(1−β),
(3.10) (n+ 1)(n+ 2)(1 + 2λ)(2a22−a3) =q2(1−β) +
1 + 3λ
(1 +λ)2q 2 1(1−β)
2.
From equations (3.7) and (3.9), we get
(3.11) p1=−q1,
also we get
(3.12) 2(n+ 1)2(λ+ 1)2a22= (1−β)2(p21+q21).
Now adding (3.8) to (3.10) gives
(3.13) 2(n+ 1)(n+ 2)(1 + 2λ)a22= (1−β)(p2+q2) +
1 + 3λ
(1 +λ)2(p 2
1+q12)(1−β)2,
substituting value ofp2
1+q12from (3.12) in (3.13) we get
(3.14) a22= (1−β)(p2+q2) 2(n+ 1)(1 +λ(1−n)).
Applying lemma (1.1) for the coefficientsp2andq2 we have
(3.15) |a2| ≤
s
2(1−β) (n+ 1)((1−n)λ+ 1),
Next, in order to find the bound on|a3|, by subtracting (3.10) from (3.8) we get
(3.16) 2(n+ 1)(n+ 2)(1 + 2λ)a3= (1−β)(p2−q2) + 2(n+ 1)(n+ 2)(1 + 2λ)a22,
putting value ofa2
2 from (3.12) in (3.16) we get
(3.17) a3=
(1−β)(p2−q2)
2(n+ 1)(n+ 2)(1 + 2λ)+
(1−β)2(p2 1+q21)
2(n+ 1)2(λ+ 1)2,
Applying lemma (1.1) once again for the coefficients p1, p2, q1 and q2, we readily
get
(3.18) |a3| ≤
2(1−β)
(n+ 1)(n+ 2)(1 + 2λ)+
4(1−β)2
(1 +n)2(1 +λ)2,
This completes the proof of Theorem 3.2.
Puttingn= 0 in Theorem 3.2 we have
Corollary 3.3. Let f(z) given by (1.1) be in the class MΣ(β, λ),0 ≤β <1, λ ≥
0, z∈U. Then
(3.19) |a2| ≤
r
2(1−β) 1 +λ , and
(3.20) |a3| ≤
1−β
1 + 2λ+
4(1−β)2)
(1 +λ)2 .
.
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