https://doi.org/10.26637/MJM0704/0011
Application of quasi-subordination for certain
subclasses of bi-univalent functions of complex
order
S
¸ ahsene Altınkaya
1Abstract
In this present paper, the author construct a new classSkλ,δ,α(γ,t,Ψ)of bi-univalent functions of complex order defined in the open unit disc. The second and the third coefficients of the Taylor-Maclaurin series for functions in the new subclass are determined. Several special consequences of the results are also pointed out.
Keywords
Bi-univalent functions, coefficient bounds, subordination, quasi-subordination.
AMS Subject Classification 30C45.
1Department of Mathematics, Bursa Uludag University 16059-Bursa, Turkey.
*Corresponding author:1[email protected]
Article History: Received24July2019; Accepted09September2019 c2019 MJM.
Contents
1 Introduction and Preliminaries. . . 681 2 Main Result and its Consequences. . . 683
References. . . 685
1. Introduction and Preliminaries
LetAindicate an analytic function family, which is nor-malized under the condition of f(0) = f0(0)−1=0 in∆=
{z:z∈Cand|z|<1} and given by the following
Taylor-Maclaurin series:
f(z) =z+ ∞
∑
n=2
anzn. (1.1)
Further, bySwe shall denote the class of all functions inA
which are univalent in∆. With a view to recalling the principle of subordination between analytic functions, let the functions
f andg be analytic in∆. Then we say that the function f
is subordinate togif there exists a Schwarz functionw(z), analytic in∆with
w(0) =0, |w(z)|<1 (z∈∆),
such that
f(z) =g(w(z)) (z∈∆).
We denote this subordination by
f ≺g or f(z)≺g(z) (z∈∆).
In particular, if the functiong is univalent in∆, the above subordination is equivalent to
f(0) =g(0), f(∆)⊂g(∆).
In the year 1970, Robertson [19] introduced the concept of quasi-subordination. For two analytic functions f andg, the functionf is said to be quasi-subordinate togin∆and written as
f(z)≺qg(z) (z∈∆),
if there exists an analytic function|h(z)| ≤1 such that hf((zz)) analytic in∆and
f(z)
that is, there exists a Schwarz functionw(z)such that f(z) = h(z)g(w(z)). Observe that ifh(z) =1, then f(z) =g(w(z))
so that f(z)≺g(z)in∆. Also notice that ifw(z) =z, then
f(z) =h(z)g(z) and it is said that is majorized by g and written f(z)g(z) in ∆. Hence it is obvious that
quasi-subordination is a generalization of quasi-subordination as well as majorization. (see, e.g. [19], [18], [14] for works related to quasi-subordination).
The Koebe-One Quarter Theorem [9] ensures that the
image of∆under every univalent function f ∈Acontains a
disc of radius 1/4. Thus every univalent function f has an inverse f−1satisfying f−1(f(z)) =zand f f−1(w)
=w
|w|<r0(f),r0(f)≥14,where
g(w) =f−1(w) =w−a2w2
+ 2a22−a3
w3− 5a32−5a2a3+a4
w4+· · ·. (1.2)
A function f ∈A is said to be bi-univalent in∆if both f
and f−1 are univalent in ∆. Let Σ denote the class of
bi-univalent functions in ∆ given by (1.1). For a brief
his-tory and interesting examples in the class Σ,see [25] (see also [5], [6], [13], [16]). Furthermore, judging by the re-markable flood of papers on the subject (see, for example, [11], [23] and [24]). Not much is known about the bounds on the general coefficient |an|.In the literature, there are
only a few works determining the general coefficient bounds
|an|for the analytic bi-univalent functions ([4], [10], [21],
[26]). The coefficient estimate problem for each of |an|
(n∈N\ {1,2}; N={1,2,3, ...})is still an open problem. The study of operators plays an important role in the Geo-metric Function Theory and its related fields. It is observed that this formalism brings an ease in further mathematical exploration and also helps to understand the geometric prop-erties of such operators better (see, for example [2], [3], [7], [12] and [15]). Recently, Darus and ˙Ibrahim [8] introduced a differential operator
Dk,α
λ,δ :A→A
by
Dk,α
λ,δf(z) =z+ ∞
∑
n=2
[nα+ (n−1)nα
λ]k
n+δ−1
δ
anzn
wherez∈∆andk,α∈N0=N∪ {0},λ,δ ≥0.
It should be remarked that the operatorDkλ,δ,α is a gener-alization of many other linear operators studied by earlier researchers. Namely:
• forα=1,λ=0;δ=0 orα= δ=0;λ=1, the oper-atorDk0,0,1≡Dk1,0,0≡Dkis the popular Salagean operator [22],
• forα=0, δ =0, the operatorDkλ,0,0 ≡Dkλ has been studied by Al-Oboudi (see [1]),
• forα=0, the operatorDkλ,0,δ ≡Dkλ,δ has been studied
by Darus and ˙Ibrahim (see [8]),
• fork=0, the operatorDkλ,δ,α ≡Dδ has been studied by
Ruscheweyh (see [20]).
Making use of the differential operatorDk,α
λ,δ, we introduce
a new class of analytic bi-univalent functions as follows:
Definition 1.1. A function f ∈Σ given by (1.1) is said to be in the class Skλ,δ,α(γ,t,Ψ), if the following conditions are
satisfied:
1
γ
z
Dkλ,δ,αf(z)
0
(1−t)Dkλ,δ,αf(z) +tzDkλ,δ,αf(z)
0−1
≺q(Ψ(z)−1)
and
1
γ
wDkλ,δ,αg(w)
0
(1−t)Dkλ,δ,αg(w) +twDkλ,δ,αg(w)
0−1
≺q(Ψ(w)−1),
where γ∈C\{0}, 0≤t<1;k,α∈N0=N∪ {0}, λ,δ ≥
0,z,w∈∆and the function g is given by (1.2).
On specializing the parameterst,k,δ one can define the
various new subclasses ofΣas illustrated in the following examples.
Example 1.2. For t=0andγ ∈C\{0},a function f ∈Σ
given by (1.1) is said to be in the class Skλ,δ,α(γ,Ψ), if the following conditions are satisfied:
1
γ
zDkλ,δ,αf(z)0
Dkλ,δ,αf(z) −1
≺q(Ψ(z)−1)
and
1
γ
wDkλ,δ,αg(w)
0
Dkλ,δ,αg(w) −1
≺q(Ψ(w)−1),
where k,α∈N0=N∪ {0},λ,δ ≥0,z,w∈∆and the
func-tion g is given by (1.2).
Example 1.3. For t=k=δ=0andγ∈C\{0},a function
f ∈Σgiven by (1.1) is said to be in the class SΣ(γ,Ψ), if the following conditions are satisfied:
1
γ
z f0(z)
f(z) −1
≺q(Ψ(z)−1)
and
1
γ
wg0(w)
g(w) −1
≺q(Ψ(w)−1),
2. Main Result and its Consequences
Firstly, we will state the Lemma 2.1 to obtain our result.
Lemma 2.1. (See [17]) If p∈P, then|pi| ≤1for each i,
where P is the family all functions p, analytic in∆, for which
ℜ{p(z)}>0,
where
p(z) =1+p1z+p2z2+· · ·.
Through out this paper it is assumed thatΨis analytic in
∆withΨ(0) =1 and let
Ψ(z) =1+C1z+C2z2+· · · (C1>0). (2.1)
Also let
h(z) =D0+D1z+D2z2+· · · (|h(z)| ≤1,z∈∆). (2.2)
We begin this section by finding the estimates on the coefficients|a2|and|a3|for functions in the classSλk,α,δ(γ,t,Ψ)
proposed by Definition 1.1.
Theorem 2.2. Let f of the form (1.1) be in the class Skλ,α,δ(γ,t,Ψ).
Then
|a2|
≤ |γ| |D0|C1pC1[(1−t)(1+δ)]−12
(×)
(1+δ) [2α(1+λ)]2k
(1−t)(C1−C2)−(1+t)γC21D0
+ [3α(1+2λ)]k
(2+δ)γC2 1D0
−1 2 and
|a3| ≤
|γD0|2C21
(1−t)2[2α(1+λ)]2k(1+δ)2+
|γD1|C1
(1−t)[3α(1+2λ)]k(1+δ)(2+δ)
+ |γD0|C1
(1−t)[3α(1+2λ)]k(1+δ)(2+δ).
Proof. If f ∈Skλ,δ,α(γ,t,Ψ)then, there are analytic functions
u,v:∆→∆withu(0) =v(0) =0,|u(z)|<1,|v(w)|<1 and a functionhgiven by (2.2), such that
1
γ
"
zDk,α λ,δf(z)
0
(1−t)Dk,α λ,δf(z)+tz
Dk,α λ,δf(z)
0−1
#
=h(z) (Ψ(w(z))−1)
(2.3)
and
1
γ
"
wDk,α λ,δg(w)
0
(1−t)Dk,α λ,δg(w)+tw
Dk,α λ,δg(w)
0−1
#
=h(z) (Ψ(w(z))−1).
(2.4)
Determine the functionsp1andp2inPgiven by
p1(z) =
1+u(z)
1−u(z)=1+p1z+p2z 2+· · ·
and
p2(w) =1+v(w)
1−v(w)=1+q1w+q2w 2+· · ·.
Thus,
u(z) = p1(z)−1 p1(z) +1=
1 2
p1z+
p2−
p21
2
z2+· · ·
(2.5)
and
v(w) = p2(w)−1 p2(w) +1=
1 2
q1w+
q2−
q21
2
w2+· · ·
. (2.6)
The fact thatp1andp2are analytic in∆withp1(0) =p2(0) =
1. Sinceu,v:∆→∆, the functionsp1,p2have a positive real
part in∆, and the relations|pi| ≤2 and|qi| ≤2 are true. Using
(2.5) and (2.6) together with (2.1) and (2.2) in the right hands of the relations (2.3) and (2.4), we obtain
h(z) [Ψ(u(z))−1] =12D0C1p1z
+n12D1C1p1+12D0C1
p2−
p2 1
2
+14D0C2p21 o
z2+· · ·
and
h(w) [Ψ(v(w))−1] =12D0C1q1w
+n12D1C1q1+12D0C1
q2−q
2 1
2
+14D0C2q21 o
w2+· · ·.
In the light of (2.3) and (2.4), we get
(1−t)[2α(1+λ)]k(1+ δ)
γ a2=
D0C1p1
2 , (2.7)
(1−t)[3α(1+2λ)]k(1+
δ)(2+δ)a3−(1−t)(1+t)[2α(1+λ)]2k(1+δ)2a2 2
γ
=D1C1p1
2 +
D0C1
2
p2−
p21
2
+D0C2p21
4
(2.8)
and
−(1−t)[2α(1+λ)]k(1+δ)
γ a2=
D0C1q1
2 , (2.9)
(1−t)[3α(1+2λ)]k(1+
δ)(2+δ)(2a22−a3)−(1−t)(1+t)[2α(1+λ)]2k(1+δ)2a2 2
γ
=D1C1q1
2 +
D0C1
2
q2−
q2 1
2
+D0C2q21
(2.10)
Now, (2.7) and (2.9) give
p1=−q1 (2.11)
and
8(1−t)2[2α(1+
λ)]2k(1+δ)2a22=γ2D20C12 p21+q21.
(2.12)
Adding (2.8) and (2.10), we get
2(1−t)[3α(1+2λ)]k(1+
δ)(2+δ)−2(1−t)(1+t)[2α(1+λ)]2k(1+ δ)2
γ a
2 2
=D0C1(p2+q2)
2 +
D0(C2−C1)(p21+q21)
4 .
(2.13)
By using (2.11), (2.12) and Lemma 2.1 in (2.13), we obtain the desired result.
Next, to find the bound on|a3|, by using subtracting (2.10) and (2.8), we have
2(1−t)[3α(1+2λ)]k(1+δ)(2+δ)
γ a3−a
2 2
=D0C1(p2−q2)
2 +
D1C1(p1−q1)
2 .
(2.14)
It follows from (2.11), (2.12) and (2.14) that
a3=
γ2D2
0C21(p21+q21)
8(1−t)2[2α(1+λ)]2k(1+δ)2+
γD1C1(p1−q1)
4(1−t)[3α(1+2λ)]k(1+δ)(2+δ)
+ γD0C1(p2−q2)
4(1−t)[3α(1+2λ)]k(1+δ)(2+δ).
Applying Lemma 2.1 once again for the coefficientsp2and
q2, we readily get
|a3| ≤
|γD0|2C12
(1−t)2[2α(1+λ)]2k(1+δ)2+
|γD1|C1
(1−t)[3α(1+2λ)]k(1+δ)(2+δ)
+ |γD0|C1
(1−t)[3α(1+2λ)]k(1+δ)(2+δ).
This completes the proof of Theorem 2.1.
Puttingt=0 in Theorem 2.1, we have the following corol-lary.
Corollary 2.3. Let f of the form (1.1) be in the class Skλ,δ,α(γ,Ψ).
|a2| ≤
|γ||D0|C1√C1 r
(1+δ)
(1+δ)[2α(1+λ)]
2k[(C1−C2)−γC21D0]+[3α(1+2λ)]k(2+
δ)γC21D0
and
|a3| ≤
|γD0|2C21
[2α(1+λ)]2k(1+δ)2+
|γD1|C1
[3α(1+2λ)]k(1+δ)(2+δ)
+ |γD0|C1
[3α(1+2λ)]k(1+δ)(2+δ).
Corollary 2.4. Let f of the form (1.1) be in the class SΣ(γ,Ψ).
Then
|a2| ≤
|γD0|C1 √
C1 q
C1−C2+γC12D0
and
|a3| ≤ |γD0|2C12+
(|D1|+|D0|)|γ|C1
2 .
For the functionΨis given by
Ψ(z) =
1+
z
1−z ξ
=1+2ξz+2ξ2z2+· · ·(0<ξ ≤1)
which gives
C1=2ξ, C2=2ξ2,
Theorem 2.1 reduces to:
Corollary 2.5. Let f ∈Skλ,δ,αhγ,t, 11+−zzξi .Then
|a2| ≤2|γ| |D0|ξ[(1−t)(1+δ)]−12
(1+δ)[2α(1+λ)]2k[(1−t)(1−ξ)
−2(1+t)γ ξD0] +2[3α(1+2λ)]k(2+δ)γ ξ D0
−12
and
|a3| ≤ 4
|γD0|2ξ2
(1−t)2[2α(1+λ)]2k(1+δ)2+
2|γD1|ξ
(1−t)[3α(1+2λ)]k(1+δ)(2+δ)
+ 2|γD0|ξ
(1−t)[3α(1+2λ)]k(1+δ)(2+δ).
If we set
Ψ(z) =1+Az
1+Bz=1+(A−B)z−B(A−B)z
2+· · ·(−1≤B≤A<1)
which gives
C1= (A−B), C2=−B(A−B),
Theorem 2.1 reduces to:
Corollary 2.6. Let f ∈Skλ,δ,αγ,t,11++AzBz .Then
|a2| ≤ |γ| |D0|(A−B) [(1−t)(1+δ)]−12
(×)
(1+δ) [2α(1+λ)]2k[(1−t)(1+B)−(1+t)γ(A−B)D 0]
+ [3α(1+2λ)]k
(2+δ)γ(A−B)D0
−1 2
and
|a3| ≤ |γD0|
2(A−B)2
(1−t)2[2α(1+λ)]2k(1+δ)2+
|γD1|(A−B)
(1−t)[3α(1+2λ)]k(1+δ)(2+δ)
+ |γD0|(A−B)
Finally, if we set
Ψ(z) =1+ (1−2β)z
1−z =1+2(1−β)z+2(1−β)z
2+· · · (0<
ξ≤1)
which gives
C1=C2=2(1−β),
Theorem 2.1 reduces to:
Corollary 2.7. Let f∈Sk,α λ,δ h
γ,t,1+(11−−z2β)zi.Then
|a2| ≤ |γ| |D0|(A−B) [(1−t)(1+δ)]− 1 2
× [3
α(1+2λ)]k(2+δ)γD
0−(1+δ) [2α(1+λ)]2k(1+t)γD0
−12
and
|a3| ≤ 4
|γD0|2(1−β)2
(1−t)2[2α(1+λ)]2k(1+δ)2+
2|γD1|(1−β) (1−t)[3α(1+2λ)]k(1+δ)(2+δ)
+ 2|γD0|(1−β) (1−t)[3α(1+2λ)]k(1+δ)(2+δ).
Acknowledgment
This research is supported by the Scientific and Techno-logical Research Council of Turkey (TUBITAK 2214A) under grant number:1059B141600188.
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