Kyungpook Mathematical Journal 2018; 58(4): 697-709  
Coefficient Bounds for Bi-spirallike Analytic Functions
Madan Mohan Soren*
Department of Mathematics, Berhampur University, Bhanja Bihar-760007, Ganjam, Odisha, India
e-mail : soren85@rediffmail.com

Akshaya Kumar Mishra
Director, Institute of Mathematics and Applications, Andharua-751003, Bhubaneswar, Odisha, India
e-mail : akshayam2001@yahoo.co.in
*Corresponding Author.
Received: January 10, 2018; Revised: October 13, 2018; Accepted: October 24, 2018; Published online: December 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In the present paper, we introduce and investigate two new subclasses, namely; the class of strongly αbi-spirallike functions of order β and αbi-spirallike functions of order ρ, of the function class ∑; of normalized analytic and bi-univalent functions in the open unit disk U={z:zand z<1}.

We find estimates on the coefficients |a2|, |a3| and |a4| for functions in these two subclasses.

Keywords: univalent functions, bi-univalent functions, bi-spirallike functions, Taylor-Maclaurin series, coeffcient bounds.
1. Introduction and Definitions

Let be the class of analytic functions f(z) in the open unit disk

U={z:zand z<1}

and represented by the normalized series:

f(z)=z+n=2anzn         (zU).

We denote by the family of univalent functions in . (see, for details,[4, 27]). It is well known that every function has an inverse f−1, defined by

f-1(f(z))=z         (zU)

and

f(f-1(w))=w         (w<r0(f);   r0(f)14)[ 4].

The inverse function f−1(w) is given by

f-1(w)=w-a2w2+(2a22-a3)w3-(5a23-5a2a3+a4)w4+.

The function is said to be bi-univalent in if (i) and (ii)f−1(w) has an univalent analytic continuation to |w| < 1. Let ∑ be the class of bi-univalent functions in . Initial pioneering work on the class ∑ were done in [9, 16]. Srivastava et al. [26] mentioned some interesting examples of functions in the class ∑. Recently, Mishra and Soren [14] were add two more examples which are well demonstrated there in.

Špaček [19] and Libera [11] introduced the families of α-spirallike functions (-π2<α<π2) and α-spirallike functions of orderρ(-π2<α<π2,   0ρ1) respectively. Libera [11] completely settled the coefficient estimate problem for α-spirallike functions of order ρ. In this paper we introduce the families of α-bispirallike functions of order ρ and strongly α-bi-spirallike functions of order β. We find estimates for |a2|, |a3| and |a4| for functions, of the form (1.1), in both these classes. Through out in this section also, we continue to denote by g the analytic continuation of the inverse of the function f to . We now have the following definitions:

Definition 1.1

The function f(z), given by (1.1), is said to be a member of α-SPΣβ, the class of strongly α-bi-spirallike functions of orderβ(απ2,0β<1), if each of the following conditions are satisfied:

fΣ         and         |arg(eiαzf(z)f(z))|<βπ2         (zU)

and

|arg(eiαwg(w)g(w))|<βπ2         (wU).

Definition 1.2

The function f(z), given by (1.1), is said to be a member of , the class of α-bi-spirallike functions of orderρ(απ2,0ρ<1), if each of the following conditions are satisfied:

fΣ         and         (eiαzf(z)f(z))>ρcos α         (zU)

and

(eiαwg(w)g(w))>ρcos α         (wU).

Furthermore, let ℘ be the class of analytic functions p(z) of the form:

p(z)=1+k=1ckzk         (zU)

and satisfy ℜ(p(z)) >0 ( ). We shall need this class to describe the classes α-SPΣβ and .

As follow up of the work of Mishra and Soren [14], at present there is renewed interest in the study of the class ∑ and its many new subclasses. For example see [1, 2, 3, 5, 7, 8, 10, 17, 18, 20, 21, 29, 30, 31]. Many researchers are still working upon finding an upper bound for an for the functions in subclasses of ∑. However, not much was known about the bound of the general coefficients an (n ≥ 4) of subclasses of bi-univalent functions up until the publication of the article by Mishra and Soren [14]. See [6, 13, 12, 15, 22, 23, 24, 25, 28]. For a brief history on the developments regarding the class ∑ see [26].

Motivated by the aforementioned work [14], in the present paper we have introduced two new subclasses of the function class ∑ and we find estimates for |a2|, |a3| and |a4| for functions, of the form (1.1), when fα-SPΣβ and .

2. Coefficient Bounds for the Class of Bi-spirallike Functions

We state and prove the following:

Theorem 2.1

Let the function f(z), represented by the series (1.1), be in the class α-SPΣβ(απ2,0β<1). Then

a22β(1+β)cos(α/β),a3{βcos(αβ),         0β13,4β21+βcos(αβ),         13β<1

and

a4{2β3(1-2316β2-3β-11+β3cos(α/β))cos(α/β),         0β<3+73322β3(1+2316β2-3β-11+β3cos(α/β))cos(α/β),         3+7332β<252β3(15β5β+4+2316β2-3β-11+β3cos(α/β))cos(α/β),         25β<1.
Proof

We write

f(z)=f(z)ze-iαh(z)         (zU;-βπ2<α<βπ2)

where h(z) is analytic in and satisfies

h(0)=eiαand arg h(z)<βπ2         (zU).

It can be checked that the function q(z) defined by:

h(z)1β=cos(αβ)q(z)+isin(αβ)         (zU)

is a member of the class ℘. Suppose that

q(z)=1+c1z+c2z2+         (zU.)

By comparing coefficients in (2.4), we have

a2=βc1e-i(αβ)cos(αβ),2a3-a22=βc2e-i(αβ)cos(αβ)+β(β-1)2+c12e-2i(αβ)cos2(αβ)

and

3a4-3a2a3+a23=βc3e-i(αβ)cos(αβ)+β(β-1)c1c2e-2i(αβ)cos2(αβ)+β(β-1)(β-2)6c13e-3i(αβ)cos3(αβ).

Similarly, we take

g(w)=g(w)we-iαH(w)         (wU;-βπ2<α<βπ2)

where H(w) is analytic in and satisfies

H(0)=eiαand arg H(w)<βπ2         (wU).

The function p(w) defined by

H(w)1β=cos(αβ)p(w)+isin(αβ)         (wU)

is a member of the class ℘. If p(w)=1+l1w+l2w2+         (wU),then again by comparing the coefficients in (2.8), we have the following:

-a2=βl1e-i(αβ)cos(αβ),3a22-2a3=βl2e-i(αβ)cos(αβ)+β(β-1)2l12e-2i(αβ)cos2(αβ)

and

-(10a23-12a2a3+3a4)=βl3e-i(αβ)cos(αβ)+β(β-1)l1l2e-2i(αβ)cos2(αβ)+β(β-1)(β-2)6l13e-3i(αβ)cos3(αβ).

From (2.5) and (2.9), gives

l1=-c1

We shall obtain a refined estimate on |c1| for use in the estimates of |a3| and |a4|. For this purpose we first add (2.6) with (2.10); then use the relations (2.12) and get the following:

2a22=β(c2+l2)e-i(αβ)cos(αβ)+β(β-1)2(c12+l12)e-2i(αβ)cos2(αβ).

Putting a2=βc1e-i(αβ)cos(αβ) from (2.5), we have after simplification:

c12=c2+l2(1+β)e-i(αβ)cos(αβ).

By applying the familiar inequalities |c2| ≤ 2 and |l2| ≤ 2 we get:

c14(1+β)cos(αβ)=2(1+β)cos(αβ)

and

a2βc1cos(α/β)=2β(1+β)cos(α/β).

We have thus obtained (2.1).

We next find a bound on |a3|. For this we substract (2.10) from (2.6) and get

4a3=4a22+β(c2-l2)e-i(αβ)cos(αβ)+β(β-1)2(c12-l12)e-2i(αβ)cos2(αβ).

The relation c12=l12 from (2.12), reduces the above expression to

4a3=4a22+β(c2-l2)e-i(αβ)cos(αβ).

Next putting that a2=βc1e-i(αβ)cos(αβ) and using (2.13), we obtain

4a3=4β2c12e-2i(αβ)cos2(αβ)+β(c2-l2)e-i(αβ)cos(αβ)=4β2(c2+l2(1+β)e-i(αβ)cos(αβ))e-2i(αβ)cos2(αβ)+β(c2-l2)e-i(αβ)cos(αβ)=β1+β[(5β+1)c2+(3β-1)l2]e-i(αβ)cos(αβ).

Therefore, the inequalities |c2| ≤ 2 and |l2| ≤ 2 give the following:

4a3{2β1+β(5β+1+1-3β)cos (αβ)=4βcos(αβ),         0β132β1+β(5β+1+3β-1)cos (αβ)=16β21+βcos(αβ),         13β<1

which simplifies to:

a3{βcos(αβ),         0β134β21+βcos(αβ),         13β<1.

This is precisely the assertion of (2.2).

We shall next find an estimate on |a4|. At first we shall derive a relation connecting c1, c2, c3, l2 and l3. To this end, we first add the equations (2.7) and (2.11) and get

-9a23+9a2a3=β(c3+l3)e-i(αβ)cos(αβ)+β(β-1)(c1c2+l1l2)e-2i(αβ)cos2(αβ)+β(β-1)(β-2)6(c13+l13)e-3i(αβ)cos3(αβ).

By putting l1 = −c1 the above expression reduces to the following:

-9a23+9a2a3=β(c3+l3)e-i(αβ)cos(αβ)+β(β-1)c1(c2-l2)e-2i(αβ)cos2(αβ).

Substituting a3=a22+β4(c2-l2)e-i(αβ)cos(αβ) from (2.15) into (2.16) we get after simplification:

9βa24(c2-l2)e-i(αβ)cos(αβ)=β(c3+l3)e-i(αβ)cos(αβ)+β(β-1)c1(c2-l2)e-2i(αβ)cos2(αβ).

Since a2=βc1e-i(αβ)cos(αβ), (see 2.5) we have

9β24c1(c2-l2)e-2i(αβ)cos2(αβ)=β(c3+l3)e-i(αβ)cos(αβ)+β(β-1)c1(c2-l2)e-2i(αβ)cos2(αβ).

Or equivalently:

c1(c2-l2)=4(c3+l3)5β+4ei(αβ)cos(αβ).

We wish to express a4 in terms of the first three coefficients of q(z) and p(w). Now substracting (2.11) from (2.7), we get

6a4=-11a23+15a2a3+β(c3-l3)e-i(αβ)cos(αβ)+β(β-1)(c1c2-l1l2)e-2i(αβ)cos2(αβ)+β(β-1)(β-2)6(c13-l13)e-3i(αβ)cos3(αβ).

Observing that l1 = −c1 we have c13-l13=2c13 and therefore

6a4=-9a23+9a2a3-2a23+6a2a3+β(c3-l3)e-(αβ)cos(αβ)+β(β-1)c1(c2+l2)e-2i(αβ)cos2(αβ)+β(β-1)(β-2)3c13e-3i(αβ)cos2(αβ).

We replace -9a23+9a2a3 by the right hand side of (2.16), put a3=β2c12e-2i(αβ)cos2(αβ)+β4(c2-l2)e-i(αβ)cos(αβ) (see (2.15)) and a2=βc1e-i(αβ)cos(αβ).

This gives

6a4=β(c3+l3)e-i(αβ)cos(αβ)+β(β-1)c1(c2-l2)e-2i(αβ)cos2(αβ)-2β3c13e-3i(αβ)cos3(αβ)+6βc1e-i(αβ)cos(αβ)   (β2c12e-2i(αβ)cos2(αβ)+β4(c2-l2)e-i(αβ)cos(αβ))+β(c3-l3)e-i(αβ)cos(αβ)+β(β-1)c1(c2+l2)e-2i(αβ)cos2(αβ)+β(β-1)(β-2)3c13e-3i(αβ)cos3(αβ)=2βc3e-i(αβ)cos(αβ)+β(5β-2)2c1(c2-l2)e-2i(αβ)cos2(αβ)+β(β-1)c1(c2+l2)e-2i(αβ)cos2(αβ)+13β3-3β2+2β3c13e-3i(αβ)cos3(αβ).

Next, replacing c1(c2l2) by the expression in the right hand side of (2.17) and c12 by (2.13) we finally get

6a4=2βc3e-i(αβ)cos(αβ)+β(5β-2)24(c3+l3)5β+4e-i(αβ)cos(αβ)+β(β-1)c1(c2+l2)e-2i(αβ)cos2(αβ)+13β3-3β2+2β3c1(c2+l2)1+βe-2i(αβ)cos2(αβ)=2βc3e-i(αβ)cos(αβ)+2β(5β-2)5β+4(c3+l3)e-i(αβ)cos(αβ)+16β3-3β2-β3(1+β)c1(c2+l2)e-2i(αβ)cos2(αβ)=β[4(5β+1)5β+4c3+2(5β-2)5β+4l3+16β2-3β-13(1+β)c1(c2+l2)e-i(αβ)cos(αβ)]e-i(αβ)cos(αβ).

This gives

a4β6{|4(5β+1)5β+4|c3+|2(5β-2)5β+4|l3+|16β2-3β-13(1+β)|c1(c2+l2)cos(αβ)}cos(αβ).

Therefore,

a4{2β3(1-2316β2-3β-11+β3cos(α/β))cos(α/β),         0β<3+73322β3(1+2316β2-3β-11+β3cos(α/β))cos(α/β),         3+7332β<252β3(15β5β+4+2316β2-3β-11+β3cos(α/β))cos(α/β),         25β<1.

We get the assertion (2.3). The proof of Theorem 2.1 is, thus, completed.

Remark 2.2

Taking α = 0 in the above Theorem 2.1, we readily arrive at Mishra and Soren [14] of Theorem 2.1.

Theorem 2.3

Let f(z), given by (1.1), be in the class SPΣα(ρ)   (απ2,0ρ<1). Then

a22(1-ρ)cos α,a32(1-ρ)cosα

and

a42(1-ρ)cos α3[1+132(1-ρ)cos α].
Proof

Let fSPΣα(ρ). Then by Definition 1.2, we have

eiαzf(z)f(z)=Q1(z)cos α+isin α

and

eiαwg(w)g(w)=P1(w)cos α+isin α

respectively, where ℜ(Q1(z)) > ρ,

Q1(z)=1+c1z+c2z2+         (zU)

and ℜ(P1(w)) > ρ,

P1(w)=1+l1w+l2w2+         (wU).

As in the proof of Theorem 2.1, by suitably comparing coefficients in (2.21) and (2.22) we have

a2eiα=c1cos α,(2a3-a22)eiα=c2cos α,(3a4-3a2a3+a23)eiα=c3cos α

and

-a2eiα=l1cos α,(3a22-2a3)eiα=l2cos α,-(10a23-12a2a3+3a4)eiα=l3cos α.

In order to express c1 interms of c2 and l2 we first add (2.24) and (2.27) and get

2a22=(c2+l2)cos αeiα.

Again putting a2e = c1 cos α from (2.23) we have

2c12cos2αe2iα=(c2+l2)cos αeiα.

Or equivalently:

c12=(c2+l2)eiα2cos α.

The familiar inequalities |c2| ≤ 2(1 − ρ), |l2| ≤ 2(1 − ρ) yield

c124(1-ρ)2cos α=2(1-ρ)cos α

which implies that

c12(1-ρ)cosα

and

a2c1cos α2(1-ρ)cos αcos α=2(1-ρ)cos α.

This proves (2.18).

Following the lines of proof of Theorem 2.1, with appropriate changes, we get that

4a3=(3c2+l2)cos αeiα.

The inequalities |c2| ≤ 2(1 − ρ), |l2| ≤ 2(1 − ρ), yield

a32(1-ρ)cosα.

This is precisely the estimate (2.19).

We shall next find an estimate on |a4|. By substracting (2.28) from (2.25) we get

6a4=-11a23+15a2a3+(c3-l3)cos αeiα.

A substitution of the value of a2 from the relation (2.23) gives

6a4=-11c13cos3αe3iα+15c1cos αeiαa3+(c3-l3)cos αeiα.

Therefore, using the inequalities |c3| ≤ 2(1 − ρ), |l3| ≤ 2(1 − ρ), the estimate for |c1| from (2.31) and the estimate for |a3| from (2.32), we get

6a411c13cos3α+15c1cos αa3+c3-l3cos α11cos3α2(1-ρ)cos α2(1-ρ)cos α+15cos α2(1-ρ)cos α2(1-ρ)cos α+4(1-ρ)cos α4(1-ρ)cos α[1+132(1-ρ)cos α].

Or equivalently,

a42(1-ρ)cos α3[1+132(1-ρ)cos α].

We get the assertion (2.20). This completes the proof of the Theorem 2.3.

Remark 2.4

Taking α = 0 in the above Theorem 2.3, we readily arrive at Mishra and Soren [14] of Theorem 2.3.

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