Kyungpook Mathematical Journal 2018; 58(4): 697-709
Coefficient Bounds for Bi-spirallike Analytic Functions
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.

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 $U={z:z∈ℂ and ∣z∣<1}.$

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

$U={z:z∈ℂ and ∣z∣<1}$

and represented by the normalized series:

$f(z)=z+∑n=2∞anzn (z∈U).$

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

$f-1(f(z))=z (z∈U)$

and

$f(f-1(w))=w (∣w∣[ 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 order$ρ (-π2<α<π2, 0≤ρ≤1)$ 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 order$β(∣α∣≤π2,0≤β<1)$, if each of the following conditions are satisfied:

$f∈Σ and |arg (eiαzf′(z)f(z))|<βπ2 (z∈U)$

and

$|arg (eiαwg′(w)g(w))|<βπ2 (w∈U).$

### 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 order$ρ(∣α∣≤π2,0≤ρ<1)$, if each of the following conditions are satisfied:

$f∈Σ and ℜ (eiαzf′(z)f(z))>ρ cos α (z∈U)$

and

$ℜ (eiαwg′(w)g(w))>ρ cos α (w∈U).$

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

$p(z)=1+∑k=1∞ckzk (z∈U)$

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 $f∈α-SPΣβ$ 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 $α-SPΣβ (∣α∣≤π2,0≤β<1)$. Then

$∣a2∣≤2β(1+β)cos(α/β),$$∣a3∣≤{βcos (αβ), 0≤β≤13,4β21+βcos (αβ), 13≤β<1$

and

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

We write

$f′(z)=f(z)ze-iαh(z) (z∈U;-βπ2<α<βπ2)$

where h(z) is analytic in and satisfies

$h(0)=eiα and ∣arg h(z)∣<βπ2 (z∈U).$

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

$h(z)1β=cos (αβ) q(z)+i sin (αβ) (z∈U)$

is a member of the class ℘. Suppose that

$q(z)=1+c1z+c2z2+⋯ (z∈U.)$

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

$g′(w)=g(w)we-iαH(w) (w∈U;-βπ2<α<βπ2)$

where H(w) is analytic in and satisfies

$H(0)=eiα and ∣arg H(w)∣<βπ2 (w∈U).$

The function p(w) defined by

$H(w)1β=cos (αβ) p(w)+i sin (αβ) (w∈U)$

is a member of the class ℘. If $p(w)=1+l1w+l2w2+⋯ (w∈U),$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:

$∣c1∣≤4(1+β) cos (αβ)=2(1+β) cos (αβ)$

and

$∣a2∣≤β∣c1∣cos(α/β)=2β(1+β)cos(α/β).$

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:

$4∣a3∣≤{2β1+β (5β+1+1-3β) cos (αβ)=4β cos (αβ), 0≤β≤132β1+β (5β+1+3β-1) cos (αβ)=16β21+β cos (αβ), 13≤β<1$

which simplifies to:

$∣a3∣≤{β cos (αβ), 0≤β≤134β21+βcos (αβ), 13≤β<1.$

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

$∣a4∣≤β6{|4(5β+1)5β+4|∣c3∣+|2(5β-2)5β+4|∣l3∣+|16β2-3β-13(1+β)|∣c1∣ ∣(c2+l2)∣cos (αβ)} cos (αβ).$

Therefore,

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

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 $SPΣα(ρ) (∣α∣≤π2,0≤ρ<1)$. Then

$∣a2∣≤2(1-ρ) cos α,$$∣a3∣≤2(1-ρ) cos α$

and

$∣a4∣≤2(1-ρ) cos α3 [1+132(1-ρ) cos α].$
Proof

Let $f∈SPΣα(ρ)$. Then by Definition 1.2, we have

$eiαzf′(z)f(z)=Q1(z) cos α+i sin α$

and

$eiαwg′(w)g(w)=P1(w) cos α+i sin α$

respectively, where ℜ(Q1(z)) > ρ,

$Q1(z)=1+c1z+c2z2+⋯ (z∈U)$

and ℜ(P1(w)) > ρ,

$P1(w)=1+l1w+l2w2+⋯ (w∈U).$

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

$a2eiα=c1 cos α,$$(2a3-a22)eiα=c2 cos α,$$(3a4-3a2a3+a23)eiα=c3 cos α$

and

$-a2eiα=l1 cos α,$$(3a22-2a3)eiα=l2 cos α,$$-(10a23-12a2a3+3a4)eiα=l3 cos 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

$2c12cos2 αe2iα=(c2+l2)cos αeiα.$

Or equivalently:

$c12=(c2+l2)eiα2 cos α.$

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

$∣c12∣≤4(1-ρ)2 cos α=2(1-ρ)cos α$

which implies that

$∣c1∣≤2(1-ρ)cos α$

and

$∣a2∣≤∣c1∣cos α≤2(1-ρ)cos αcos α=2(1-ρ) cos α.$

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

$∣a3∣≤2(1-ρ) cos α.$

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

$6a4=-11c13cos3 αe3iα+15c1cos αeiαa3+(c3-l3)cos αeiα.$

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

$6∣a4∣≤11∣c13∣cos3 α+15∣c1∣cos α∣a3∣+∣c3-l3∣cos α≤11 cos3 α2(1-ρ)cos α2(1-ρ)cos α+15 cos α2(1-ρ)cos α2(1-ρ) cos α+4(1-ρ) cos α≤4(1-ρ) cos α[1+132(1-ρ) cos α].$

Or equivalently,

$∣a4∣≤2(1-ρ) cos α3[1+132(1-ρ) cos α].$

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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