Kyungpook Mathematical Journal 2018; 58(4): 677-688
Coefficient Estimates for a Subclass of Bi-univalent Functions Defined by Sălăgean Type q-Calculus Operator
Prakash Namdeo Kamble
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India
e-mail : kamble.prakash69@gmail.com

Mallikarjun Gurullingappa Shrigan∗
Department of Mathematics, Dr. D Y Patil School of Engineering and Technology, Pune 412205, India
e-mail : mgshrigan@gmail.com
*Corresponding Author.
Received: March 7, 2018; Revised: September 25, 2018; Accepted: October 2, 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 this paper, we introduce and investigate a new subclass of bi-univalent functions defined by Sălăgean q-calculus operator in the open disk . For functions belonging to the subclass, we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|. Some consequences of the main results are also observed.

Keywords: analytic functions, bi-univalent functions, coefficient bounds, Sălăgean q-differential operator , Sălăgean derivative.
1. Introduction

Let denote the family of functions analytic in the open unit disk

$U={z:z∈ℂ and ∣z∣<1},$

which are normalized by the condition:

$f(0)=f′(0)-1=0$

and given by the following Taylor-Maclaurin series:

$f(z)=z+∑k=2∞ak zk.$

Also let be the class of functions of the form given by (1.1), which are univalent in . The Koebe one-quarter theorem [7] ensures that the image of under every univalent function contains a disk of radius $14$. Hence every function has an inverse f−1, defined by

$f-1(f(z))=z, (z∈U)$

and

$f-1(f(w))=w, (∣w∣

where

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

A function is said to be bi-univalent in if both f and f−1 are univalent in . Let ∑ denote the class of bi-univalent functions in given by the Taylor-Maclaurin series expansion (1.1). For a brief history and interesting examples of functions in the class ∑, see [28] (see also [4]). From the work of Srivastava et al. [28], we choose to recall the following examples of functions in the class ∑:

$z1-z, -log(1-z), 12log(1+z1-z).$

However, familiar Koebe function is not a member of ∑.

The class of bi-univalent functions was investigated by Lewin [13], who proved that |a2| < 1.51. In 1981, Styer and Wright [30] showed that |a2| > 4/3. Subsequently, Brannan and Clunie [3] improved Lewin’s result to $∣a2∣≤2$. Netanyahu [14], showed that $maxf∈Σ∣a2∣=43$. In 1985, Branges [2] proved Bieberbach conjecture which showed that

$∣an∣≤n; (n∈N-1),$

N being positive integer.

The problem of finding coefficient estimates for the bi-univalent functions has received much attention in recent years. In fact, the aforecited work of Srivastava et al. [28] essentially revived the investigation of various subclasses of bi-univalent function class ∑ in recent years and that it leads to a flood of papers on the subject (see, for e.g., [6, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29]); it was followed by such works as those by Tang et al . [31], Xu et al . [32, 33] and Lashin [12], and others (see, for e.g., [1, 5, 8]). The coefficient estimate problem involving the bound of |an|(n ∈ ℕ {1, 2}) for each f ∈ ∑ is still an open problem.

In the field of geometric function theory, various subclasses of the normalized analytic function class have been studied from different view points. The q-calculus as well as the fractional calculus provide important tools that have been used in order to investigate various subclasses of . Historically speaking, the firm footing of the usage of the q-calculus in the context of geometric function theory which was actually provided and q-hypergeometric functions were first used in geometric function theory in a book chapter by Srivastava (see, for details, [18, pp. 347 et seq.]). Ismail et al. [10] introduced the class of generalized complex functions via q-calculus on some subclasses of analytic functions. Recently, Purohit and Raina [16] investigated applications of fractional q-calculus operator to define new classes of functions which are analytic in unit disk (see, for details, [9]).

For 0 < q < 1, the q-derivative of a function f given by (1.1) is defined as

$Dqf(z)={f(qz)-f(z)(q-1)zfor z≠0,f′(0)for z=0.$

We note that $limq→1-Dqf(z)=f′(z)$. From (1.2), we deduce that

$Dqf(z)=1+∑k=2∞[k]qakzk-1,$

where as q → 1

$[k]q=1-qk1-q=1+q+…+qk→k.$

Making use of the q-differential operator for function , we introduced the Sălăgean q-differential operator as given below

$Dq0f(z)=f(z)Dq1f(z)=zDqf(z)Dqnf(z)=zDq(Dqn-1f(z))Dqnf(z)=z+∑k=2∞[k]qnakzk (n∈ℕ0,z∈U).$

We note that limq → 1

$Dnf(z)=z+∑k=2∞knakzk (n∈ℕ0,z∈U),$

the familiar Sălăgean derivative [17].

Recently, Kamble and Shrigan [11] introduce the following two subclasses of the bi-univalent function class ∑ and obtained estimate on first two Taylor-Maclaurin coefficients |a2| and |a3| for functions in these subclasses as follows.

### Definition 1.1.([11])

For 0 < α ≤ 1, 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0, a function f(z) given by (1.1) is said to be in the class$ℋΣq,μ(n,α,λ)$ if the following conditions are satisfied

$f∈Σ and |arg((1-λ) (Dqnf(z)z)μ+λ(Dqnf(z))′(Dqnf(z)z)μ-1)|<απ2$

and

$|arg((1-λ) (Dqng(w)w)μ+λ(Dqng(w))′(Dqng(w)w)μ-1)|<απ2,$

where the function g is given by

$g(w)=w-a2w2+(2a22-a3)w3-(5a23-5a2a3+a4)w4+…$

and $Dqn$ is the Sălăgean q-differential operator.

### Theorem 1.2.([11])

Let f(z) given by (1.1) be in the function class$ℋΣq,μ(n,α,λ)$. Then

$∣a2∣≤2αα(2(2λ+μ)[3]qn-(λ2+2λ+μ)[2]q2n)+(λ+μ)2[2]q2n$

and

$∣a3∣≤4α2(λ+μ)2[2]q2n+2α(2λ+μ)[3]qn,$

where 0 < α ≤ 1, 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0.

### Definition 1.3.([11])

For 0 ≤ β < 1, 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0, a function f(z) given by (1.1) is said to be in the class$ℋΣq,μ(n,β,λ)$ if the following conditions are satisfied

$f∈Σ and Re{(1-λ) (Dqnf(z)z)μ+λ(Dqnf(z))′(Dqnf(z)z)μ-1}>β$

and

$Re{(1-λ) (Dqng(w)w)μ+λ(Dqng(w))′(Dqng(w)w)μ-1}>β.$

### Theorem 1.4.([11])

Let f(z) given by (1.1) be in the function class$ℋΣq,μ(n,β,λ)$. Then

$∣a2∣≤min{4(1-β)∣2[3]qn+(μ-1)[2]q2n(2λ+μ)∣,2(1-β)(λ+μ)[2]qn}$

and

$∣a3∣≤min{4(1-β)2(λ+μ)2[2]q2n+2(1-β)(2λ+μ)[3]qn,(1-β) {∣4[3]qn+[2]q2n(μ-1)∣-[2]q2n(∣μ-1∣)}∣2[3]qn+(μ-1)[2]q2n∣(2λ+μ)[3]qn},$

where 0 ≤ β < 1, 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0.

### Remark 1.5

By appropriately specializing the parameters in Definition 1.1 and 1.3, we can get several known subclasses of the bi-univalent function class ∑. For example:

• For n = 0 and q → 1, we obtain the bi-univalent function classes $ℋΣ1,μ(0,α,λ)=NΣμ(α,λ) and ℋΣ1,μ(0,β,λ)=NΣμ(β,λ)$(see [ 21]);

• For μ = 1, n = 0 and q → 1, we obtain the bi-univalent function classes $ℋΣ1,1(0,α,λ)=ℬΣ(α,λ) and ℋΣ1,1(0,β,λ)=ℬΣ(β,λ)$(see [ 8]);

• For μ = 1 and q → 1 we obtain the bi-univalent function classes $ℋΣ1,1(n,α,λ)=ℬΣ(n,α,λ) and ℋΣ1,1(n,β,λ)=ℬΣ(n,β,λ)$(see [ 15]);

• For μ = 1, n = 0, λ = 1 and q → 1, we obtain the bi-univalent function classes $ℋΣ1,1(0,α,1)=ℋΣα and ℋΣ1,1(0,β,1)=ℋΣ(β)$(see [ 28]);

• For μ = 0, n = 0, λ = 1 and q → 1, we obtain the bi-univalent function classes $ℋΣ1,0(0,α,1)=SΣ*(α) and ℋΣ1,0(0,β,1)=SΣ*(β)$(see [ 4]);

This paper is a sequel to some of the aforecited works (especially see [11, 32, 33]). Here we introduce and investigate the general subclass $ℋΣh,p(λ,μ,n,q) (0 of the analytic function class , which is given by Definition 1.6 below.

### Definition 1.6

Let h, be analytic functions and

$min{Re(h(z)),Re(p(z))}>0 (z∈U) and h(0)=p(0)=1.$

Also let the function f given by (1.1), be in the analytic function class . We say that$f∈ℋΣh,p(λ,μ,n,q) (0

if the following conditions satisfied:

$f∈Σ and (1-λ) (Dqnf(z)z)μ+λ(Dqnf(z))′(Dqnf(z)z)μ-1∈h(U) (z∈U)$

and

$(1-λ) (Dqng(w)w)μ+λ(Dqng(w))′(Dqng(w)w)μ-1∈p(U) (w∈U),$

where the function g is given by (1.9).

If $f∈ℋΣh,p(λ,μ,n,q)$, then

$f∈Σ and |arg((1-λ) (Dqnf(z)z)μ+λ(Dqnf(z))′(Dqnf(z)z)μ-1)|<απ2$

and

$|arg((1-λ) (Dqng(w)w)μ+λ(Dqng(w))′(Dqng(w)w)μ-1)|<απ2$

or

$f∈Σ and Re{(1-λ) (Dqnf(z)z)μ+λ(Dqnf(z))′(Dqnf(z)z)μ-1}>β$

and

$Re{(1-λ) (Dqng(w)w)μ+λ(Dqng(w))′(Dqng(w)w)μ-1}>β.$

where the function g is given by (1.9).

Our paper is motivated and stimulated especially by the work of Srivastava et al. [21, 28]. Here we propose to investigate the bi-univalent function subclass $ℋΣh,p(λ,μ,n,q)$ of the function class ∑ and find estimates on the initial coefficients |a2| and |a3| for functions in the new subclass of the function class ∑ using Sălăgean q-differential operator.

### 2. A Set of General Coefficient Estimates

In this section, we derive estimates on the initial coefficients |a2| and |a3| for functions in subclass $ℋΣh,p(λ,μ,n,q)$ given by Definition 1.6.

### Theorem 2.1

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function class$ℋΣh,p(λ,μ,n,q)$. Then

$∣a2∣≤min{∣h′(0)∣2+∣p′(0)∣22(λ+μ)2[2]q2n,∣h″(0)∣+∣p″(0)∣2(2λ+μ)∣(μ-1)[2]q2n+2[3]qn∣}$

and

$∣a3∣≤min{∣h′(0)∣2+∣p′(0)∣22(λ+μ)2[2]q2n+∣h″(0)∣+∣p″(0)∣4(2λ+μ)[3]qn,∣(μ-1)[2]q2n+4[3]qn∣ ∣h″(0)∣+∣μ-1∣[2]q2n∣p″(0)∣4(2λ+μ)[3]qn∣(μ-1)[2]q2n+2[3]qn∣},$

where 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0.

Proof

It follows from (1.16) and (1.17) that

$(1-λ) (Dqnf(z)z)μ+λ(Dqnf(z))′(Dqnf(z)z)μ-1=h(U)$

and

$(1-λ) (Dqng(w)w)μ+λ(Dqng(w))′(Dqng(w)w)μ-1=p(U)$

Comparing the coefficients of z and z2 in (2.3) and (2.4), we have

$(λ+μ) [2]qna2=h1,$$(μ-1) (λ+μ2) [2]q2na22+(2λ+μ) [3]qna3=h2,$$-(λ+μ) [2]qna2=p1$

and

$-(2λ+μ) [3]qna3+(4[3]qn+(μ-1) [2]q2n) (λ+μ2) a22=p2.$

From (2.5) and (2.7), we obtain

$h1=-p1$

and

$2(λ+μ)2[2]q2na22=h12+p12.$

Also, from (2.6) and (2.8), we find that

${(μ-1) [2]q2n+2[3]qn} (2λ+μ)2a22=h2+p2.$

Therefore, we find from the equations (2.10) and (2.11) that

$∣a2∣≤∣h′(0)∣2+∣p′(0)∣22(λ+μ)2[2]q2n$

and

$∣a2∣≤∣h″(0)∣+∣p″(0)∣2(2λ+μ)∣(μ-1) [2]q2n+2[3]qn,$

respectively. So we get the desired estimate on the coefficients |a2| as asserted in (2.1).

Next, in order to find the bound on the coefficient |a3|, we subtract (2.8) from (2.6), we get

$2(2λ+μ) [3]qna3-2[3]qn(2λ+μ)a22=h2-p2.$

Upon substituting the value of $a22$ from (2.10) into (2.12), we arrive at

$a3=h12+p122(λ+μ)2[2]q2n+h2-p22(2λ+μ) [3]qn.$

We thus find that

$∣a3∣≤∣h′(0)∣2+∣p′(0)∣22(λ+μ)2[2]q2n+∣h″(0)∣+∣p″(0)∣4(2λ+μ) [3]qn.$

On the other hand, upon substituting the value of $a22$ from (2.11) into (2.12), we arrive at

$a3={(μ-1) [2]q2n+4[3]qn}h2+(μ-1) [2]q2np22(2λ+μ) [3]qn{(μ-1) [2]q2n+2[3]qn}.$

Consequently, we have

$∣a3∣≤∣(μ-1) [2]q2n+4[3]qn∣ ∣h″(0)∣+∣μ-1∣ [2]q2n∣p″(0)∣4(2λ+μ) [3]qn ∣(μ-1) [2]q2n+2[3]qn∣.$

This evidently completes the proof of Theorem 2.1.

3. Corollaries and Consequences

By Setting μ = 1, q → 1 and n = 0 in Theorem 2.1, we deduce the following consequence of Theorem 2.1.

### Corollary 3.1

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function class$ℬΣh,p(λ) (λ≥1)$. Then

$∣a2∣≤min{∣h′(0)∣2+∣p′(0)∣22(1+λ)2,∣h″(0)∣+∣p″(0)∣4(1+2λ)}$

and

$∣a3∣≤min{∣h′(0)∣2+∣p′(0)∣22(1+λ)2+∣h″(0)∣+∣p″(0)∣4(1+2λ),∣h″(0)∣2(1+2λ)}.$

By Setting μ = 0, λ = 1, q → 1 and n = 0 in Theorem 2.1, we deduce the following.

### Corollary 3.2.([5])

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function class$ℬΣh,p$. Then

$∣a2∣≤min{∣h′(0)∣2+∣p′(0)∣22,∣h″(0)∣+∣p″(0)∣4}$

and

$∣a3∣≤min{∣h′(0)∣2+∣p′(0)∣28+∣h″(0)∣+∣p″(0)∣8,3∣h″(0)∣+∣p″(0)∣8}.$

### Remark 3.3

Corollary 3.2 is an improvement of the following estimates obtained by Xu et al . [33].

### Corollary 3.4.([33])

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function class$ℬΣh,p(λ) (λ≥1)$. Then

$∣a2∣≤∣h″(0)∣+∣p″(0)∣4(1+2λ)$

and

$∣a3∣≤∣h″(0)∣2(1+2λ).$

By Setting λ = 1, μ = 1, q → 1 and n = 0 in Theorem 2.1, we deduce the following Corollary 3.5.

### Corollary 3.5.([32])

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function class$ℋΣh,p$. Then

$∣a2∣≤∣h″(0)∣+∣p″(0)∣12$

and

$∣a3∣≤∣h″(0)∣6.$
4. Concluding Remarks and Observations

The main objective in this paper has been to derive first two Taylor-Maclaurin coefficient estimates for functions belonging to a new subclass $ℋΣh,p(λ,μ,n,q)$ of analytic and bi-univalent function in the open unit disk . Indeed, by using Sălăgean q-calculus operator, we have successfully determined the first two Taylor-Maclaurin coefficient estimates for functions belonging to a new subclass $ℋΣh,p(λ,μ,n,q)$.

By means of corollaries and consequences which we discuss in the preceding section by suitable specializing the parameters λ and μ, we have also shown already that the results presented in this paper would generalize and improve some recent works of Xu et al . [32, 33] and other authors.

Acknowledgements

We thank the referees for their insightful suggestions and scholarly guidance to revise and improve the results as in present form.

References
1. Altinkaya, Ş, and Yalçin, S (2016). Faber polynomial coefficient bounds for a subclass of bi-univalent functions. Stud Univ Babeş-Bolyai Math. 61, 37-44.
2. Branges, LD (1985). A proof of the Bieberbach conjecture. ActaMath. 154, 137-152.
3. Brannan, DA, and Clunie, JG 1980. Aspect of contemporary complex analysis., Proceedings of the NATO Advanced Study Institute, July 1–20, 1979, pp.79-95.
4. Brannan, DA, and Taha, TS (1988). On some classes of bi-univalent functions. Mathematical Analysis and its Applications(Kuwait, 1985). Oxford: Pergamon Press, Elsevier science Limited, pp. -Array
5. Bulut, S (2013). Coefficient estimates for a class of analytic and bi-univalent functions. Novi Sad J Math. 43, 59-65.
6. Çağlar, M, Deniz, E, and Srivastava, HM (2017). Second Hankel determinant for certain subclasses of bi-univalent functions. Turkish J Math. 41, 694-706.
7. Duren, PL (1983). Univalent Functions. Grundlehren der Mathematischen Wissenschaften, Band. New York, Berlin, Heidelberg and Tokyo: Springer-Verlag
8. Frasin, BA, and Aouf, MK (2011). New subclasses of bi-univalent functions. Appl Math Lett. 24, 1569-1573.
9. Gasper, G, and Rahman, M (1990). Basic Hypergeometric Series. Cambridge: Cambridge University Press
10. Ismail, MEH, Merkes, E, and Styer, D (1990). A generalization of starlike functions. Complex Variables Theory Appl. 14, 77-84.
11. Kamble, PN, and Shrigan, MG (2018). Initial coefficient estimates for bi-univalent functions. Far East J Math. 105, 271-282.
12. Lashin, AY (2016). On certain subclasses of analytic and bi-univalent functions. J Egyptian Math Soc. 24, 220-225.
13. Lewin, M (1967). On a coefficient problem for bi-univalent functions. Proc Amer Math Soc. 18, 63-68.
14. Netanyahu, E (1969). The minimal distance of the image boundary from the origin and second coefficient of a univalent function in |z| < 1. Arch Rational Mech Anal. 32, 100-112.
15. Porwal, S, and Darus, M (2013). On a new subclass of bi-univalent functions. J Egyptian Math Soc. 21, 190-193.
16. Purohit, SD, and Raina, RK (2011). Certain subclasses of analytic functions associated with fractional q-calculus operators. Math Scand. 109, 55-70.
17. Sălăgean, GS 1983. Subclasses of univalent functions., Complex Analysis - Fifth Romanian Finish Seminar, Bucharest, pp.362-372.
18. Srivastava, HM (1989). Univalent functions, fractional calculus, and associated generalized hypergeometric functions. Univalent Functions; Fractional Calculus; and Their Applications, Srivastava, HM, and Owa, S, ed. Array: Array, pp. 329-354
19. Srivastava, HM, Altinkaya, Ş, and Yaļcin, S (2018). Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator. Filomat. 32, 503-516.
20. Srivastava, HM, and Bansal, D (2015). Coefficient estimates for a subclass of analytic and bi-univalent functions. J Egyptian Math Soc. 23, 242-246.
21. Srivastava, HM, Bulut, S, Çağlar, M, and Yağmur, N (2013). Coefficient estimates for a general subclass of analytic and bi-univalent functions. Filomat. 27, 831-842.
22. Srivastava, HM, Eker, SS, and Ali, RM (2015). Coefficient bounds for a certain class of analytic and bi-univalent functions. Filomat. 29, 1839-1845.
23. Srivastava, HM, Eker, SS, Hamadi, SG, and Jahangiri, JM (2018). Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator. Bull Iranian Math Soc. 44, 149-157.
24. Srivastava, HM, Gaboury, S, and Ghanim, F (2015). Coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions. Acta Univ Apulensis Math Inform. 41, 153-164.
25. Srivastava, HM, Gaboury, S, and Ghanim, F (2016). Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions. Acta Math Sci Ser B (Engl Ed). 36, 863-871.
26. Srivastava, HM, Gaboury, S, and Ghanim, F (2017). Coefficient estimates for some general subclasses of analytic and bi-univalent functions. Afr Mat. 28, 693-706.
27. Srivastava, HM, Joshi, SB, Joshi, SS, and Pawar, H (2016). Coefficient estimates for certain subclasses of meromorphically bi-univalent functions. Palest J Math. 5, 250-258.
28. Srivastava, HM, Mishra, AK, and Gochhayat, P (2010). Certain subclasses of analytic and bi-univalent functions. Appl Math Lett. 23, 1188-1192.
29. Srivastava, HM, Müge Sakar, F, and Özlem Güney, H (2018). Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination. Filomat. 32, 1313-1322.
30. Styer, D, and Wright, J (1981). Results on bi-univalent functions. Proc Amer Math Soc. 82, 243-248.
31. Tang, H, Srivastava, HM, Sivasubramanian, S, and Gurusamy, P (2016). The Fekete-Szegö functional problems for some classes of m-fold symmetric bi-univalent functions. J Math Inequal. 10, 1063-1092.
32. Xu, Q-H, Gui, Y-C, and Srivastava, HM (2012). Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl Math Lett. 25, 990-994.
33. Xu, Q-H, Xiao, H-G, and Srivastava, HM (2012). A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems. Appl Math Comput. 218, 11461-11465.