1. Introduction
Let A denote the class of functions of the form(1)fz=z+∑n=2∞anznwhich are analytic in the open unit disc U={z:z∈C and |z|<1}. Further, by S we denote the family of all functions in A which are univalent in U. Let h(z) be an analytic function in U and |h(z)|≤1, such that(2)hz=h0+h1z+h2z2+h3z3+⋯,where all coefficients are real. Also, let φ be an analytic and univalent function with positive real part in U with φ(0)=1, φ′(0)>0 and φ maps the unit disk U onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor’s series expansion of such function is of the form(3)φz=1+B1z+B2z2+B3z3+⋯,where all coefficients are real and B1>0. Throughout this paper we assume that the functions h and φ satisfy the above conditions one or otherwise stated.

For two functions f and g are analytic in U, we say that the function f(z) is subordinate to g(z) in U and write (4)fz≺gz z∈Uif there exists a Schwarz function w(z), analytic in U, with (5)w0=0,wz<1 z∈U,such that (6)fz=gwz z∈U.In particular, if the function g is univalent in U, the above subordination is equivalent to (7)f0=g0,fU⊂gU.

For two analytic functions f and g, the function f is quasi-subordinate to g in the open unit disc U if there exist analytic functions h and w, with |h(z)|≤1, w(0)=0, and |w(z)|<1, such that f(z)/h(z) is analytic in U and written as (8)fzhz≺gz z∈U.We also denote the above expression by (9)fz≺q gz z∈Uand this is equivalent to (10)fz=hzgwz z∈U.

Observe that if h(z)≡1, then f(z)=g(w(z)), so that f(z)≺g(z) in U. Also notice that if w(z)=z, then f(z)=h(z)g(z) and it is said that f is majorized by g and written by f(z)≪g(z) in U. Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization (see [1]).

In [2] Ma and Minda introduced the unified classes S∗(φ) and K(φ) given below:(11)S∗φ≔f:f∈A, zf′zfz≺φz; z∈U,Kφ≔f:f∈A, 1+zf′′zf′z≺φz; z∈U.

For the choice(12)φz=1+1-2αz1-z 0≤α<1or(13)φz=1+z1-zβ 0<β≤1the classes S∗(φ) and K(φ) consist of functions known as the starlike (resp., convex) functions of order α or strongly starlike (resp., convex) functions of order β, respectively.

Recently, El-Ashwah and Kanas [3] introduced and studied the following two subclasses:(14)Sq∗γ,φ≔f:f∈A, 1γzf′zfz-1≺q φz-1; z∈U, 0≠γ∈C,Kqγ,φ≔f:f∈A, 1γzf′′zf′z≺q φz-1; z∈U, 0≠γ∈C.We note that when h(z)≡1, the classes Sq∗(γ,φ) and Kq(γ,φ) reduce, respectively, to the familiar classes S∗(γ,φ) and K(γ,φ) of Ma-Minda starlike and convex functions of complex order γ (γ∈C∖{0}) in U (see [4]). For γ=1, the classes Sq∗(γ,φ) and Kq(γ,φ) reduce to the classes Sq∗(φ) and Kq(φ), respectively, that are analogous to Ma-Minda starlike and convex functions, introduced by Mohd and Darus [5].

It is well known that every function f∈S has an inverse f-1, defined by (15)f-1fz=z z∈U,ff-1w=w w<r0f;r0f≥14,where(16)f-1w=w+∑n=2∞bnwn w<r0f,where(17)bn=-1n+1n!Aijand |Aij| is the (n-1)th order determinant whose entries are defined in terms of the coefficients of f(z) by the following:(18)Aij=i-j+1n+j-1ai-j+2,i+1≥j;0,i+1<j.For initial values of n, we have(19)b2=-a2,b3=2a22-a3,b4=5a2a3-5a23-a4,and so on. A function f∈A is said to be biunivalent in U if both f and f-1 are univalent in U. Let σ denote the class of biunivalent functions in U given by (1). For a brief history and interesting examples of functions which are in (or which are not in) the class σ, together with various other properties of the biunivalent function class σ, one can refer to the work of Srivastava et al. [6] and references therein. Recently, various subclasses of the biunivalent function class σ were introduced and nonsharp estimates on the first two coefficients |a2| and |a3| in the Taylor–Maclaurin series expansion (1) were found in several recent investigations (see, e.g., [7–17]). But the problem of finding the coefficient bounds on |an| (n=3,4,…) for functions f∈σ is still an open problem.

Motivated by the above mentioned works, we define the following subclass of function class σ.

A function f∈σ given by (1) is said to be in the class Mq,σδ,λ(γ,φ), 0≠γ∈C, δ≥0, if the following quasi-subordination conditions are satisfied:(20)1γ1-δzFλ′zFλz+δ1+zFλ′′zFλ′z-1≺q φz-1 z∈U,1γ1-δwGλ′wGλw+δ1+wGλ′′wGλ′w-1≺q φw-1 w∈U,where(21)Fλz=1-λfz+λzf′z,Gλw=1-λgw+λwg′w 0≤λ≤1,and the function g is the extension of f-1 to U.

It is interesting to note that the special values of δ, γ, λ, and φ and the class Mq,σδ,λ(γ,φ) unify the following known and new classes.

Remark 1.
Setting λ=0 in the above class, we have (22)Mq,σδ,0γ,φ≔Mq,σδγ,φ.In particular, for γ=1, we have (23)Mq,σδ1,φ≔Mq,σδφwhich was introduced and studied by Goyal and Kumar [18, Definition 2.3, p. 541]. Also, we note that for h(z)≡1 the class Mq,σδ(φ)≔Mσδ(φ) was introduced and studied by Ali et al. [7] (see also [19]). If we take φ(z) by (12) in the class Mσδ(φ), we are led to the class which we denote, for convenience, by Mσδ(α), introduced and studied by Li and Wang [12, Definition 3.1., p. 1500], and upon replacing φ by (13) in the class Mσδ(φ), we have Mσδ(β); this class was introduced and studied by Li and Wang [12, Definition 2.1., p. 1497].

Remark 2.
Taking λ=0 and δ=0 in the class Mq,σδ,λ(γ,φ), we have (24)Mq,σ0,0γ,φ≔Sq,σ∗γ,φ.In particular, for γ=1, we have (25)Sq,σ∗1,φ≔Sq,σ∗φ.

The class Sq,σ∗(φ) is particular case of the class Mq,σδ(φ), when δ=0 and it was introduced and studied by Goyal and Kumar [18, Definition 2.3, p. 541]. We note that, for h(z)≡1, the class Sq,σ∗(γ,φ)≔Sσ∗(γ,φ) was introduced and studied by Deniz [10]. Further, for h(z)≡1, the class Sq,σ∗(φ)≔Sσ∗(φ) was introduced by Ali et al. [7] and Srivastava et al. [16]. For φ(z) given by (12), the class Sσ∗(α) was introduced by Brannan and Taha [20] and studied by Bulut [8], Çaglar et al. [9], Li and Wang [12], and others.

Remark 3.
Setting λ=0 and δ=1 in the class Mq,σδ,λ(γ,φ), we have (26)Mq,σ1,0γ,φ≔Kq,σγ,φ.In particular, for γ=1, we get (27)Kq,σ1,φ≔Kq,σφ.

The class Kq,σ(φ) is particular case of the class Mq,σδ(φ), when δ=1 and it was introduced and studied by Goyal and Kumar [18, Definition 2.3, p. 541]. We note that, for h(z)≡1, the class Kq,σ(γ,φ)≔Kσ(γ,φ) was introduced and studied by Deniz [10]. Further, for h(z)≡1, the class Kq,σ(φ)≔Kσ(φ) was considered by Ali et al. [7]. For φ(z) given by (12), we get the class Kσ(α), introduced by Brannan and Taha [20] and studied by Li and Wang [12] and others.

Remark 4.
Taking δ=0, we have the class Mq,σ0,λ(γ,φ)≡Pq,σ(γ,λ,φ) as defined below.

A function f∈σ is said to be in the class Pq,σ(γ,λ,φ), 0≠γ∈C, 0≤λ≤1, if the following quasi-subordinations hold:(28)1γzf′z+λz2f′′z1-λfz+λzf′z-1 ≺q φz-1,1γwg′w+λw2g′′w1-λgw+λwg′w-1 ≺q φw-1,where g(w)=f-1(w). A function in the class Pq,σ(γ,λ,φ) is called both bi-λ-convex functions and bi-λ-starlike functions of complex order γ of Ma-Minda type. For h(z)≡1, the class Pq,σ(γ,λ,φ)≔Pσ(γ,λ,φ) was introduced and studied by Deniz [10].

Remark 5.
Putting δ=1, we have the class Mq,σ1,λ(γ,φ)≡Kq,σ(γ,λ,φ) as defined below.

A function f∈σ is said to be in the class Kq,σ(γ,λ,φ), 0≠γ∈C, 0≤λ≤1, if the following quasi-subordinations hold:(29)1γzf′z+1+2λz2f′′z+λz3f′′′zzf′z+λz2f′′z-1≺q φz-1,1γwg′w+1+2λw2g′′w+λw3g′′′wwg′w+λw2g′′w-1≺q φw-1,where g(w)=f-1(w).

Remark 6.
For h(z)≡1, the class Mq,σδ,λ(γ,φ)≔Mσ(δ,λ,γ,φ) was introduced in [21].

In this paper we introduce the unified biunivalent function class Mq,σδ,λ(γ,φ) as defined above and obtain the coefficient estimates for Taylor-Maclaurin coefficients |a2| and |a3| for functions belonging to Mq,σδ,λ(γ,φ). Some interesting applications of the results presented here are also discussed.

In order to derive our results, we need the following lemma.

Lemma 7 (see [<xref ref-type="bibr" rid="B18">22</xref>]).
If p∈P, then |pi|≤2 for each i, where P is the family of all functions p, analytic in U, for which (30)Rpz>0 z∈U,where (31)pz=1+p1z+p2z2+⋯ z∈U.

2. Coefficient Estimates for the Class <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M215"><mml:msubsup><mml:mrow><mml:mi mathvariant="script">M</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>δ</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:msubsup><mml:mfenced separators="|"><mml:mrow><mml:mi>γ</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>φ</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>
Theorem 8.
Let f(z) given by (1) be in the class Mq,σδ,λ(γ,φ), 0≤λ<1, 0≠γ∈C, and δ≥0. Then(32)a2≤γh0B1B1γ21+2δ1+2λ-1+3δ1+λ2h0B12-1+δ21+λ2B2-B1,(33)a3≤γh1B121+2δ1+2λ+γh0B2-B11+δ1+2λ-λ21+3δ+γh0B11+3δ1+λ2+3+5δ1+2λ-λ21+3δ41+2δ1+2λ1+δ1+2λ-λ21+3δ.

Proof.
Since f∈Mq,σδ,λ(γ,φ), there exist two analytic functions r,s:U→U, with r(0)=0=s(0), such that(34)1γ1-δzFλ′zFλz+δ1+zFλ′′zFλ′z-1=hzφrz-1,1γ1-δwGλ′wGλw+δ1+wGλ′′wGλ′w-1=hwφsw-1.Define the functions u and v by(35)uz=1+rz1-rz=1+u1z+u2z2+u3z3+⋯,vz=1+sz1-sz=1+v1z+v2z2+v3z3+⋯or equivalently(36)rz=uz-1uz+1=12u1z+u2-u122z2+u3+u12u122-u2-u1u22z3+⋯,sz=vz-1vz+1=12v1z+v2-v122z2+v3+v12v122-v2-v1v22z3+⋯.

Using (36) in (34), we have(37)1γ1-δzFλ′zFλz+δ1+zFλ′′zFλ′z-1=hzφuz-1uz+1-1,1γ1-δwGλ′wGλw+δ1+wGλ′′wGλ′w-1=hwφqw-1qw+1-1.

Again using (36) along with (3), it is evident that(38)hzφuz-1uz+1-1=12h0B1u1z+12h1B1u1+12h0B1u2-12u12+14h0B2u12z2+⋯,hwφqw-1qw+1-1=12h0B1v1w+12h1B1v1+12h0B1v2-12v12+14h0B2v12w2+⋯.

It follows from (37) and (38) that(39)1γ1+δ1+λa2=12h0B1u1,(40)1γ21+2δ1+2λa3-1+3δ1+λ2a22=12h1B1u1+12h0B1u2-12u12+14h0B2u12,(41)-1γ1+δ1+λa2=12h0B1v1,(42)1γ41+2δ1+2λ-1+3δ1+λ2a22-21+2δ1+2λa3=12h1B1v1+12h0B1v2-12v12+14h0B2v12.From (39) and (41), we find that(43)a2=γh0B1u121+δ1+λ=-γh0B1v121+δ1+λ;it follows that(44)u1=-v1,(45)81+δ21+λ2a22=h02B12γ2u12+v12.Adding (40) and (42), we have(46)a221γ41+2δ1+2λ-21+3δ1+λ2=h0B12u2+v2+h0B2-B14u12+v12.Substituting (43) and (44) into (46), we get(47)a22=γ2h02B13u2+v24γ21+2δ1+2λ-1+3δ1+λ2h0B12-41+δ21+λ2B2-B1.Applying Lemma 7 in (47), we get desired inequality (32). Subtracting (40) from (42) and a computation using (44) finally lead to(48)a3=a22+γh1B1u141+2δ1+2λ+γh0B1u2-v281+2δ1+2λ.Again applying Lemma 7, (48) yields desired inequality (33). This completes the proof of Theorem 8.

In light of Remarks 1–5, we have following corollaries.

Corollary 9.
If f∈Sq,σ∗(γ,φ), 0≠γ∈C, then(49)a2≤γh0B1B1γh0B12-B2+B1,a3≤γh1B12+γh0B1+B2-B1.

Remark 10.
Corollary 9 reduces to [23, Corollary 2.3, p. 82].

Corollary 11.
If f∈Kq,σ(γ,φ), 0≠γ∈C, then(50)a2≤γh0B1B12γh0B12-4B2-B1,a3≤γh1B16+γh0B1+B2-B12.

Corollary 12.
If f∈Mq,σδ(γ,φ), 0≠γ∈C, and δ≥0, then(51)a2≤γh0B1B1γh0B121+δ-B2-B11+δ2,a3≤γh1B12+4δ+γh0B1+B2-B11+δ.

Corollary 13.
If f∈Pq,σ(γ,λ,φ), 0≠γ∈C, and 0≤λ≤1, then(52)a2≤γh0B1B1γ1+2λ-λ2h0B12-1+λ2B2-B1,a3≤γh1B12+4λ+γh0B2-B11+2λ-λ2+γh0B11+λ2+3+6λ-λ241+2λ1+2λ-λ2.

Corollary 14.
If f∈Kq,σ(γ,λ,φ), 0≠γ∈C, and 0≤λ≤1, then(53)a2≤γh0B1B1γ2+4λ-4λ2h0B12-41+λ2B2-B1,(54)a3≤γh1B16+12λ+γh0B2-B12+4λ-4λ2+γh0B11+λ2+2+4λ-λ231+2λ2+4λ-4λ2.

Remark 15.
Taking h(z)≡1 in Corollary 9, we get estimates in [10, Corollary 2.3, p. 54] and setting h(z)≡1 in Corollary 11 we have bounds in [10, Corollary 2.2, p. 53]. For h(z)≡1 and γ=1, the inequalities obtained in Corollary 11 coincide with [7, Corollary 2.2, p. 349]. For h(z)≡1 and γ=1, the estimates in Corollary 12 reduce to a known result in [7, Theorem 2.3, p. 348]. Further, for h(z)≡1, γ=1, and φ given by (12) the inequalities in Corollary 12 reduce to a result proven earlier by [12, Theorem 3.2, p. 1500] and for h(z)≡1, γ=1, and φ given by (13) the inequalities in Corollary 12 would reduce to known result in [12, Theorem 2.2, p. 1498]. Also, for h(z)≡1, the estimates in Corollary 13 provide improvement over the estimates derived by Deniz [10, Theorem 2.1, p. 32]. For h(z)≡1, the results obtained in this paper coincide with results in [21]. Furthermore, various other interesting corollaries and consequences of our results could be derived similarly by specializing φ.