# Second Hankel Determinants for Some Subclasses of Biunivalent Functions Associated with Pseudo-Starlike Functions

Authors: K. Rajya Laxmi; R. Bharavi Sharma
Journal: Journal of Complex Analysis
 (2017)

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## Abstract

We introduce second Hankel determinant of biunivalent analytic functions associated withλ-pseudo-starlike function in the open unit disc Δ subordinate to a starlike univalent function whose range is symmetric with respect to the real axis.


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## Body

## 1. Introduction

LetA be the class of all analytic functions f of the form(1)fz=z+∑n=2∞anzn,in the open unit disc Δ={z:z<1}. Let S be the subclass of A consisting of univalent functions. Let P be the family of analytic functions p(z) in Δ such that p(0)=1 and Rp(z)>0(z∈Δ) of the form(2)pz=1+∑n=1∞cnzn.For any two functions f and g analytic in Δ, we say that the function f is subordinate to g in Δ and we write it as f(z)≺g(z), if there exists an analytic function w, in Δ with w(0)=0, wz<1z∈Δ such that f(z)=gwz. In view of Koebe 1/4 theorem, every function f∈S has an inverse f-1, defined by(3)f-1fz=z,z∈Δ,ff-1w=ww<r0f;r0f≥14.In fact, the inverse function is given by(4)f-1w=w-a2w2+2a22-a3w3-5a23-5a2a3+a4w4+⋯.A function f∈A is said to be biunivalent in Δ if both f and f-1 are univalent in Δ. Let Σ denote the class of all biunivalent functions defined in the unit disc Δ. We notice that Σ is nonempty. The behavior of the coefficients is unpredictable when the biunivalency condition is imposed on the function f∈A. In 1967, Lewin [1] introduced the class Σ of biunivalent functions and investigated second coefficient in Taylor-Maclaurin series expansion for every f∈Σ. Subsequently, in 1967, Brannan and Clunie [2] introduced bistarlike functions and biconvex functions similar to the familiar subclasses of univalent functions consisting of strongly starlike, strongly convex, starlike, and convex functions and so on and obtained estimates on the initial coefficients conjectured that a2≤√2 for bistarlike functions and a2≤1 for biconvex functions. Only the last estimate is sharp; equality occurs only for f(z)=z/(1-z) or its rotation. Since then, various subclasses of biunivalent functions class Σ were introduced and nonsharp estimates on the first two coefficients a2 and a3 in Taylor-Maclaurin series expansion were found in several investigations. The coefficient estimate problem for each of an is still an open problem. In 1976, Noonan and Thomas [3] defined qth Hankel determinants of f for q≥1 and n≥1 which is stated as follows:(5)Hqn=anan+1an+2⋯an+q-1an+1an+2an+3⋯an+qan+2an+3an+4⋯an+q+1--------------------------an+q-1an+qan+q+1⋯an+2q-2.Easily one can observe that H2(1)=a3-a22 is a special case of the well known Fekete-Szegö functional a3-μa22 where μ is real, for μ=1. Now for q=2, n=2, we get second Hankel determinant(6)H22=a2a3a3a4=a2a4-a32.In particular, sharp upper bounds on H2(2) were obtained by the authors of articles [4–6] for various subclasses of analytic and univalent functions. In 2013, Babalola [7] determined the second Hankel determinant with Fekete-Szegö parameter a2a4-λa32 for some subclasses of analytic functions. Let ϕ be an analytic function with positive real part in Δ such that ϕ(0)=1, ϕ′(0)>0 which is symmetric with respect to the real axis. Such a function has a Maclaurin series expansion of the form ϕ(z)=1+B1z+B2z2+B3z3+⋯(B1>0).Researchers like Duren [8], Singh [9], and so on have studied various subclasses of usual known Bazilevic˘ function of order α denoted by B(α) which satisfy the geometric condition Re⁡f(z)α-1f′z/zα-1>0, where α is nonnegative real number, different ways of perspectives of convexity, radii of convexity and starlikeness, inclusion properties, and so on. The class B(α) reduces to the starlike function and bounded turning function whenever α=0 and α=1, respectively. This class is extended to B(α,β) which satisfy the geometric condition Re⁡(f(z)α-1f′(z)/zα-1)>β, where α is nonnegative real number and 0≤β<1. Recently, Babalola [7] defined new subclass λ-pseudo-starlike functions of order β(0≤β<1) which satisfy the condition Re⁡(z[f′z]λ/f(z))>β, (λ≥1∈R,0≤β<1,z∈Δ) and is denoted by Lλ(β). Babalola [7] proved that all pseudo-starlike functions are Bazilevic˘ of type (1-1/λ), order β(1/λ), and univalent in the open unit disc Δ. For λ=2 we note that functions in L2(β) are defined by Re⁡f′z(zf′z/fz)>β which is a product combination of geometric expressions for bounded turning and starlike functions. Note that the singleton subclass L∞(β) of S contains the identity map. In 2016, Joshi et al. [10] defined two new subclasses of biunivalent functions using pseudo-starlike functions, one is LBΣλα class of strong λ-bi-pseudo-starlike functions of order α and other is LBΣ(λ,β)  λ-bi-pseudo-starlike functions of order β in the open unit disc. Many researchers [11–15] have estimated the second Hankel determinants for some subclasses of biunivalent functions. Motivated by the above-mentioned work, in this paper we have introduced λ-bi-pseudo-starlike functions subordinate to a starlike univalent function whose range is symmetric with respect to the real axis and estimated second Hankel determinants.Definition 1.
A functionf∈Σ is said to be in the class LBΣλ(ϕ), λ≥1, if it satisfies the following conditions:(7)zf′zλfz≺ϕz,z∈Δ,wg′wλgw≺ϕw,w∈Δ,
whereg is an extension of f-1 to Δ.(1)
Ifϕ(z)=(1+z)/(1-z), then the class LBΣλ(ϕ) reduces to the class LBΣλ and satisfies the following conditions:(8)Re⁡zf′zλfz>0,z∈Δ,Re⁡wg′wλgw>0,w∈Δ,whereg is an extension of f-1 to Δ.(2)
Ifϕ(z)=(1+(1-2α)z)/(1-z), 0≤α<1, then the class LBΣλ(ϕ) reduces to the class LBΣλ(α) and satisfies the following conditions:(9)Re⁡zf′zλfz>α,z∈Δ,Re⁡wg′wλgw>α,w∈Δ,whereg is an extension of f-1 to Δ.(3)
Ifϕ(z)=((1+z)/(1-z))β, then the class LBΣλ(ϕ) reduces to the class LBΣλ(β), 0<β≤1 and satisfies following conditions:(10)arg⁡zf′zλfz<βπ2,z∈Δ,arg⁡wg′wλgw<βπ2,w∈Δ,whereg is an extension of f-1 to Δ.(4)
Ifλ=1, then the class LBΣλ(ϕ) reduces to the class of bistarlike functions STΣ(ϕ) and satisfies the following conditions:(11)zf′zfz≺ϕz,z∈Δ,wg′wgw≺ϕw,w∈Δ,whereg is an extension of f-1 to Δ.Several choices ofϕ would reduce the class STΣ(ϕ) to some well known subclasses of Σ.(1)
For the functionϕ given by ϕ(z)=(1+1-2αz)/(1-z), 0≤α<1, the class STΣ(ϕ) reduces to the class STΣ(α) and satisfies the following conditions:(12)Re⁡zf′zfz>α,z∈Δ,Re⁡wg′wgw>α,w∈Δ,whereg is an extension of f-1 to Δ and this class is called class of bistarlike function of order α.(2)
For the functionϕ given by ϕ(z)=(1+z)/(1-z), the class STΣ(ϕ) reduces to the class STΣ and satisfies the following conditions:(13)Re⁡zf′zfz>0,z∈Δ,Re⁡wg′wgw>0,w∈Δ,whereg is an extension of f-1 to Δ and this class is called class of bistarlike function.


## 2. Preliminary Lemmas

LetP denote the class of functions consisting of p, such that(14)pz=1+c1z+c2z2+c3z3+⋯=1+∑n=1∞cnznwhich are analytic in the open unit disc Δ and satisfy R{p(z)}>0 for any z∈Δ.Lemma 2 (see [8]).
Ifp∈P, then cn≤2 for each n≥1 and the inequality is sharp for the function (1+z)/(1-z).Lemma 3 (see [16]).
The power series forp(z)=1+∑n=1∞cnzn given in (14) converges in the open unit disc Δ to a function in P if and only if the Toeplitz determinants(15)Dn=2c1c2c3------cnc-12c1c2------cn-1c-2c-12c1------cn-1------------------------------------------c-nc-n+1c-n+2c-n+3--------2;∀n∈Nand c-k=ck¯ are all nonnegative. They are strictly positive except for p(z)=∑k=1mρkpo(exp⁡itkz), ρk>0, tk real, and tk≠tj, for k≠j, where po(z)=(1+z)/(1-z); in this case Dn>0 for n<(m-1) and Dn=0 for n≥m.We may assume without any restriction thatc1>0, on using Lemma 3 for n=2 and n=3, respectively, we have(16)D2=2c1c2c1¯2c1c2¯c1¯2=8+2Rc12c2-2c22-4c12≥0which is equivalent to(17)2c2=c12+x4-c12,for somex,x≤1.If we consider the determinant(18)D3=2c1c2c3c1¯2c1c2c2¯c1¯2c1c3¯c2¯c1¯2≥0we get the following inequality:(19)4c3-4c1c2+c134-c12+c12c2-c122≤24-c122-22c2-c122.From (17) and (19), it is obtained that(20)4c3=c13+2c14-c12x-c14-c12x2+2c14-c121-x2zfor some z, z≤1.Another required result is the optimal value of quadratic expression. Standard computations show that(21)max0≤t2≤4⁡Pt2+Qt+R=R,Q≤0,P≤-Q4,16P+4Q+R,Q≥0,P≥-Q8orQ≤0,P≥-Q4,4PR-Q24P,Q>0,P≤-Q8.


## 3. Main Results

Theorem 4.
Iff∈LBΣλϕ and is of the form (1) then we have the following. (1)
a2≤B1/(2λ-1).(2)
a3≤B12/(2λ-1)2+B1/(3λ-1).(3)
a4≤B13-4λ3+13λ-3/3(2λ-1)3(4λ-1)+5B12/2(2λ-1)(3λ-1)+4B1/(4λ-1)+B3/(4λ-1).Proof.
Sincef∈LBΣλ(ϕ), there exist two Schwartz functions u(z), v(w) in Δ with u(0)=0, v(0)=0 and uz≤1, vw≤1 such that(22)zf′zλfz=ϕuz,wg′wλgw=ϕvw.Define two functions p(z), q(w) such that(23)pz=1+uz1-uz=1+c1z+c2z2+c3z3+⋯,qw=1+vw1-vw=1+d1w+d2w2+d3w3+⋯.Then(24)ϕpz-1pz+1=1+B1c1z2+B12c2-c122+B2c124z2+B12c134-c1c2+c3+B242c1c2-c13+B38c13z3+⋯,ϕqw-1qw+1=1+B1d1w2+B12d2-d122+B2d124w2+B12d134-d1d2+d3+B242d1d2-d13+B38d13w3+⋯.Then (22) becomes(25)zf′zλfz=ϕ1+uz1-uz,wg′wλgw=ϕ1+vw1-vw.Now equating the coefficients in (25)(26)2λ-1a2=B1c12,(27)3λ-1a3--2λ2+4λ-1a22=B12c2-c122+B2c124,(28)4λ-1a4--6λ2+11λ-2a2a3+4λλ-1λ-23+-2λ2+4λ-1a23=B12c134-c1c2+c3+B242c1c2-c13+B38c13,(29)-2λ-1a2=B1d12,(30)2λ2+2λ-1a22-3λ-1a3=B12d2-d122+B2d124,(31)-4λ-1a4+6λ2+9λ-3a2a3-4λλ-1λ-23+10λ2+2λ-2a23=B12d134-d1d2+d3+B242d1d2-d13+B38d13.Now from (26) and (29)(32)c1=-d1,a2=B1c122λ-1.Now from (27) and (30)(33)a3=B12c1242λ-12+B1c2-d243λ-1.Now from (28) and (31)(34)a4=B13c13-4λ3+13λ-3242λ-134λ-1+5B12c1c2-d2162λ-13λ-1+B244λ-1c1c2+d2-c13+B144λ-1c132-c1c2+d2+c3-d3+B3c1384λ-1and with the help of the above Lemma 2, we get the required results.Theorem 5.
IfLBΣλ(ϕ) is of the form (1) then(35)a3-μa22≤B1hμ3λ-1;hμ≥1,B13λ-1;0≤hμ≤1.Proof.
Now adding (27) and (30), we get that(36)4λ2-2λa22=B12c2+d2+c12+d12B2-B14.Now from (26) and (29), we get that(37)a22=B12c12+d1282λ-12.Now from (36) and (37)(38)a3-μa22=B143λ-1c21+hμ+d2-1+hμ,where h(μ)=B12(1-μ)(3λ-1)/[λ(2λ-1)B12+(B1-B2)(2λ-1)2] which completes the proof of the theorem.Theorem 6.
IfLBΣλ(ϕ) is of the form (1) then we have the following. (1)
If4ξ1≤ξ3, ξ2≤B1/2(3λ-2)2 then a2a4-a32≤B12/(3λ-1)2.(2)
If4ξ1≥ξ3, ξ1-ξ2/2-B1(1/2λ-14λ-1-1/(3λ-1)2)≥0 or 4ξ1≤ξ3, ξ2≥B1/2(3λ-2)2 then a2a4-a32≤2B1ξ2.(3)
4ξ1>ξ3, ξ1-ξ2/2-B1(1/(2λ-1)(4λ-1)-1/(3λ-1)2)≤0; then(39)a2a4-a32≤B1/88B1ξ2/3λ-12-4B1ξ1/2λ-14λ-1-B12/2λ-124λ-12-4ξ12ξ2-ξ1-B1/21/2λ-14λ-1-1/3λ-12.where(40)ξ1=B1243λ-12λ-12+B22λ-14λ-1,ξ2=λ2λ+1B1362λ-134λ-1+B322λ-14λ-1,ξ3=B143λ-12-22λ-14λ-1.Proof.
Using the values ofa2,a3,a4 from the above theorem, one can obtain(41)a2a4-a32=B18-c1422λ-14λ-1λB134λ2-132λ-13-B1-B3+2B2+c2B12c2-d243λ-12λ-12+B2-B1c2+d22λ-14λ-1+c1B1c3-d32λ-14λ-1-B1c2-d2222λ-12.According to Lemma 3 we get that(42)2c2=c12+x4-c12,2d2=d12+y4-d12⇓sssssc2-d2=4-c12x-y2;c2+d2=c12+4-c12x+y2,c3-d3=c132+c14-c12x+y2-c14-c12x2+y24+4-c1221-x2z-1-y2w.For some z, w with z≤1, w≤1. using (42), we have(43)a2a4-a32≤B18c14λ2λ+1B1362λ-134λ-1+B322λ-14λ-1+c1B14-c122λ-14λ-1+c12B124-c1283λ-12λ-12+c124-c12B222λ-14λ-1x+y+B1c12-2c4-c1244λ-12λ-1x2+y2+B14-c12283λ-12x+y2.Since p∈P, c1≤2. Letting c1=c we may assume without any restriction that c∈[0,2]. Thus for γ1=x≤1 and γ2=y≤1, we obtain(44)a2a4-a32≤T1+T2γ1+γ2+T3γ12+γ22+T4γ1+γ22=Fγ1,γ2,where(45)T1=B18c4λ2λ+1B1362λ-134λ-1+B322λ-14λ-1+cB14-c24λ-12λ-1,T2=B18c2B124-c283λ-12λ-12+c24-c2B224λ-12λ-1,T3=B12c2-2c4-c2324λ-12λ-1,T4=B124-c22643λ-12.Now we need to maximize F(γ1,γ2) in the closed square S=[0,1]×[0,1] for c∈[0,2]. We must investigate the maximum of F(γ1,γ2) according to c∈(0,2), c=2, and c=0 taking into account the sign of Fγ1γ1Fγ2γ2-(Fγ1γ2)2.
First, letc∈(0,2). Since T3<0 and T3+2T4>0, we conclude that Fγ1γ1Fγ2γ2-(Fγ1γ2)2<0. Thus the function F cannot have a local maximum in the interior of the square S. Now, we investigate the maximum of F on the boundary of the square S.
Forγ1=0 and 0≤γ2≤1 (similarly γ2=0 and 0≤γ1≤1), we obtain(46)F0,γ2=Gγ2=T1+T2γ2+T3+T4γ22.(i) The CaseT3+T4≥0. In this case 0≤γ2≤1 and for any fixed c with 0<c<2, it is clear that G′γ2=2T3+T4γ2+T2>0; that is, Gγ2 is an increasing function. Hence for any fixed c∈0,2 the maximum of Gγ2 occurs at γ2=1 and(47)max⁡Gγ2=G1=T1+T2+T3+T4.(ii) The CaseT3+T4<0. Since 2(T3+T4)+T2≥0 for 0≤γ2≤1 and for any fixed c with 0<c<2, it is clear that 2T3+T4+T2<2T3+T4γ2+T2<T2 and so G′γ2>0. Hence for any fixed c∈[0,2) the maximum of G(γ2) occurs at γ2=1. Also for c=2 we obtain(48)Fγ1,γ2=B1λ2λ+1B1332λ-134λ-1+B32λ-14λ-1.Taking into account the value of (48) and case (i) and case (ii), for 0≤γ2≤1 and for any fixed c with 0≤c≤2, (49)max⁡Gγ2=G1=T1+T2+T3+T4.For γ1=1 and 0≤γ2≤1 (similarly γ2=1 and 0≤γ1≤1), we obtain(50)F1,γ2=Hγ2=T3+T3γ22+T2+2T4γ2+T1+T2+T3+T4.Similar to the above case of T3+T4, we get that(51)max⁡Hγ2=H1=T1+2T2+2T3+4T4.Since G(1)≤H(1) for c∈[0,2], max⁡F(γ1,γ2)=F(1,1) on the boundary of the square S. Thus the maximum of F occurs at γ1=1 and γ2=1 in the closed square S.
LettingK:[0,2]→R,(52)Kc=max⁡Fγ1,γ2=F1,1=T1+2T2+2T3+4T4.Substituting the values of T1,T2,T3,T4 in the above equation,(53)Kc=B18c4λ2λ+1B1362λ-134λ-1+B322λ-14λ-1-B1243λ-12λ-12+B22λ-14λ-1-B1212λ-14λ-1-13λ-12+c2B122λ-123λ-1+4B22λ-14λ-1-B143λ-12-22λ-14λ-1+8B13λ-12.Let(54)P=B18λ2λ+1B1362λ-134λ-1+B322λ-14λ-1-B1243λ-12λ-12+B22λ-14λ-1-B1212λ-14λ-1-13λ-12,Q=B18B122λ-123λ-1+4B22λ-14λ-1-B143λ-12-22λ-14λ-1,R=B123λ-12.Then K(c)=Pt2+Qt+R, where t=c2.
Then with help of optimal value of quadratic expression, we get the required result. This completes the proof of the theorem.Corollary 7.
Iff∈LBΣλ and is of the form (1) then(55)a2a4-a32≤44λ2λ+132λ-134λ-1+12λ-14λ-1.Corollary 8.
Iff∈LBΣλ(α) and is of the form (1) then(56)a2a4-a32≤41-α24λ1-α22λ+132λ-134λ-1+12λ-14λ-1,α∈0,τ,1-α2ρ11-α2-181-αρ2+ρ32λ-14λ-1ρ41-α2-3ρ21-α+ρ5,α∈τ,1,where(57)ρ1=4λ-116λ2λ+12λ-1-34λ-1,ρ2=2λ-13λ-14λ-1,ρ3=32λ-1242λ-14λ-1-93λ-12,ρ4=4λ2λ+13λ-12,ρ5=32λ-134λ-1-62λ-123λ-12,τ=1-32λ-14λ-1+2λ-194λ-12+962λ+13λ-1216λ2λ+13λ-1.Corollary 9.
Iff∈LBΣλ(β) and is of the form (1) then(58)a2a4-a32≤4β23λ-12,βζ2≤ζ4,β2ζ1≤τ5,4ββ3ζ1+β32λ-14λ-1,βζ2≥ζ4;ζ1β2-ζ2β2+162λ-14λ-1≥0orβζ2≤ζ4,β2ζ1≥τ5,β2β2τ1-τ2-βτ3+τ4β2ζ1-βζ2+ζ3,βζ2>ζ4,ξ1β2-ζ2β2+162λ-14λ-1≤0,
where(59)τ1=16λ2λ+132λ-133λ-124λ-1+832λ-13λ-124λ-1,τ2=13λ-122λ-14+42λ-124λ-12-42λ-133λ-14λ-1,τ3=22λ-133λ-14λ-1+42λ-124λ-12,τ4=432λ-13λ-124λ-1-12λ-124λ-12,τ5=13λ-12-132λ-14λ-1,ζ1=4λ2λ+132λ-134λ-1+232λ-14λ-1,ζ2=12λ-123λ-1+22λ-14λ-1,ζ3=13λ-12-232λ-14λ-1,ζ4=23λ-12-12λ-14λ-1.Corollary 10.
Iff∈STΣ(ϕ) and is of the form (1) then we have the following. (1)
If4(B12/8+B2/3)≤B1/3, (B13+B3)/6≤B1/8 then a2a4-a32≤B12/4.(2)
If4(B12/8+B2/3)≥B1/3,(B13+B3)/6-(1/2)(B12/8+B2/3)-B1/12≥0 or 4(B12/8+B2/3)≤B1/3, (B13+B3)/6≥B1/8 then a2a4-a32≤B1(B13+B3)/3.(3)
4(B12/8+B2/3)>B1/3, (B13+B3)/6-(1/2)(B12/8+B2/3)-B1/12≤0; then(60)a2a4-a32≤B18B1/3B13+B3-4B1/3B12/8+B2/3-B12/9-4B12/8+B2/32B13+B3/6-B12/8+B2/3-B1/24.The above result is obtained by taking λ=1 in Theorem 6, which is the second Hankel determinant of bistarlike function.Corollary 11.
Iff∈STΣ(α) and is of the form (1) then(61)a2a4-a32≤41-α234α2-8α+5;ifα∈0,29-13732,1-α213α2-14α-716α2-26α+5;ifα∈29-13732,1.Corollary 12.
Iff∈STΣ and is of the form (1) then(62)a2a4-a32≤203.



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