A function is said to be bi-Bazilevic in a given domain if both the function and its inverse map are Bazilevic there. Applying the Faber polynomial expansions to the meromorphic Bazilevic functions, we obtain the general coefficient bounds for bi-Bazilevic functions. We also demonstrate the unpredictability of the behavior of early coefficients of bi-Bazilevic functions.

1. Introduction

Let Σ denote the family of meromorphic functions g of the form
(1)g(z)=z+b0+∑n=1∞bn1zn,
which are univalent in Δ:={z:1<|z|<∞}. The coefficients of h=g-1, the inverse map of the function g∈Σ, are given by the Faber polynomial expansion:
(2)h(w)=g-1(w)=w+B0+∑n=1∞Bnwn=w-b0-∑n≥11nKn+1n(b0,b1,…,bn)1wn;w∈Δ,
where
(3)Kn+1n=nb0n-1b1+n(n-1)b0n-2b2+12n(n-1)(n-2)b0n-3(b3+b12)+n(n-1)(n-2)(n-3)3!b0n-4(b4+3b1b2)+∑j≥5b0n-jVj
and Vj with 5≤j≤n is a homogeneous polynomial of degree j in the variables b1,b2,…,bn (see [1], p. 349 or [2–4]).

For 0≤α<1, 0≤β<1, g∈Σ, and h=g-1, let B(α;β) denote the class of bi-Bazilevic functions of order α and type β (see Bazilevic [5]) if and only if
(4)Re((zg(z))1-βg′(z))>α;z∈Δ,Re((wh(w))1-βh′(w))>α;w∈Δ.

Estimates on the coefficients of classes of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer [4] obtained the estimate |b2|≤2/3 for meromorphic univalent functions g with b0=0 and Duren ([6] or [7]) proved that if b1=b2=⋯=bk=0 for 1≤k<n/2, then |bn|≤2/(n+1). Schober [8] considered the case where b0=0 and obtained the estimate |B2n-1|≤(2n-2)!/(n!(n-1)!) for the odd coefficients of the inverse function h=g-1 subject to the restrictions B2=B4=⋯=Bn-2=0 if n is even or that B2=B4=⋯=Bn-3=0 if n is odd. Kapoor and Mishra [9] considered the inverse function h=g-1, where g∈B(α;0), and obtained the bound 2(1-α)/(n+1) if ((n-1)/n)≤α<1. This restriction imposed on α is a very tight restriction since the class B(α;0) shrinks for large values of n. More recently, Hamidi et al. [10] (also see [11]) improved the coefficient estimate given by Kapoor and Mishra in [9]. The real difficulty arises when the bi-univalency condition is imposed on the meromorphic functions g and its inverse h=g-1. The unexpected and unusual behavior of the coefficients of meromorphic functions g and their inverses h=g-1 prove the investigation of the coefficient bounds for bi-univalent functions to be very challenging. In this paper we extend the results of Kapoor and Mishra [9] and Hamidi et al. [10, 11] to a larger class of meromorphic bi-univalent functions, namely, B(α;β). We conclude our paper with an examination of the unexpected behavior of the early coefficients of meromorphic bi-Bazilevic functions which is the best estimate yet appeared in the literature.

2. Main Results

Applying a result of Airault [12] or [1, 3] to meromorphic functions g of the form (1), for real values of p we can write
(5)(g(z)z)pzg′(z)g(z)=1+∑n=0∞(1-n+1p)Kn+1phhhhhhhh×(b0,b1,b2,…,bn)1zn+1,
where
(6)Kn+1p=p!(p-(n+1))!(n+1)!b0n+1+p!(p-n)!(n-1)!b0n-1b1+p!(p-n+1)!(n-2)!b0n-2b2+p!(p-n+2)!(n-3)!b0n-3[b3+p-n+22b12]+p!(p-n+3)!(n-4)!b0n-4[b4+(p-n+3)b1b2]+∑j≥5b0n-jVj
and Vj is a homogeneous polynomial of degree j:5≤j≤n in the variables b1,b2,…,bn. A simple calculation reveals that the first three terms of Kn+1p(b0,b1,b2,…,bn) may be expressed as
(7)(1-1p)K1p=(p-1)b0,(1-2p)K2p=(p-2)(p-12b02+b1),(1-3p)K3p=(p-3)×((p-1)(p-2)6b03+(p-1)b0b1+b2).
In general, for any real number p, an expansion of Knp=Knp(a2,a3,…,an) (e.g., see [2, equation (4)] or [3]) is given by
(8)Knp=pan+p(p-1)2Dn2+p!(p-3)!3!Dn3+⋯+p!(p-n)!n!Dnn,
where
(9)Ds+mm=Ds+mm(a1,a2,…,as+m)=∑s=1∞m!(a1)μ1⋯(as+m)μs+mμ1!⋯μs+m!
and a1=1. Here we note that the sum is taken over all nonnegative integers μ1,…,μs+m satisfying μ1+μ2+⋯+μs+m=m and μ1+2μ2+⋯+(s+m)μs+m=s+m. Evidently, Dnn(a1,a2,…,as+m)=a1n, [12]. A similar Faber polynomial expansion formula holds for the coefficients of h, the inverse map of g (e.g., see [1, p. 349]).

The Faber polynomials introduced by Faber [13] play an important role in various areas of mathematical sciences, especially in geometric function theory (Gong [14] Chapter III and Schiffer [4]). The recent interest in the calculus of the Faber polynomials, especially when it involves h=g-1, the inverse of g (see [1, 3, 12, 15]), beautifully fits our case for the meromorphic bi-univalent functions. As a result, we are able to state and prove the following.

Theorem 1.

For 0≤α<1 and 0≤β<1 let g ∈B(α;β) and h=g-1∈B(α;β). If b1=b2=⋯=bn-1=0 for n being odd or if b0=b1=⋯=bn-1=0 for n being even, then
(10)|bn|≤2(1-α)n+1-β.

Proof.

For g∈B(α;β) and for h=g-1∈B(α;β), there exist positive real part functions p(z)=1+∑n=1∞cnz-n and q(w)=1+∑n=1∞dnw-n in Δ so that
(11)(zg(z))1-βg′(z)=α+(1-α)p(z)=1+(1-α)∑n=1∞cnzn,(12)(wh(w))1-βh′(w)=α+(1-α)q(w)=1+(1-α)∑n=1∞dnwn.
Note that, according to the Caratheodory lemma (e.g., [7]), |cn|≤2 and |dn|≤2.

On the other hand, by the Faber polynomial expansion, we observe that
(13)(zg(z))1-βg′(z)=(g(z)z)β(zg′(z)g(z))=1+∑n=0∞(1-n+1β)hhhhhhhh×Kn+1β(b0,b1,…,bn)1zn+1,(14)(wh(w))1-βh′(w)=(h(w)w)β(wh′(w)h(w))=1+∑n=0∞(1-n+1β)hhhhhhhh×Kn+1β(B0,B1,…,Bn)1wn+1.
Comparing the corresponding coefficients of (11) and (13) we obtain
(15)(1-α)cn+1=(1-n+1β)Kn+1β(b0,b1,…,bn).
Similarly, from (12) and (14) we obtain
(16)(1-α)dn+1=(1-n+1β)Kn+1β(B0,B1,…,Bn).
For the case b1=b2=⋯=bn-1=0 (n = odd), (15) and (16), respectively, upon using a simple algebraic manipulation and the fact that Bn=-bn, reduce to
(17)(β-1)⋯(β-(n+1))(n+1)!b0n+1+(β-(n+1))bn=(1-α)cn+1,(18)(β-1)⋯(β-(n+1))(n+1)!b0n+1-(β-(n+1))bn=(1-α)dn+1.
Multiplying (18) by −1 and adding it to (17) we obtain
(19)+2(β-(n+1))bn=(1-α)(cn+1-dn+1).
For the other case b0=b1=b2=⋯=bn-1=0(n = even) (15) and (16), respectively, reduce to
(20)+(β-(n+1))bn=(1-α)cn+1,(21)-(β-(n+1))bn=(1-α)dn+1.

Solving either of (19), (20), or (21) for bn, taking the absolute values, and applying the Caratheodory lemma we obtain |bn|≤2(1-α)/(n+1-β).

Relaxing the coefficient restrictions imposed on Theorem 1, we experience the unpredictable behavior of the coefficients of bi-univalent functions.

Theorem 2.

Let g∈B(α;β), 0≤α<1, 0≤β<1, be bi-univalent in Δ. Then

(i) For n∈{0,1,2} (15) yields
(25)(1-α)c1=(β-1)b0,(26)(1-α)c2=(β-1)(β-2)2!b02+(β-2)b1,(27)(1-α)c3=(β-1)(β-2)(β-3)3!b03+(β-3)(β-1)b0b1+(β-3)b2.
Similarly, for n∈{0,1,2} (16) yields
(28)(1-α)d1=-(β-1)b0,(29)(1-α)d2=(β-1)(β-2)2!b02-(β-2)b1,(30)(1-α)d3=-(β-1)(β-2)(β-3)3!b03+(β-3)(β-2)b0b1-(β-3)b2.
From either of the relations (25) or (28) we obtain
(31)|b0|≤2(1-α)1-β.
On the other hand, adding (26) and (29) yields
(32)(1-α)(c2+d2)=(β-1)(β-2)b02.
Solve the above equation for b0, take the absolute values of both sides, and apply the Caratheodory lemma to obtain
(33)|b0|≤(1-α)(|c2|+|d2|)(1-β)(2-β)≤4(1-α)(1-β)(2-β).
Now the bounds given in Theorem 2 (i) for |b0| follow upon noting that
(34)2(1-α)1-β≤4(1-α)(1-β)(2-β)if12-β≤α<1.

(ii) Multiply (29) by −1 and adding it to (26) we obtain
(35)(1-α)(c2-d2)=2(β-2)b1.
Solve the above equation for b1, take the absolute values of both sides, and apply the Caratheodory lemma to obtain the bound |b1|≤2(1-α)/(2-β).

(iii) From (29) we have
(36)b1-b02=1-α2-β(d2-(2-β)(3-β)2(1-α)b02).

Substituting for b0=((1-α)/(1-β))d1 and taking the absolute values of both sides we obtain
(37)|b1-b02|=1-α2-β|d2-(1-α)(2-β)(3-β)2(1-β)d12|=1-α2-β|d2+λd12|.

Using the fact |d2+λd12|≤2+λ|d1|2 if λ≥-1/2 which is due to the first author [16, Lemma 1] and noting that λ≥-1/2 if α≥(5-4β+β2)/(6-5β+β2), we obtain
(38)|b1-b02|=1-α2-β(2-(1-α)(2-β)(3-β)2(1-β)|d1|2).

Now substituting back for d1=((1-β)/(1-α))b0 we obtain
(39)|b1-b02|=1-α2-β(2-(1-β)(2-β)(3-β)2(1-α)|b0|2).

Remark 3.

For the special case B(α;0) we obtain the class of meromorphic bi-starlike functions. Consequently, the bound given for |b0| by our Theorem 2 (i) is an improvement to that given in Hamidi et al. [11, Theorem 2.i.].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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