Coefficient Estimates for Certain Classes of Bi-Univalent Functions

Abstract

A function analytic in the open unit disk 𝔻 is said to be bi-univalent in 𝔻 if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in 𝔻 makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions.

1. Introduction

Let 𝒜 denote the class of functions 𝑓 which are analytic in the open unit disk 𝔻∶={𝑧∈ℂ∶|𝑧|<1} and normalized by 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛. (1)

Let 𝒮 denote the class of functions 𝑓∈𝒜 that are univalent in 𝔻 and let 𝒫 be the class of functions 𝑝(𝑧)=1+∑^∞_𝑛=1𝑝_𝑛𝑧^𝑛 that are analytic in 𝔻 and satisfy the condition Re(𝑝(𝑧))>0 in 𝔻. By the Caratheodory lemma (e.g., see [1]) we have |𝑝_𝑛|≤2.

For 0≤𝛼<1 and 𝜆≥1 we let 𝒟(𝛼;𝜆) denote the family of analytic functions 𝑓∈𝒜 so that ()𝑓(𝑧)Re1−𝜆𝑧+𝜆𝑓^′(𝑧)>𝛼,𝑧∈𝔻. (2)

We note that 𝒟(0;1) is the class of bounded boundary turning functions and also that 𝒟(𝛼;𝜆)⊂𝒟(𝛽;𝜆) if 0≤𝛽<𝛼. For 𝑓∈𝒜, the class 𝒟(𝛼;𝜆)⊂𝒮 and was first defined and investigated by Ding et al. [2].

It is well known that every function 𝑓∈𝒮 has an inverse 𝑓⁻¹ satisfying 𝑓⁻¹(𝑓(𝑧))=𝑧 for 𝑧∈𝔻 and 𝑓(𝑓⁻¹(𝑤))=𝑤 for |𝑤|<1/4, according to Kobe One Quarter Theorem (e.g., see [1]).

A function 𝑓∈𝒜 is said to be bi-univalent in 𝔻 if both 𝑓∈𝒮 and 𝑔=𝑓⁻¹∈𝒮. Finding bounds for the coefficients of classes of bi-univalent functions dates back to 1967 (see Lewin [3]). But the interest on the bounds for the coefficients of classes of bi-univalent functions picked up by the publications of Brannan and Taha [4], Srivastava et al. [5], Frasin and Aouf [6], Ali et al. [7], and Hamidi et al. [8]. The bi-univalency condition imposed on the functions 𝑓∈𝒜 makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions, as Ali et al. [7] also remarked that finding the bounds for |𝑎_𝑛| when 𝑛≥4 is an open problem. Here in this paper we let 𝑓∈𝒟(𝛼;𝜆) and 𝑔=𝑓⁻¹∈𝒟(𝛼;𝜆) and use the Faber polynomial coefficient expansions to provide bounds for the general coefficients |𝑎_𝑛| of such functions with a given gap series. We also obtain estimates for the first two coefficients |𝑎₂| and |𝑎₃| of these functions as well as providing an estimate for their coefficient body (𝑎₂,𝑎₃). The bounds provided in this paper prove to be better than those estimates provided by Srivastava et al. [5] and Frasin and Aouf [6].

2. Main Results

Using the Faber polynomial expansion of functions 𝑓∈𝒜 of the form (1), the coefficients of its inverse map 𝑔=𝑓⁻¹ may be expressed as, [9], 𝑔(𝑤)=𝑓⁻¹(𝑤)=𝑤+^∞_𝑛=21𝑛𝐾^−𝑛_𝑛−1𝑎₂,𝑎₃,…𝑤^𝑛, (3) where 𝐾^−𝑛_𝑛−1=(−𝑛)!()!()!𝑎−2𝑛+1𝑛−1^2𝑛−1+(−𝑛)!(2(!()!𝑎−𝑛+1))𝑛−3^2𝑛−3𝑎₃+(−𝑛)!()!()!𝑎−2𝑛+3𝑛−4^2𝑛−4𝑎₄+(−𝑛)!(2(!()!𝑎−𝑛+2))𝑛−5^2𝑛−5𝑎₅+(−𝑛+2)𝑎²₃+(−𝑛)!()!()!𝑎−2𝑛+5𝑛−6^2𝑛−6𝑎₆+(−2𝑛+5)𝑎₃𝑎₄+_𝑗≥7𝑎₂^𝑛−𝑗𝑉_𝑗, (4) such that 𝑉_𝑗 with 7≤𝑗≤𝑛 is a homogeneous polynomial in the variables 𝑎₂,𝑎₃,…,𝑎_𝑛 [10]. In particular, the first three terms of 𝐾^−𝑛_𝑛−1 are 12𝐾₁⁻²=−𝑎₂,13𝐾₂⁻³=2𝑎²₂−𝑎₃,14𝐾₃⁻⁴=−5𝑎³₂−5𝑎₂𝑎₃+𝑎₄. (5)

In general, for any 𝑝∈ℕ, an expansion of 𝐾^𝑝_𝑛 is as, [9, page 183], 𝐾^𝑝_𝑛=𝑝𝑎_𝑛+𝑝𝑝−12𝐷²_𝑛+𝑝!𝐷𝑝−3!3!^3𝑛+⋯+𝑝!𝐷𝑝−𝑛!𝑛!^𝑛𝑛, (6) where 𝐷^𝑝_𝑛=𝐷^𝑝_𝑛(𝑎₂,𝑎₃,…) and by [11] or [12], 𝐷^𝑚_𝑛𝑎₁,𝑎₂,…,𝑎_𝑛=^∞_𝑚=1𝑚!𝑎₁^𝜇1⋯𝑎_𝑛^𝜇𝑛𝜇₁!⋯𝜇_𝑛!, (7) while 𝑎₁=1, and the sum is taken over all nonnegative integers 𝜇₁,…,𝜇_𝑛 satisfying 𝜇₁+𝜇₂+⋯+𝜇_𝑛𝜇=𝑚,₁+2𝜇₂+⋯+𝑛𝜇_𝑛=𝑛. (8)

Evidently, 𝐷^𝑛_𝑛(𝑎₁,𝑎₂,…,𝑎_𝑠+𝑚)=𝑎^𝑛₁, [13].

Theorem 1. For 0≤𝛼<1 and 𝜆≥1 let 𝑓∈𝒟(𝛼;𝜆) and 𝑔∈𝒟(𝛼;𝜆). If 𝑎_𝑘=0; 2≤𝑘≤𝑛−1, then 𝑎_𝑛≤2(1−𝛼)1+(𝑛−1)𝜆;𝑛≥4. (9)

Proof. For analytic functions 𝑓 of the form (1) we have ()𝑓(𝑧)1−𝜆𝑧+𝜆𝑓^′(𝑧)=1+^∞_𝑛=2(1+(𝑛−1)𝜆)𝑎_𝑛𝑧^𝑛−1, (10) and for its inverse map, 𝑔=𝑓⁻¹, we have ()𝑔(𝑤)1−𝜆𝑤+𝜆𝑔^′(𝑤)=1+^∞_𝑛=2(1+(𝑛−1)𝜆)𝑏_𝑛𝑤^𝑛−1=1+^∞_𝑛=2(()𝜆)×11+𝑛−1𝑛𝐾^−𝑛_𝑛−1𝑎₂,𝑎₃,…,𝑎_𝑛𝑤^𝑛−1. (11)
On the other hand, since 𝑓∈𝒟(𝛼;𝜆) and 𝑔=𝑓⁻¹∈𝒟(𝛼;𝜆), by definition, there exist two positive real part functions 𝑝(𝑧)=1+∑^∞_𝑛=1𝑐_𝑛𝑧^−𝑛 and 𝑞(𝑤)=1+∑^∞_𝑛=1𝑑_𝑛𝑤^−𝑛 where Re𝑝(𝑧)>0 and Re𝑞(𝑤)>0 in 𝒟 so that ()𝑓(𝑧)1−𝜆𝑧+𝜆𝑓^′(𝑧)=1+(1−𝛼)^∞_𝑛=1𝐾¹_𝑛𝑐₁,𝑐₂,…,𝑐_𝑛𝑧^𝑛,()𝑔(𝑤)1−𝜆𝑤+𝜆𝑔^′(𝑤)=1+(1−𝛼)^∞_𝑛=1𝐾¹_𝑛𝑑₁,𝑑₂,…,𝑑_𝑛𝑤^𝑛. (12)(13)
Comparing the corresponding coefficients of (10) and (12) yields (1+𝜆(𝑛−1))𝑎_𝑛=(1−𝛼)𝐾¹_𝑛−1𝑐₁,𝑐₂,…,𝑐_𝑛−1, (14) and similarly, from (11) and (13) we obtain 1𝑛(1+(𝑛−1)𝜆)𝐾^−𝑛_𝑛−1𝑏₀,𝑏₁,…,𝑏_𝑛=(1−𝛼)𝐾¹_𝑛−1𝑑₁,𝑑₂,…,𝑑_𝑛−1. (15)
Note that for 𝑎_𝑘=0; 2≤𝑘≤𝑛−1 we have 𝑏_𝑛=−𝑎_𝑛 and so (1+(𝑛−1)𝜆)𝑎_𝑛=(1−𝛼)𝑐_𝑛−1,−(1+(𝑛−1)𝜆)𝑎_𝑛=(1−𝛼)𝑑_𝑛−1. (16)
Now taking the absolute values of either of the above two equations and applying the Caratheodory lemma, we obtain 𝑎_𝑛≤(1−𝛼)𝑐_𝑛−1|()𝜆|=(𝑑1+𝑛−11−𝛼)_𝑛−1|()𝜆|≤2()1+𝑛−11−𝛼()𝜆.1+𝑛−1 (17)

Theorem 2. For 0≤𝛼<1 and 𝜆≥1 let 𝑓∈𝒟(𝛼;𝜆) and 𝑔∈𝒟(𝛼;𝜆). Then one has the following (𝑖)𝑎₂≤2(1−𝛼)1+2𝜆,0≤𝛼<1+2𝜆−𝜆²2();2()1+2𝜆1−𝛼,1+𝜆1+2𝜆−𝜆²2()(𝑎1+2𝜆≤𝛼<1.𝑖𝑖)₃≤2(1−𝛼).(𝑎1+2𝜆𝑖𝑖𝑖)₃−2𝑎²₂≤2(1−𝛼).1+2𝜆 (18)

Proof. Replacing 𝑛 by 2 and 3 in (14) and (15), respectively, we deduce (1+𝜆)𝑎₂=(1−𝛼)𝑐₁,(1+2𝜆)𝑎₃=(1−𝛼)𝑐₂,−(1+𝜆)𝑎₂=(1−𝛼)𝑑₁,(1+2𝜆)2𝑎²₂−𝑎₃=(1−𝛼)𝑑₂. (19)(20)(21)(22)
Dividing (19) or (21) by (1+𝜆), taking their absolute values, and applying the Caratheodory lemma, we obtain 𝑎₂≤(1−𝛼)𝑐₁=(𝑑1+𝜆1−𝛼)₁≤2()1+𝜆1−𝛼.1+𝜆 (23)
Adding (20) to (22) implies 2(1+2𝜆)𝑎²₂=(1−𝛼)𝑐₂+𝑑₂ (24) or 𝑎²₂=(1−𝛼)𝑐₂+𝑑₂2().1+2𝜆 (25)
An application of Caratheodory lemma followed by taking the square roots yields 𝑎₂≤²(1−𝛼)𝑐₂+𝑑₂2()≤1+2𝜆²2(1−𝛼).1+2𝜆 (26)
Now the bounds given in Theorem 2 (i) for |𝑎₂| follow upon noting that if (1+2𝜆−𝜆²)/2(1+2𝜆)≤𝛼<1, then 2(1−𝛼)≤1+𝜆²2(1−𝛼).1+2𝜆 (27)
Dividing (20) by (1+2𝜆), taking the absolute values of both sides, and applying the Caratheodory lemma yield 𝑎₃=(1−𝛼)𝑐₂≤2()1+2𝜆1−𝛼.1+2𝜆 (28)
Dividing (22) by (1+2𝜆), taking the absolute values of both sides, and applying the Caratheodory lemma, we obtain 𝑎₃−2𝑎²₂≤2(1−𝛼).1+2𝜆 (29)

Corollary 3. For 0≤𝛼<1 let 𝑓∈𝒟(𝛼;1) and 𝑔∈𝒟(𝛼;1). Then one has the following (𝑖)𝑎₂≤2(1−𝛼)31,0≤𝛼<3;11−𝛼,3(𝑎≤𝛼<1.𝑖𝑖)₃≤2(1−𝛼)3. (30)

Remark 4. The above two estimates for |𝑎₂| and |𝑎₃| show that the bounds given in Theorem 2 are better than those given by Srivastava et al. ([5, page 1191, Theorem 2] and Frasin and Aouf [6, page 1572, Theorem 3.2]).