Bounds of Hankel Determinant for a Class of Univalent Functions

Abstract

The authors study the coefficient condition for the class ℌ_𝛼 defined as the family of analytic functions 𝑓,𝑓(0)=0 and 𝑓′(0)=1, which satisfy ℜ[(1−𝛼)𝑓′(𝑧)+𝛼(1+(𝑧𝑓′′(𝑧))/(𝑓′(𝑧)))]>𝛼,𝑧∈𝐸, where 𝛼 is a real number.

1. Introduction

Let 𝒜 be the class of functions of the following form: 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛, (1.1) which are analytic in the unit disc 𝐸={𝑧∶|𝑧|<1}, and let 𝑆 be the subclass of 𝒜 consisting of functions which are univalent in 𝐸. A function 𝑓∈𝒜 is said to be close to convex in the open unit disc 𝐸 if there exists a convex function 𝑔 (not necessarily normalized) such that ℜ𝑓^′(𝑧)𝑔^′(𝑧)>0,𝑧∈𝐸. (1.2) For fixed real numbers 𝛼, let ℌ_𝛼 denote the family of functions 𝑓 in 𝒜 which satisfy ℜ(1−𝛼)𝑓^′(𝑧)+𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>𝛼,𝑧∈𝐸. (1.3)

In 2005, V. Singh et al. [1] established that, for 0<𝛼<1, functions in ℌ_𝛼 satisfy ℜ𝑓^′(𝑧)>0 in 𝐸 and so are close to convex in 𝐸.

In [2], Noonan and Thomas defined the Hankel determinant 𝐻_𝑞(𝑛) of the function 𝑓 for 𝑞≥1 and 𝑛≥1 by 𝐻_𝑞(𝑛)=𝑎_𝑛𝑎_𝑛+1⋯𝑎_{𝑛+𝑞−1}𝑎_𝑛+1𝑎_𝑛+2⋯𝑎_𝑛+𝑞𝑎⋮⋮⋮⋮_{𝑛+𝑞−1}𝑎_𝑛+𝑞⋯𝑎_{𝑛+2(𝑞−1)}. (1.4)

The determinant has been investigated by several authors with the subject of inquiry ranging from rate of growth of 𝐻_𝑞(𝑛) as 𝑛→∞, to the determination of precise bounds on 𝐻_𝑞(𝑛) for specific 𝑞 and 𝑛 for some special classes of functions. In a classical theorem, Fekete and Szegö [3] considered the Hankel determinant of 𝑓∈𝑆 for 𝑞=2 and 𝑛=1𝐻₂(1)=𝑎₁𝑎₂𝑎₂𝑎₃. (1.5)

The well-known result due to them states that if 𝑓∈𝑆, then 𝑎₃−𝜇𝑎₂²≤3−4𝜇if𝜇≤0,1+2exp−2𝜇1−𝜇if0≤𝜇<1,4𝜇−3if𝜇≥1, (1.6) where 𝑎₁=1 and 𝜇 is a real number. In the present paper, we obtain a sharp bound for 𝐻₂(2)=|𝑎₂𝑎₄−𝑎²₃| when 𝑓∈ℌ_𝛼.

2. Preliminary Results

We denote by 𝑃 the family of all functions 𝑝(𝑧) given by 𝑝(𝑧)=1+𝑐₁𝑧+𝑐₂𝑧²+⋯ (2.1) analytic in 𝐸 for which ℜ{𝑝(𝑧)}>0 for 𝑧∈𝐸. It is well known that for 𝑝∈𝑃, |𝑐_𝑘|≤2 for each 𝑘.

Lemma 2.1 (See [4]). The power series for p(z) given in (2.1) converges in 𝐸 to a function in 𝑃 if and only if the Toeplitz determinants 𝐷_𝑛=2𝑐₁𝑐₂⋯𝑐_𝑛𝑐₋₁2𝑐₁⋯𝑐_𝑛−1𝑐⋮⋮_−𝑛𝑐_−𝑛+1𝑐_−𝑛+2⋯2 (2.2)𝑛=1,2,3… and 𝑐_−𝑘 = 𝑐_𝑘 are all nonnegative. They are strictly positive except for 𝑝(𝑧)=^𝑚_𝑘=1𝜌_𝑘𝑝₀𝑒^𝑖𝑡𝑘𝑧,𝜌_𝑘>0,𝑡_𝑘real, (2.3) and 𝑡_𝑘≠𝑡_𝑗 for 𝑘≠𝑗; in this case, 𝐷_𝑛>0 for 𝑛<𝑚−1 and 𝐷_𝑛=0 for 𝑛≥𝑚.

Lemma 2.2 (See [5, 6]). Let 𝑝∈𝑃. Then 2𝑐₂=𝑐₁²+𝑥4−𝑐₁²,4𝑐₃=𝑐₁³+2𝑥𝑐₁4−𝑐₁²−𝑥²𝑐₁4−𝑐₁²+2𝑧1−|𝑥|²4−𝑐₁² (2.4) for some 𝑥,𝑧 such that |𝑥|≤1 and |𝑧|≤1.

3. Main Result

Theorem 3.1. Let 𝛼, 0≤𝛼≤1, be a real number. If 𝑓∈ℌ_𝛼, then 𝑎₂𝑎₄−𝑎₃²≤49(1−𝛼)²(1+𝛼)²0≤𝛼≤𝛼₀,𝐾(𝛼)𝛼₀≤𝛼≤1, (3.1) where 𝛼₀=0.4276891324… is the root of the equation 10𝛼³−5𝛼²+12𝛼−5=0 and 𝐾(𝛼)=(1−𝛼)²72(1+𝛼)²()()−11+2𝛼321+2𝛼210𝛼³−5𝛼²+12𝛼−5²4𝛼⁵−6𝛼⁴−18𝛼³+29𝛼²−20𝛼−1. (3.2)

Proof. Since 𝑓∈ℌ_𝛼, it follows from (1.3) that there exists a function 𝑝∈𝑃 such that (1−𝛼)𝑓^′(𝑧)+𝛼1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)=𝛼+(1−𝛼)𝑝(𝑧). (3.3) Equating coefficients in (3.3) yields 2𝑎₂=(1−𝛼)𝑐₁,3𝑎₃(1+𝛼)=(1−𝛼)𝑐₂+𝛼(1−𝛼)²𝑐₁²,4𝑎₄(1+2𝛼)=(1−𝛼)𝑐₃+3𝛼(1−𝛼)²()𝑐1+𝛼¹𝑐₂+𝛼(1−𝛼)³(2𝛼−1)()𝑐1+𝛼¹³. (3.4)
Thus, we can easily establish that 𝑎₂𝑎₄−𝑎₃²=(1−𝛼)²72(1+𝛼)²()×9()1+2𝛼1+𝛼²𝑐₁𝑐₃−8(1+2𝛼)𝑐₂²−𝛼(1−𝛼)(11−5𝛼)𝑐₁²𝑐₂+𝛼(1−𝛼)²2𝛼²+𝛼−9𝑐₁⁴. (3.5)
Using (2.4), in view of Lemma 2.2, we obtain that 𝑎₂𝑎₄−𝑎₃²=(1−𝛼)²72(1+𝛼)²()𝑐1+2𝛼4¹⁴9𝛼²+2𝛼+1+4𝛼2𝛼²+𝛼−9(1−𝛼)²(+1+2𝛼1−𝛼)(11−5𝛼)24−𝑐₁²𝑐₁²𝑥9𝛼²(+9+2𝛼+1+𝛼1−𝛼)(11−5𝛼)2(1+𝛼)²4−𝑐₁²𝑐₁1−|𝑥|²1𝑧−4𝑥²4−𝑐₁²×32(1+2𝛼)+𝑐₁²9𝛼²+2𝛼+1. (3.6)
Since 𝑝∈𝑃, so |𝑐₁|≤2. Letting 𝑐₁=𝑐, we may assume without restriction that 𝑐∈[0,2]. Thus, applying the triangle inequality on (3.6), with 𝜌=|𝑥|≤1, we obtain 𝑎₂𝑎₄−𝑎₃²≤(1−𝛼)²72(1+𝛼)²()𝑐1+2𝛼4⁴9𝛼²+2𝛼+1+4𝛼2𝛼²+𝛼−9(1−𝛼)²(+1+2𝛼1−𝛼)(11−5𝛼)24−𝑐²𝑐²𝜌9𝛼²(+9+2𝛼+1+𝛼1−𝛼)(11−5𝛼)2(1+𝛼)²4−𝑐²1𝑐+44−𝑐²(𝑐−2)𝜌²×𝑐9𝛼²+2𝛼+1−16(2𝛼+1)≡𝐹𝜌. (3.7)
Differentiating 𝐹(𝜌), we get the following: 𝐹^′𝜌=(1−𝛼)²72(1+𝛼)²()11+2𝛼24−𝑐²𝑐²9𝛼²(+1+2𝛼+1+𝛼1−𝛼)(11−5𝛼)24−𝑐²(𝑐−2)𝜌𝑐9𝛼²+2𝛼+1−16(2𝛼+1). (3.8)
Using elementary calculus, one can show that 𝐹^′(𝜌)>0 for 𝜌>0. It implies that 𝐹 is an increasing function, and, thus, the upper bound for 𝐹(𝜌) corresponds to 𝜌=1, in which case 𝐹𝜌≤(1−𝛼)²72(1+𝛼)²()×𝑐1+2𝛼₄44𝛼2𝛼²+𝛼−9(1−𝛼)²−29𝛼²+2𝛼+1+𝑐²39𝛼²+2𝛼+1+2𝛼(1−𝛼)(11−5𝛼)−8(2𝛼+1)+32(2𝛼+1)≡𝐺(𝑐). (3.9)
Then, 𝐺^′(𝑐)=(1−𝛼)²72(1+𝛼)²()𝑐1+2𝛼2𝑐²2𝛼2𝛼²+𝛼−9(1−𝛼)²−9𝛼²3+2𝛼+1+29𝛼²+2𝛼+1+2𝛼(1−𝛼)(11−5𝛼)−8(2𝛼+1). (3.10)
Setting 𝐺^′(𝑐)=0, since 0≤𝑐≤2, we have 𝑐₀=−10𝛼³−5𝛼²+12𝛼−54𝛼⁵−6𝛼⁴−18𝛼³+29𝛼²−20𝛼−1 (3.11) provided 𝛼≥𝛼₀, where 𝛼₀=0.4276891324… is the root of the equation 10𝛼³−5𝛼²+12𝛼−5=0.

Case 1. When 0≤𝛼≤𝛼₀, then the maximum value of 𝐺(𝑐) corresponds to 𝑐=0. Therefore, we have max_0≤𝑐≤2𝐺(𝑐)(0)=4=𝐺9(1−𝛼)²(1+𝛼)². (3.12)

Case 2. When 𝛼₀≤𝛼≤1, the maximum value of 𝐺(𝑐) corresponds to 𝑐=𝑐₀. Therefore, we have max_0≤𝑐≤2𝐺(𝑐)=𝐺𝑐₀=𝐾(𝛼), (3.13) where 𝐾(𝛼) is given by (3.2). This completes the proof of the Theorem.

Setting 𝛼=0 in above theorem, we get the following result of Janteng et al. [7].

Corollary 3.2. If an analytic function 𝑓 is such that ℜ{𝑓^′(𝑧)}>0, 𝑧∈𝐸, then 𝑎₂𝑎₄−𝑎₃²≤49. (3.14) The result is sharp.