Coefficient inequalities for a subclass of Bazilevič functions

1 Definitions and preliminaries

Denote by A the class of analytic functions f defined for z∈D={z:|z|<1}, and normalized so that f(0) = 0 and f′(0) = 1, and by S the subclass of A consisting of functions that are univalent in D={z:|z|<1}. Let f be given by

Then, for α > 0, it was shown by Bazilevič [1] that if f∈A and is given by eq. (1), then there exists starlike functions g such that

it follows that f∈S. We denote this class of Bazilevič functions by ℬ(α), so that ℬ(α)⊂S when α > 0.

The case α = 0 was subsequently considered by Sheil-Small [2], who showed that ℬ(α)⊂S when α ≥ 0.

Taking g(z) ≡ z gives the class ℬ1(α) of Bazilevič functions, which has been the subject of much recent research. We note that ℬ1(0) is the class S⁎ of starlike functions, and ℬ1(1) the well-known class ℛ of functions whose derivative has positive real part in D.

Thus, f∈ℬ1(α), if and only if, for z∈D

Various properties have been obtained for functions in ℬ1(α). Among other results, Singh [3] found sharp estimates for the moduli of the first four coefficients and obtained the solution to the Fekete–Szegö problem. Sharp bounds for the second Hankel determinant, the initial coefficients of the function log(f(z)/z), and the initial coefficients of the inverse function f⁻¹ were obtained in [4], and distortion theorems and some length–area results were also obtained in [5,6,7].

We now define the subclass ℬ1(α,λ) of ℬ1(α), which was introduced in 1996 by Ponnusamy and Singh [8, Theorem 3]. In this article, the authors determine condition on λ so that functions in ℬ1(α,λ) are starlike in D (see also [9, Theorem 3] for an extension of this result). Later in [10, Theorems 1 and 2], the authors considered complex values of α and obtained condition on λ such as the functions in ℬ1(α,λ) are spirallike in D. Two of the present authors in [11] studied the class ℬ1(α,λ) in the case λ = 1. Therefore, it is natural to consider the investigation of the problems discussed in this article for the complex values of α in the context of the investigation from [10].

We note that f∈ℬ1(0,1) reduces to the class of bounded starlike functions considered by Singh [12].

Although the aforementioned definition requires that α ≥ 0, choosing α = −1 gives the class U(λ) of univalent functions defined for z∈D by

The class U(λ) has been the focus of a great deal of research in recent years (see e.g. [13,14], and for a summary of some known results, see [15]). Although the classes ℬ1(α,λ) for α ≥ 0 and U(λ) have similar structural representations, they are fundamentally different in many ways, and we shall see in the following analysis that the methods used in this study cannot be applied to the class U(λ). It is also interesting to note that the only known negative value of α which gives a subset of S appears to be α = −1. See [16,17] and references therein for recent investigation, which also deals with the case α = −1 for meromorphic functions.

In this study, we give sharp bounds for the modulus of the coefficients a_n for f∈ℬ1(α,λ) when 2 ≤ n ≤ 5, together with other related results, noting that when f∈U(λ), sharp bounds have been found only for some initial coefficients.

First note that from eq. (2), we can write

for z∈D, where ω is the Schwarz function.

Next, recall the class P of functions with positive real part in D, so that h∈P, if, and only if, Re h(z) > 0 for z∈D.

We write

Thus, as we can write

eq. (3) can be written as

We shall use the following results concerning the coefficients of h∈P.

2 Initial coefficients

We first give sharp bounds for some initial coefficients for f∈ℬ1(α,λ), extending those given in [11].

The inequality for |a₂| is trivial.

For a₃, we apply Lemma 1.2 with

Since 0 ≤ μ ≤ 2 for α ≥ 0 and 0 < λ ≤ 1, the inequality for |a₃| follows.

For a₄, we use Lemma 1.4 with

and

Since 0 ≤ B ≤ 1, and B(2B − 1) ≤ D ≤ B, when α ≥ 0 and 0 < λ ≤ 1, the inequality for |a₄| follows.

For a₅, we apply Lemma 1.5 with α₁, α₂, β₁ and β₂ the respective coefficients of a₅ in eq. (7). Since 0 < α₁ < 1 and 0 < α₂ < 1, for α ≥ 0 and 0 < λ ≤ 1, then by expanding both sides and subtracting, it is easily seen that the conditions (6) of Lemma 1.5 are satisfied (the detailed proof of this step can be found in [21]), and so the inequality for |a₅| follows. The inequality of |a_i| is sharp on choosing c_i = 2 when 2 ≤ i ≤ 5, and c_j = 0 when i ≠ j.□

3 Inverse coefficients

Since ℬ1(α,λ)⊂S, inverse functions f⁻¹ exist, and so we can write

valid in some disk |w| ≤ r₀(f). It is an easy exercise to show that

We first prove the following.

The inequality for |A₂| is trivial, since |c₁| ≤ 2.

For A₃, we use Lemma 1.2 with

and the inequalities for |A₃| easily follow. The inequality for |A₂| is sharp when c₁ = 2. The first and second inequalities for |A₃| are sharp on choosing c₁ = 0 and c₂ = 2. The third inequality for |A₃| is sharp when c₁ = c₂ = 2.□

When λ = 1, obtaining sharp bounds for |A₄| follows relatively easily from an application of Lemmas 1.3 and 1.4 [11]. However, finding sharp bounds when 0 < λ ≤ 1 appears to be a much more difficult problem, as the next theorem demonstrates.

The inequalities for |A₄| for f∈ℬ1(α,λ) are complicated, and in the interest of brevity, we omit many of the detailed calculations. Also, to simplify the analysis and presentation of the results, we define Γ_i(α) for i = 1, 2, 3 as follows:

We also denote the positive real root of the equation 21 + 17α − 2α³ = 0 by α1⁎=3.40366…, α2⁎=12(1+33)=3.37228…, and the positive real root of the equation 4 + 9α − α³ = 0 by α3⁎=3.20147….

Also,

All the inequalities are sharp.

We note that using the aforementioned lemmas, Theorem 3.2 proves that sharp inequalities for |A₄| are established for α ≥ 0 and 0 < λ ≤ 1, apart from the intervals 0≤α≤α3⁎ and Γ₃(α) < λ ≤ 1, where the methods used fail.

To find the maximum of the modulus of eq. (13), we first use Lemmas 1.3 and 1.4.

Let

and

To see that eq. (11) holds in cases (i)–(iv), we use Lemma 1.4, noting that a long computation shows that both 0 ≤ B ≤ 1 and B(2B − 1) ≤ D ≤ B are valid in all cases. This proves inequality (11).

For inequality (12), we write eq. (13) as

In this case, we can therefore apply Lemma 1.4, provided that both 0 ≤ B ≤ 1 and D − B ≥ 0 are valid, and again a long computation shows that these inequalities are valid in cases (v) and (vi).

A simple calculation shows that

and so we obtain from Lemma 1.4 and the inequality |c₁| ≤ 2 that

We are therefore left to prove eq. (12) in case (vii), where we use Lemma 1.3, with

Then, μ > 1 when 0≤α<α2⁎, and Γ₂(α) < λ ≤ 1, and D − μ ≥ 0 when α3⁎≤α<α2⁎ and Γ₂(α) < λ ≤ 1, and so both inequalities are satisfied when α3⁎≤α<α2⁎ and Γ₂(α) < λ ≤ 1.

Thus, writing

Lemma 1.3, and the inequality |c₁| ≤ 2, gives

provided α3⁎≤α<α2⁎ and Γ₂(α) < λ ≤ 1, which gives inequality (12) in case (vii).□

4 The logarithmic coefficients

The logarithmic coefficients γ_n of f are defined in D by

Differentiating eq. (15) and equating coefficients give

Using the same techniques as in the proof of Theorem 2.1, it is possible to prove the following (proofs can be found in [21]).

5 The second Hankel determinant

The qth Hankel determinant of f is defined for q ≥ 1 and n ≥ 1 as follows and has been extensively studied (see e.g. [22,23,24,25])

We prove the following, noting that the result is valid for α ≥ 0.

Next applying Lemma 1.1, noting that H₂(2) is rotationally invariant, and again writing c1:=c, so that 0 ≤ c ≤ 2, it follows that

Taking the modulus in eq. (19), and noting that |η| ≤ 1, gives

Since the derivative of ψ(α, λ, |ζ|, c) with respect to |ζ| is positive, we deduce from eq. (20) that

Thus, we must find the maximum value of ψ₁(α, λ, c), when 0 ≤ c ≤ 2.

Elementary calculus shows that ψ′1(α,λ,c)=0 has three roots in 0 ≤ c ≤ 2, but the only valid root is at c = 0.

Since ψ1(α,λ,0)=λ2(2+α)2 and ψ1(α,λ,2)=λ4|1−α|12(1+α)3, the proof of the theorem is complete on noting that ψ₁(α, λ, 0) ≥ ψ₁(α, λ, 2) when α ≥ 0 and 0 < λ ≤ 1.

The inequality is sharp on choosing c₁ = 0 and c₂ = c₃ = 2 in eq. (18).□

6 A Fekete–Szegö theorem

We finally give a sharp Fekete–Szegö inequality for B₁(α,λ) omitting the proof, which is a straightforward application of Lemma 1.2.