On the Difference of Coefficients of Bazilevič Functions

Let f be analytic in the unit disk D={z∈C:|z|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$\end{document}, and S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}$$\end{document} be the subclass of normalized univalent functions given by f(z)=z+∑n=2∞anzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$\end{document} for z∈D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\in {\mathbb {D}}$$\end{document}. We give bounds for ||a3|-|a2||\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$| |a_3|-|a_2| | $$\end{document} for the subclass B(α,iβ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal B}(\alpha ,i \beta )$$\end{document} of generalized Bazilevič functions when α≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ge 0$$\end{document}, and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} is real.


Introduction
Let A denote the class of analytic functions f in the unit disk D = {z ∈ C : |z| < 1} normalized by f (0) = 0 = f (0) − 1. Then for z ∈ D, f ∈ A has the following representation f (z) = z + ∞ n=2 a n z n . (1.1) Let S denote the subclass of all univalent (i.e., one-to-one) functions in A.
In 1985, de Branges [2] solved the famous Bieberbach conjecture by showing that if f ∈ S, then |a n | ≤ n for n ≥ 2, with equality when f (z) = k(z) := z/(1 − z) 2 , or a rotation. It was therefore natural to ask if for f ∈ S, the inequality ||a n+1 | − |a n || ≤ 1 is true when n ≥ 2. This was shown not to be the case even when n = 2 [4], and that the following sharp bounds hold.
where λ 0 is the unique value of λ in 0 < λ < 1, satisfying the equation 4λ = e λ . Hayman [6] showed that if f ∈ S, then ||a n+1 | − |a n || ≤ C, where C is an absolute constant. The exact value of C is unknown, best estimate to date being C = 3.61 . . . [5], which because of the sharp estimate above when n = 2, cannot be reduced to 1.
Denote by S * the subclass of S consisting of starlike functions, i.e. functions f which map D onto a set which is star-shaped with respect to the origin. Then it is well-known that a function f ∈ S * if, and only if, for z ∈ D It was shown in [8], that when f ∈ S * , then ||a n+1 | − |a n || ≤ 1 is true when n ≥ 2.
Next denote by K the subclass of S consisting of functions which are close-toconvex, i.e. functions f which map D onto a close-to-convex domain. Then again it is well-known that a function f ∈ K if, and only if, there exists g ∈ S * such that Koepf [7] showed that if f ∈ K, then ||a n+1 | − |a n || ≤ 1, when n = 2, but establishing this inequality when n ≥ 3 remains an open problem.
In 1955, Bazilevič [1] extended the notion of starlike and close-to-convex functions by showing that if f ∈ A, and is given by (1.1), then if α > 0 and β ∈ R, f given by where g ∈ S * , and p ∈ P, the class of functions with positive real part in D, then functions defined by (1.3) form a subset of S. Such functions are known as Bazilevič functions. We note that in the original definition of Bazilevič functions [1], Bazilevič assumed that α > 0, however Sheil-Small [10], subsequently showed that when α = 0, such functions also belong to S, and satisfy where p ∈ P.
Although much is known about the initial coefficients of functions in B 1 (α), there appears to be no published information concerning the difference of coefficients. We also note that B 1 (1, 0) reduces to the class of functions in R such that their derivatives have positive real part for z ∈ D, and that the class B 1 (1, iβ) has been little studied.

Preliminary Lemmas
Denote by P, the class of analytic functions p with positive real part on D given by (2.1) We will use the following properties for the coefficients of functions P, given by (2.1).

Lemma 2.2 [3]
If p ∈ P, then and for some ζ i ∈ D, i ∈ {1, 2}. For ζ 1 ∈ T, the boundary of D, there is a unique function p ∈ P with p 1 as in (2.2), namely, For ζ 1 ∈ D and ζ 2 ∈ T, there is a unique function p ∈ P with p 1 and p 2 as in (2.2) and (2.3), namely, We will also need the following well-known result.

Lemma 2.3 [7, Lem. 3]
Let g ∈ S * and be given by g(z) = z + ∞ n=2 b n z n . Then for any λ ∈ C, The inequality is sharp when g

The class B(˛, iˇ)
We begin by proving the following inequalities for f ∈ B(α, iβ).
Thus in order to establish the upper bound in (3.1), we use (3.10) and (3.11), and need to show that and We first obtain (3.12). Since

12) holds provided
Clearly A 1 ≤ A 2 is true when α = 0. For α > 0, using the inequalities it follows that (3.14) We next note that the following is valid provided α ∈ [0, ( A similar process to the above gives which proves inequality (3.13), and so the proof of Theorem 3.1 is complete.

The inequality is sharp when both the functions f and g are the Koebe function.
We end this section by noting from the definition, since B 1 (0, iβ) ≡ B(0, iβ), the following is an immediate consequence of Theorem 4.1 below.

Theorem 3.2 Let f ∈ B(0, iβ), and be given by
Both inequalities are sharp.
Thus the proof of the inequalities for |a 3 | − |a 2 | is complete.