The Fifth Coefﬁcient of Bazileviˇc Functions

: Let f ∈ A , the class of normalized analytic functions deﬁned in the unit disk D , and be given by f ( z ) = z + ∑ ∞ n = 2 a n z n for z ∈ D . This paper presents a new approach to ﬁnding bounds for | a n | . As an application, we ﬁnd the sharp bound for | a 5 | for the class B 1 ( α ) of Bazileviˇc functions when α ≥ 1.


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 Denote by S, the subset of A consisting of univalent functions in D.
We remark at the outset that in a great number of the more familiar subclasses of S, sharp bounds have been found for the coefficients |a n |, when 2 ≤ n ≤ 4, but bounds when n = 5 and beyond are much more difficult to obtain.(See, e.g., [1]).
Denote by S * , the class of starlike functions defined as follows.
Definition 1.Let f ∈ A. Then f ∈ S * if, and only if, for z ∈ D, An application of the method introduced in this paper to estimate the fifth coefficient of functions in A, concerns the B 1 (α) Bazilevič functions defined as follows.
Definition 2. Let f ∈ A. Then f ∈ B 1 (α) if, and only if, for α ≥ 0, and z ∈ D, We note that B 1 (0) = S * , and each of the above classes are necessarily subclasses of S. Apart from α = 0, where |a n | ≤ n for n ≥ 2, we also note that sharp bounds for |a n | are known for f ∈ B 1 (α) when α ≥ 0 only when 2 ≤ n ≤ 4, [2], and only partial solutions are known for |a n | when n ≥ 5 [3,4] for α ≥ 1.
It was conjectured in [4], that when α ≥ 1, the sharp bound for |a n | when n ≥ 2 is given by and a partial solution to this problem in the case n = 5 was given in [3].
In this paper, we illustrate our method by giving a complete solution to finding the sharp bound for |a 5 | when f ∈ B 1 (α) for α ≥ 1.

Auxiliary Results
Denote by P, the class of analytic functions p with positive real part on D given by Lemma 1 ([5]).If the functions belong to P, then the same is true of the function We first outline the method of proof.Let p ∈ P be in the form (3), and with B i ∈ C, i ∈ {1, 2, 3, 4}.Assume that there exists q ∈ P of the form q Then 1 + G(z) ∈ P, and v n = b n p n /2, n ∈ N.
Assume that the function q where Ψ and A 4 are given by ( 5) and (6), respectively.
Since the system of equations in (B) has many solutions, we now place some restrictions on the parameters sufficient for our purpose.
We fix Then if τ ∈ [−1, 1], q ∈ P and is given by q We also assume that ζ i , i ∈ {1, 2, 3} take real values.Then the identities for u i , i ∈ {1, 2, 3} become Thus we are able to conclude the following.(C) Let p ∈ P be in the form (3).

The Fifth Coefficient of B 1 (α) Bazilevič Functions
Lemma 4. ([9] Cohn's rule) Let t(z) = a 0 + a 1 z + • • • + a n z n be a polynomial of degree n and Let r and s be the number of zeros of t inside and on the unit circle |z| = 1, respectively.If is a polynomial of degree n − 1 and has r 1 = r − 1 and s 1 = s number of zeros inside and on the unit circle |z| = 1, respectively.
We now use the above method to find the sharp bound for |a 5 | when α ≥ 1.
Thus it is enough to show that |Ψ| ≤ 2.
When α = 1, it is clear that |Ψ| ≤ 2 holds trivially, and so we can assume that α > 1. Put , and Now let ω ∈ B be defined by (12).Thus the function k defined by k = L • ω, where L is given by (8), belongs to P, with k(0) = 1. Setting and With n (n ∈ N) as in (4), we obtain Now define q by and We shall show that q belongs P. Let where A and B are polynomials of degree 3, and defined by  Let f ∈ B 1 (α) be the function defined by (13).Then, by equating coefficients in (13), we get a 2 = a 3 = a 4 = 0 and a 5 = 2/(4 + α), which shows that this result is sharp.This completes the proof of Theorem 1.