Gamma-Bazilevič Functions

and by S , the subclass of A consisting of functions which are univalent in D. A function f ∈ S is said to be convex if f maps D onto a convex set, and starlike if f maps D onto a set star-shaped with respect to the origin. Let C and S∗ denote the classes of convex and starlike functions in S respectively. Then f ∈ C if and only if Re (1 + (z f ′′(z)/ f ′(z))) > 0 for z ∈ D. Similarly, f ∈ S∗ if and only if Re (z f ′(z)/ f (z)) > 0 for z ∈ D. For α ∈ R, the classMα of α-convex functions defined by,


Introduction and Definitions
Denote by A the class of normalized analytic functions f , defined in the unit disk D, and given by f (z) = z + ∞ ∑ n=2 a n z n , (1) and by S, the subclass of A consisting of functions which are univalent in D.
A function f ∈ S is said to be convex if f maps D onto a convex set, and starlike if f maps D onto a set star-shaped with respect to the origin. Let C and S * denote the classes of convex and starlike functions in S respectively. Then f ∈ C if and only if Re (1 + (z f (z)/ f (z))) > 0 for z ∈ D. Similarly, f ∈ S * if and only if Re (z f (z)/ f (z)) > 0 for z ∈ D.
For α ∈ R, the class M α of α-convex functions defined by, for z ∈ D and f (z) z f (z) = 0 is well known. Introduced by Miller, Mocanu and Reade [1], many interesting properties for functions in M α have been found (See e.g., [2,3]).
Denote by M γ the analogue of M α in term of powers, defined for γ ∈ R by for z ∈ D. The class M γ was introduced in [4], and many interesting properties of functions in M γ have been found. It was shown in [4] that M γ is a subset of S * . Further, sharp bounds for |a 2 |and|a 3 | were obtained, together with the sharp Fekete-Szegö theorem. Other result can be found in [5,6]. The purpose of this paper is to introduce an analogue of M γ for Bazilevič functions. We first recall the Bazilevič functions B 1 (α) introduced by Singh in 1973, which form a natural subset of S as follows [7]. Definition 1. Let f ∈ A. Then for α ≥ 0, f ∈ B 1 (α) if, and only if, for z ∈ D, We next introduce the Gamma-Bazilevič functions as follows, noting that we restrict our definition γ ≥ 0 merely for convenience.
Definition 2. Let f ∈ A, with f (z) = 0 and f (z) = 0. For γ ≥ 0 and α ≥ 0, a function f ∈ A is said to be Gamma-Bazilevič if, for z ∈ D, We denote this class by B γ 1 (α).
We also note that when α = 1 and γ = 0, we obtain the class R of functions whose derivative has a positive real part, and that when α = 0 and γ = 0 we obtain the starlike functions, and when α = 0 and γ = 1 we obtain the convex functions.
We also note that when γ = 1, we obtain the following new class B γ 1 (1), which forms a subset of R.

Preliminaries
We begin by stating two Lemmas which we will use in what follows.
Let P be the class of function h satisfying Re h(z) > 0 for z ∈ D, with expansion We shall use the following results concerning the coefficients c n of h ∈ P, which can be found in [9].

Lemma 2.
If h ∈ P and be given by (2), then |c n | ≤ 2 for n ≥ 1, and
where p(z 0 ) = −ia and k ≤ − 1 2 a + 1 a . Therefore, we have a contradiction. There is thus no point

Initial Coefficients
We first find expressions for a 2 and a 3 in terms of the coefficients of h ∈ P. It follows from Definition 2 that we can write, where h ∈ P. Equating coefficients in (3) gives , We now extend coefficient results given in [6] for the coefficients of M γ and the results of Singh [7] for B 1 (α), noting that the bounds for |a 2 | and |a 3 | hold for all γ ≥ 0 and α ≥ 0.
Proof. The first inequality in Theorem 2 follows at once from (4) since |c 1 | ≤ 2. For |a 3 |, from (4) we use Lemma 2, and write Then in Lemma 2, let so that applying Lemma 2 gives the inequalities for |a 3 |. The inequality for |a 2 | is sharp when c 1 = 2. The first inequality for |a 3 | is sharp when c 1 = 0 and c 2 = 2, and the second inequality for |a 3 | is sharp when c 1 = c 2 = 2, which completes the proof of Theorem 2.
Proof. From (4) we obtain Applying Lemma 2, µ ∈ [0, 2] whenever gives the second inequality. When µ outside [0, 2], Lemma 2 gives the first inequality when and the third inequality when The second inequality is sharp when c 1 = 0 and c 2 = 2. The first and third inequalities are sharp when c 1 = c 2 = 2. This completes the proof of Theorem 3.

Logarithmic Coefficients
The logarithmic coefficients g n of f are defined in D by Differentiating (5) and equating coefficients gives ).
The result for |g 2 | follows at once from the above Fekete-Szegö theorem in the case µ = 1/2. For the first inequality, we use the second inequality in Theorem 3, and for the second inequality we use the first inequality in Theorem 3.
We note that the inequality for |g 1 | is sharp when c 1 = 2. The first inequality for |g 2 | is sharp when c 2 = 2 and c 1 = 0, and the second inequality is sharp when choosing c 1 = c 2 = 2. This completes the proof of Theorem 4.

Inverse Coefficients
For any univalent function f there exists an inverse function f −1 defined on some disc |ω| < r 0 ( f ), with Taylor expansion Suppose that B γ 1 (α) −1 is the set of inverse functions f −1 of B γ 1 (α), given by (6). Then f ( f −1 (ω)) = ω, and equating coefficients gives We prove the following, noting again that the inequalities for |A 2 | and |A 3 | hold for all γ ≥ 0 and α ≥ 0 thus extending results extend in [10] and [6].

Remark 2.
Clearly finding sharp bounds for |a 4 | and |A 4 | appears to be far more difficult, and requires significantly more analysis. We note that applying the often used lemmas in [9] fails to give sharp results.
We also note that even when γ = 1, the analysis for |a 4 | and |A 4 | is far from simple, and appears to require methods deeper than those used or mentioned in this paper.