Subordination of functions in subclass of Bazilevič Functions B 1(α, β)

Let f be analytic in the unit disc D = {z : |z| < 1} with f(z)=z+∑n=2∞anzn , and for α ≥ 0 and 0 < β ≤ 1, let B 1(α, ß), denote for the class of Bazilevič functions satisfying the expression |argz1−αf′(z)f(z)1−α|<βπ2 . We give sharp estimates for various coefficient problems for functions in B 1(α, β), which unify and extend well-known results for starlike functions, strongly starlike functions and functions whose derivative has positive real part in domain D.


Definitions and Some Preliminaries
We will first recall the Bazilevič functions B(α) as follows [4].
Let S be the class of analytic and normalized univalent functions f defined in z ∈ D = {z : |z| < 1} and given by the following f (z) = z + ∞ n=2 a n z n . (1) Then for α ≥ 0, f ∈ B(α) ⊂ S if, and only if, there exists a starlike function g in z ∈ D, such that Taking g(z) ≡ z gives the class B 1 (α) of Bazilevič Functions with logarithmic growth. We note that B 1 (0) is the class of starlike functions S * , and B 1 (1) the well-known class R of functions whose derivative has positive real part in D.
Thus f ∈ B 1 (α), if and only if,  2 We know that various best possible properties of these problems have been obtained for the class B 1 (α). Amongst other results in this topic, fortunately, Singh [16], found sharp estimates for the moduli of the first four coefficients, and he also obtained the solution to the expression known as the Fekete-Szegö problem. The author of [22] has recently obtained sharp bounds for the second Hankel determinant, the initial coefficients of the function with the form log f (z) z , and also obtained the initial coefficients of the inverse function f −1 . Distortion theorems and some length-area results were also obtained in London, Singh, and Thomas [13,19,20].
For 0 < β ≤ 1, let B 1 (α, β) be the set of functions f , given by (1) satisfying Then B 1 (0, β) is the class of strongly starlike functions SS * and B 1 (1, β) the class of functions such that Re f (z) lies in a sector, which are extensions of the classical sets of starlike functions S * and R respectively.
It follows from (2) that we can write where h ∈ P , the class of function satisfying Re h(z) > 0 for z ∈ D = {z : |z| < 1}. Let We shall use the following results, see e.g. [1,2,6,9]

Lemma 1
If h ∈ P with coefficients c n as above, then for some complex valued x with |x| ≤ 1 and some complex valued ζ with |ζ| ≤ 1,

Lemma 2
If p ∈ P , then

Lemma 4
Let h ∈ P with coefficients c n as above, then,
Proof. Equating coefficients in (3) gives The first inequality in Theorem 1 follows at once since |c 1 | ≤ 2.
Applying Lemma 2 now gives the inequalities for |a 3 |.
For the coefficient a 4 , we use Lemmas 2, 3, and 4.
By applying Lemma 3 gives the first and last inequalities for |a 4 |.
The inequality for |a 2 | is sharp when c 1 = 2. The first and third inequalities for |a 3 | are sharp when c 1 = 0 and c 2 = 2, and the second inequality for |a 3 | is sharp when c 1 = c 2 = 2. For |a 4 |, the first and third inequalities are sharp when c 1 = c 2 = 0 and c 3 = 2 and the second inequality is sharp when c 1 = c 2 = c 3 = 2. The proof is complete.

Theorem 3
Proof. A simple application of Lemma 2 gives the result. The proof is complete.

Remark 1
Obtaining the sharp upper bounds for |a n | for all n ≥ 5 for f ∈ B 1 (α, β) remains an open question, even in the case β = 1. It was shown in [17], Marjono [15] and Sa'adatul Fitri [18] that when β = 1, |a n | ≤ |B n |, for α = 1 N , where N ≥ 2 is a positive integer and where B n is the general coefficient of the extreme function φ for the class B 1 (α, 1) given by

The Second Hankel Determinant
The q th Hankel determinant of f is defined for q ≥ 1 and n ≥ 1 as follows, and has been extensively studied e.g. [8,9,15,17].
We prove the following, which extends the result in [22], noting that the result is valid for α ≥ 0.
The inequality is sharp.
Proof. Using (4) and simplifying, we obtain . We now use Lemma 1 to write c 2 and c 3 in terms of c 1 , and, without loss in generality, take c 2 = c, where c ∈ [0, 2]. Also for simplicity, we write X = 4 − c 2 and Z = (1 − |x| 2 )ζ, to obtain We now use the triangle inequality to obtain First assume that there is a critical point at (c 0 , |x 0 |) inside I. Since each term of the derivative of φ(c, |x|) with respect to |x| contains the expression 4 − c 2 , equating to zero gives a contradiction. Thus any maximum point must occur on the boundary of I.