Fekete-Szegö Problems for Quasi-Subordination Classes

and Applied Analysis 3 Example 1.4. The function f : D → C defined by the following:


Introduction and Motivation
Let A be the class of analytic function f in the open unit disk D {z : |z| < 1} normalized by f 0 0 and f 0 1 of the form f z z ∞ n 2 a n z n . For two analytic functions f and g, the function f is subordinate to g, written as follows: belongs to the class R q φ .
It is known that a function f ∈ A with Re f z > 0 in D is univalent. The above class of functions defined in terms of the quasi-subordination is associated with the class of functions with positive real part.
Functions in the following classes, M q α, φ and L q α, φ are analogous to the αconvex functions of Miller et al. 7 and α-logarithmically convex functions introduced by Lewandowski et al. 8 see also 9 , respectively.
It is well known see 10 that the n-th coefficient of a univalent function f ∈ A is bounded by n. The bounds for coefficient give information about various geometric properties of the function. Many authors have also investigated the bounds for the Fekete-Szegö coefficient for various classes 11-25 . In this paper, we obtain coefficient estimates for the functions in the above defined classes.
Let Ω be the class of analytic functions w, normalized by w 0 0, and satisfying the condition |w z | < 1. We need the following lemma to prove our results. Lemma 1.11 see 26 . If w ∈ Ω, then for any complex number t w 2 − tw 2 1 ≤ max{1; |t|}.

1.15
The result is sharp for the functions w z z 2 or w z z.

Main Results
Although Theorems 2.1 and 2.4 are contained in the corresponding results for the classes M q α, φ and L q α, φ , they are stated and proved separately here because of the importance of the classes.

2.1
and, for any complex number μ, Abstract and Applied Analysis 5 Proof. If f ∈ S * q φ , then there exist analytic functions ϕ and w, with |ϕ z | ≤ 1, w 0 0 and |w z | < 1 such that Since

2.6
Since ϕ z is analytic and bounded in D, we have 27, page 172 By using this fact and the well-known inequality, |w 1 | ≤ 1, we get Further, Then

6
Abstract and Applied Analysis Applying Lemma 1.11 to Observe that 2.14 and hence we can conclude that For μ 0, the above will reduce to the estimate of |a 3 |.

2.17
and, for any complex number μ, Proof. The result follows by taking w z z in the proof of Theorem 2.1.

2.19
and, for any complex number μ, Proof. Observe that when zf ∈ S * q , equality 2.3 becomes and the converse can be verified easily. By the Alexander relation, that is f ∈ C q if and only if zf ∈ S * q , we can obtain the required estimates.
then the following inequalities hold:

2.24
and, for any complex number μ, Abstract and Applied Analysis Theorem 2.6. If f ∈ A belongs to R q φ , then

2.26
and, for any complex number μ, Proof. For f ∈ R q φ , we know that by Definition 1.5 there exist analytic functions ϕ and w, with |ϕ z | ≤ 1, w 0 0 and |w z | < 1 such that

2.28
Since it follows from 2.28 and 2.5 that

2.30
Following the same argument as in Theorem 2.1, where |c 0 | ≤ 1 and |c 1 | ≤ 1, we can deduce that
then the following inequalities hold:

2.35
and, for any complex number μ, Let the class R ρ q φ consist of functions f ∈ A satisfying the quasi-subordination where ρ ∈ C \ {0}. The following corollary gives the results for f ∈ R ρ q φ .

2.38
and, for any complex number μ,