Let A be the class of functions f of the form

which are analytic and normalized by f(0)=f′(0)−1=0 in the open unit disk Δ={z∈ℂ:|z|<1}. The subclass of A of all univalent functions f in Δ is denoted by S. We denote by P the well-known class of analytic functions p with p(0) = 1 and Re(p(z))>0, z∈Δ. We also denote by B the class of analytic functions w(z) in Δ with w(0)=0 and |w(z)|<1, z∈Δ. If f and g are two functions in A, then we say that f is subordinate to g, written f(z)≺g(z), if there exists a function w∈B such that f(z)=g(w(z)) for all z∈Δ. As a special case, if the function g is univalent in Δ, then we have the following equivalence:

A function f∈S is starlike (with respect to 0) if tw∈f(Δ) whenever w∈f(Δ) and t∈[0,1]. The class of starlike functions is denoted by S*. We say that f∈S*(γ) (0≤γ<1) if and only if

The equality S*(0)=S* is well known. Recently Kargar et al. (see [4]) introduced a certain subclass of starlike functions as follows.

Definition 1.1. Let π/2≤α<π. Then the function f∈A belongs to the class M(α) if f satisfies

Consider the function ϕ as follows

It is clear that ϕ(π/2)=1−π/4≈0.2146 and

Thus, the class M(α) is a subclass of the class f∈S*(ϕ(π/2)) of starlike functions of order ϕ(π/2)=1−π/4.

By the subordination principle we have the following lemma.

Lemma 1.2. [4] Let f(z)∈A and π/2≤α<π. Then f∈M(α) if and only if

where

The function Bα(z) is convex univalent in Δ and maps Δ onto

in other words, the image of Δ is a vertical strip when π/2≤α<π. For other α, Bα(z) is convex univalent in Δ and maps Δ onto the convex hull of three points (one of which may be that point at infinity) on the boundary of Ωα. Therefore, in other cases, we obtain a trapezium, or a triangle, see [3]. Also, we have that

where

The following lemma will be useful.

Lemma 1.3. (see [9]) Let q(z)=∑ n=1∞Qnzn be analytic and univalent in Δ, and suppose that q(z) maps Δ onto a convex domain. If p(z)=∑ n=1∞Pnzn is analytic in Δ and satisfies the following subordination

then

This paper is organized as follows. In Section 2 we study the class M(α). We consider the coefficient estimates and Fekete-Szegö inequality. Also, in Section 3 we introduce a certain subclass Mσ(α) of bi--univalent functions and we estimate the initial coefficients of functions belonging to Mσ(α).

Theorem 2.1. ([10]) Let π/2≤α<π. If a function f∈A of the form (1.1) belongs to the class M(α), then

Here, we considerthe problem of finding sharp upper bounds for the Fekete-Szegö coefficient functional associated with the k-th root transform for functions in the class M(α). For a univalent function f(z) of the form (1.1), the k-th root transform is defined by

In order to prove next result, we need the following lemma due to Keogh and Merkes [5].

Lemma 2.2. [5] Let the function g(z) given by

be in the class P. Then, for any complex number µ

The result is sharp.

Theorem 2.3. Let π/2≤α<π. Suppose also that f∈M(α) and let F be the k-th root transform of f defined by (2.2). Then, for any complex number µ,

The result is sharp.

Proof. Since f∈M(α), from Lemma 1.2 and by definition of subordination, there exists a function w∈B such that

We define

and note that p∈P. Relationships (1.6) and (2.5) give us

where A₁=1 and A2=−cosα. If we equate the coefficients of z and z² on both sides of (2.4), then we get

and

Foreach f given by (1.1) and with a simple calculation we have

Moreover by (2.2) and (2.9), we obtain

By inserting (2.7) and (2.8) into (2.10), we get

and

Therefore,

Applying Lemma 2.2 in (2.11) with

gives the inequality (2.3). For the sharpness it is sufficient to consider the k-th root transforms of the function

It is clear that f∈M(α). If we take in (2.12) w(z)=z, then from (2.5) we obtain p1=p2=2 hence from (2.11) we get

If we take in (2.12) w(z)=z2, then from (2.5) we obtain p₁=0 while p₂=2 hence from (2.11) we get for this case

It shows the sharpness of (2.3) and ends the proof.

The problem of finding sharp upper bound for the coefficient functional |a3−μa22| for different subclasses of the class A is known as the Fekete-Szegö problem. Putting k=1 in the Theorem (2.3) gives us:

Corollary 2.4. Let α∈[π/2,π). Suppose also that f∈M(α). Then, for any complex number µ,

The result is sharp.

Putting α=π/2, in the Corollary 2.4, we get:

Corollary 2.5. Assume that the function f given by (1.1) satisfies in the following two-sided inequality:

then

If we take α→π− in the Corollary 2.4, then we have:

Corollary 2.6. Assume that the function f given by (1.1) satisfies in the following inequality:

then

Corollary 2.7. Letthe function f, given by (1.1), be in the class M(α). Also let the function f−1(w)=w+∑ n=2∞bnwn be the inverse of f. Then

and

We remark that every function f∈S has an inverse f−1, defined by f−1(f(z))=z (z∈Δ) and

where

Proof Comparing (2.18) with f−1(w)=w+∑ n=2∞bnwn, gives us

Applying Theorem 2.1 we get

The second inequality (2.17) follows by taking µ=-2 in the Corollary 2.4.

First, we recall that a function f∈A is said to be bi-univalent in Δ if f univalent in Δ and f^-1 has an univalent extension from |w|<r0<1 to Δ. We denote by σ the class of bi-univalent functions in the unit disk Δ.

In 1967 Lewin [6] introduced the class σ of bi-univalent functions. He obtained the bound for the second coefficient. Recently, several authors have subsequently studied similar problems in this direction (see [2, 7]). For example, Brannan and Taha [1] considered certain subclasses of bi-univalent functions, similar to the familiar subclasses of univalent functions including of strongly starlike, starlike and convex functions. They introduced bi-starlike functions and bi-convex functions and obtained estimates on the initial coefficients.

In this section we introduce by Mσ(α) a certain subclass of bi-starlike functions as follows. Also, we obtain the bound for the initial coefficients.

Definition 3.1. A function f∈σ is said to be in the class Mσ(α), if the following inequalities hold:

and

where g(w)=f−1(w) and π/2≤α<π.

For functions in the class Mσ(α), the following result is obtained.

Theorem 3.2.Let the function f∈A of the form (1.1) belongs to the class Mσ(α). Then

and

Proof. Let f∈Mσ(α) and g=f−1. Then using Lemma 1.2, there are analytic functions u,v∈B, satisfying

where Bα(.) defined by (1.4). Define the functions k and l by

or, equivalently,

and

It is clear that the functions k(z) and l(z)belong to class P and we have |ki|≤2 and |li|≤2 (i=1,2,…) (see [8]). However, clearly

From (1.6), (3.6) and (3.7), we have

and

where A₁=1 and A2=−cosα, are given by (1.7). By suitably comparing coefficients of (3.5), we get

and

From (3.11) and (3.13), we get

Also, from (3.12)-(3.15), we find that

Therefore, we have

Thisgives the bound on |a2| as asserted in (3.3). Now, further computations from (3.12) and (3.14)-(3.16) lead to

Since |ki|≤2 and |li|≤2, we have

Therefore, the proof of Theorem 3.2 is completed.

Corollary 3.3. Let the function f be in the class Mσ(π/2). Then

and

Also, if we take α→π−, in Theorem 3.2 we get