Further results for starlike functions related with Booth lemniscate

In this paper we investigate an interesting subclass $\mathcal{BS}(\alpha)$ ($0\leq \alpha<1$) of starlike functions in the unit disk $\Delta$. The class $\mathcal{BS}(\alpha)$ was introduced by Kargar et al. [R. Kargar, A. Ebadian and J. Sok\'o{\l}, {\it On Booth lemniscate and starlike functions}, Anal. Math. Phys. (2017) DOI: 10.1007/s13324-017-0187-3] which is strongly related to the Booth lemniscate. Some geometric properties of this class of analytic functions including, radius of starlikeness of order $\gamma$ ($0\leq\gamma<1$), the image of $f(\{z:|z|<r\})$ when $f\in \mathcal{BS}(\alpha)$, an special example and estimate of bounds for ${\rm Re}\{f(z)/z\}$ are studied.


Introduction
Let H denote the class of analytic functions in the open unit disk ∆ = {z ∈ C : |z| < 1} on the complex plane C. Also let A denote the subclass of H including of functions normalized by f (0) = f (0) − 1 = 0. The subclass of A consists of all univalent functions f (z) in ∆ is denoted by S. We denote by B the class of functions w(z) analytic in ∆ with w(0) = 0 and |w(z)| < 1, (z ∈ ∆). For two analytic and normalized functions f and g, we say that f is subordinate to g, written f ≺ g in ∆, if there exists a function w ∈ B such that f (z) = g(w(z)) for all z ∈ ∆. In special case, if the function g is univalent in ∆, then f (z) ≺ g(z) ⇔ (f (0) = g(0) and f (∆) ⊂ g(∆)) .
It is easy to see that for any complex numbers λ = 0 and µ, we have: The set of all functions f ∈ A that are starlike univalent in ∆ will be denoted by S * and the set of all functions f ∈ A that are convex univalent in ∆ will be denoted by K. Robertson (see [5]) introduced and studied the class S * (γ) of starlike functions of order γ ≤ 1 as follows We note that if γ ∈ [0, 1), then a function in S * (γ) is univalent. Also we say that f ∈ K(γ) (the class of convex functions of order γ) if and only if zf (z) ∈ S * (γ). In particular we put S * (0) ≡ S * and K(0) ≡ K.
Recently, Kargar et al. [3] introduced and studied a class functions related to the Booth lemniscate as follows.
Definition 1.1. (see [3]) The function f ∈ A belongs to the class BS(α), 0 ≤ α < 1, if it satisfies the condition Recall that [4], a one-parameter family of functions given by are starlike univalent when 0 ≤ α ≤ 1 and are convex for It is clear that the curve is the Booth lemniscate of elliptic type (see Fig. 1, for α = 1/3). For more details, see [3].
In this work, some geometric properties of the class BS(α) are investigated.

Main results
We start with the following lemma that gives the structural formula for the function of the considered class.
Lemma 2.1. The function f ∈ A belongs to the class BS(α), 0 ≤ α < 1, if and only if there exists an analytic function q, q(0) = 0 and q ≺ F α such that The proof is easy. Putting q = F α in Lemma 2.1 we obtain the function which is extremal function for several problems in the class BS(α). Moreover, we consider From (1.7) we conclude that f ∈ BS(α) is starlike of order α α−1 < 0, hence f may not be univalent in ∆. It may therefore be interesting to consider a problem to find the radius of starlikeness of order γ, γ ∈ [0, 1) (hence univalence) of the class BS(α), i.e. the largest radius r s (α, γ) such that each function f ∈ BS(α) is starlike of order γ in the disc |z| < r s (α, γ). For this purpose we recall the following property of the class B.
. The result is sharp.
Recently, one of the interesting problems for mathematician is to find bounds for Re{f (z)/z} (see [2,7]). In the sequel, we obtain lower and upper bounds for Re{f (z)/z}. We first get the following result for the function F (z) given by (2.3).
Proof. Let us define