The Booth Lemniscate Starlikeness Radius for Janowski Starlike Functions

Abstract

The function G α (z) = 1 + z/(1 − αz 2 ), 0 ≤ α < 1, maps the open unit disc D onto the interior of a domain known as the Booth lemniscate. Associated with this function G α is the recently introduced class BS(α) consisting of normalized analytic functions f on D satisfying the subordination zf ′ (z)/f (z) ≺ G α (z). Of interest is its connection with known classes M of functions in the sense g(z) = (1/r)f (rz) belongs to BS(α) for some r in (0, 1) and all f ∈ M. We find the largest radius r for different classes M, particularly when M is the class of starlike functions of order β, or the Janowski class of starlike functions. As a primary tool for this purpose, we find the radius of the largest disc contained in G α (D) and centered at a certain point a ∈ R.