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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.

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For a function f starlike of order α, 0 α < 1, a non-constant polynomial Q of degree n which is non-vanishing in the unit disc D and β > 0, we consider the functionand find the largest value of r ∈ (0, 1] such that r −1 F (rz) lies in various known subclasses of starlike functions such as the class of starlike functions of order λ, the classes of starlike functions associated with the exponential function, cardioid, a rational function, nephroid domain and modified sigmoid function. Our radii results are sharp. We also discuss the correlation with known radii results as special cases.

For given non-negative real numbers α k with m k=1 α k = 1 and normalized analytic functions f k , k = 1, . . . , m, defined on the open unit disc, let the functions F and F n be defined bywhere the function h is a convex univalent function with positive real part. We also consider the analogues of the classes of starlike functions with respect to symmetric, conjugate, and symmetric conjugate points. Inclusion and convolution results are proved for these and related classes. Our classes generalize several well-known classes and connection with the previous works are indicated.