MA-MINDA CONVEXITY AND STARLIKENESS FOR CERTAIN SUBCLASSES OF THE CLOSE-TO-STARLIKE FUNCTIONS
Abstract
An analytic function f (z) = z + a2z2 + · · · defined on the unit disc D is close-to-starlike if there exists a starlike function g : D → C satisfying the inequality Re(f (z)/g(z)) > 0 for all z ∈ D. We are particularly interested in the starlikeness of the class W, which consists of all functions that satisfy the close-to-starlike condition with g(z) ≡ z and the subclass Wn of W,
which contains all those functions of the form f (z) = z + an+1zn+1 + · · · . The usual starlikeness of an analytic function f requires that the range of zf ′(z)/f (z) is contained in the right half-plane. More generally, a normalized analytic function f : D → C is Ma-Minda starlike if the function
zf ′/f is subordinate to the function φ and Ma-Minda convex if the function 1 + zf ′′/f ′ is subordinate to the function φ. We have determined the sharp radius of Ma-Minda convexity/starlikeness of the class W and Wn when the range of φ is a nephroid, lune, lemniscate of Bernoulli, cardioid,
or, a particular rational function.
which contains all those functions of the form f (z) = z + an+1zn+1 + · · · . The usual starlikeness of an analytic function f requires that the range of zf ′(z)/f (z) is contained in the right half-plane. More generally, a normalized analytic function f : D → C is Ma-Minda starlike if the function
zf ′/f is subordinate to the function φ and Ma-Minda convex if the function 1 + zf ′′/f ′ is subordinate to the function φ. We have determined the sharp radius of Ma-Minda convexity/starlikeness of the class W and Wn when the range of φ is a nephroid, lune, lemniscate of Bernoulli, cardioid,
or, a particular rational function.
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