Generalized Sequences for a Subfamily of Univalent Functions

Functions ƒ(z) = z + … that are analytic in the unit disk and satisfy there the inequality Re(ƒ′(z) + zƒ″(z))>0 are known to be univalent. We investigate associated functions ƒ̃(z) = z + ∑∞k = 2ankznk formed by choosing subsequences {nk} of {n}. We determine the largest disk |z| < r0 in which we are guaranteed that ƒ̃ is univalent and close-to-convex.