Sharp Criteria for Starlike and Convex Univalent Functions
Abstract
We derive sufficient conditions for starlikeness and convexity of analytic functions in the unit disk associated with bounded derivatives. The results provide visual illustrations of the behavior of functions that satisfy or violate these criteria. Sharp inequalities are obtained and interesting examples of extremal starlike and convex functions are considered. These examples underscore the delicate balance between coefficient growth and geometric properties, and they reinforce the necessity of sharp bounds in the characterization of univalent functions.
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