Subclasses of Starlike and Convex Functions Associated with the Limaçon Domain

Abstract

Let ST L ( s ) and CV L ( s ) denote the family of analytic and normalized functions f in the unit disk D : = z : | z | 1 , such that the quantity z f ′ ( z ) / f ( z ) or 1 + z f ″ ( z ) / f ′ ( z ) respectively are lying in the region bounded by the limaçon ( u − 1 ) 2 + v 2 − s 4 2 = 4 s 2 u − 1 + s 2 2 + v 2 , where 0 s ≤ 1 / 2 . The limaçon of Pascal is a curve that possesses properties which qualify it for the several applications in mathematics, statistics (hypothesis testing problem) but also in mechanics (fluid processing applications, known limaçon technology is employed to extract electrical power from low-grade heat, etc.). In this paper we present some results concerning the behavior of f on the classes ST L ( s ) or CV L ( s ) . Some appropriate examples are given.

Keywords

• univalent functions
• subordination
• starlike and convex functions
• limaçon of Pascal