New extensions for classes of analytic functions associated with close-to-convex and starlike of order [formula omitted

Abstract

Let A be the class of functions f ( z ) with f ( 0 ) = 0 and f ′ ( 0 ) = 1 in the open unit disk U . For f ( z ) ∈ A , f ( z ) is said to be in the class A ( β 1 , β 2 , β 3 ; λ ) if it satisfies | β 1 z ( f ( z ) z ) ′ + β 2 z 2 ( f ( z ) z ) ″ + β 3 z 3 ( f ( z ) z ) ‴ | ≦ λ ( z ∈ U ) for some complex numbers β 1 , β 2 and β 3 , and for some real λ > 0 . Some interesting properties for f ( z ) ∈ A ( β 1 , β 2 , β 3 ; λ ) associated with close-to-convex functions and starlike functions of order α are discussed.