On a class of p -valent starlike functions of order α

Let Ω denote the class of functions w ( z ) , w ( 0 ) = 0 , | w ( z ) | < 1 analytic in the unit disc ⋃ = { z : | z | < 1 } . For arbitrary fixed numbers A , B , − 1 < A ≤ 1 , − 1 ≤ B < 1 and 0 ≤ α < p , denote by P ( A , B , p , α ) the class of functions p ( z ) = p + ∑ n = 1 ∞ b n z n analytic in ⋃ such that P ( z ) ϵ P ( A , B , p , α ) if and only if P ( z ) = p + [ p B + ( A − B ) ( p − α ) ] w ( z ) 1 + B w ( z ) , w ϵ Ω , z ϵ ⋃ . Moreover, let S ( A , B , p , α ) denote the class of functions f ( z ) = z p + ∑ n = p + 1 ∞ a n z n analytic in ⋃ and satisfying the condition that f ( z ) ϵ S ( A , B , p , α ) if and only if z f ′ ( z ) f ( z ) = P ( z ) for some P ( z ) ϵ P ( A , B , p , α ) and all z in ⋃ . In this paper we determine the bounds for | f ( z ) | and | arg f ( z ) z | in S ( A , B , p , α ) , we investigate the coefficient estimates for functions of the class S ( A , B , p , α ) and we study some properties of the class S ( A , B , p , α ) .