Abstract:

Let A_n(p)(p,n ∈ N = {1,2,3,...}) denote the class of functions of the form f(z) = z^p+a_p+nz^p+n+... which are analytic in the unit diskE = {z:|z| < 1}. A function f(z) in A(p) is said to be in the class S_p(a,b) if it satisfies zf'(z)...bz for some a and b(-1 ≤ b < a ≤ 1). In this paper, using the method of differential subordinations, we give new criteria for f(z) to be in the classes S_p(a,b)(-1 ≤ b < a ≤ 1), that is, if f(z) ∈ A(p) satisfies f(z) = 0 in 0 < |z| < 1,f'(z) = 0 when μ=1, and zf'(z)..., then f(z) ∈S_p(a,b). Our results improve or extend some results due to Owa, Nunokawa, Padmanabhan, Silverman, Obradovic, Yang and

 others^[3-10]. 

Key words: analytic function, starlikeness, subordination