New sufficient conditions for <i>p</i>-valent functions

Let A(p) be the class of functions f(z) of the form f(z) = z(p )+ a(p+1)z(p+1)+a(p+2)z(p+2 )+ , (p is an element of N = {1,2,3, ... }) that are analytic in the open unit disc U = {z is an element of C:|z| < 1}. For f(z) is an element of A(p), Nunokawa considered some conditions such that f(z) is p-valent in U. Applying the results by Nunokawa, we discuss some interesting properties for functions f(z) is an element of A(p). Also, we give some examples for our results.