Let A(n) be the class of functions $f(z)=a_nz^n+a_{n+1^{z^{n+1}}+...(ninmathbb{N})}$ which are analytic in the open unit disk U, where an ̸= 0. For f(z) 2 A(n) , Miller and Mocanu in 1978 showed a veryinteresting result for f(z) . Applying the result due to Miller and Mocanu, we would like to consider some new resultsfor such functions. Our results in this paper are generalizations for results by Nunokawa in 1992.