Properties of functions concerned with Caratheodory functions

Let ${\mathcal{P}}_n$ denote the class of analytic functions $p(z)$ of the form $p(z)=1+c_n{z}^n+c_{n+1}{z}^{n+1}+\dots$ in the open unit disc $\mathbb{U}$. Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p(z)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.