Some Properties for Subordinations of Analytic Functions

Let the class of functions of $f\left(z\right)$ of the form $f\left(z\right)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k$, which are denoted by $\mathcal{A}$ and called analytic functions in the open-unit disk. There are many interesting properties of the functions $f\left(z\right)$ in the class $\mathcal{A}$ concerning the subordinations. Applying the three lemmas for $f\left(z\right)\in \mathcal{A}$ provided by Miller and Mocanu and by Nunokawa, we consider many interesting properties of $f\left(z\right)\in \mathcal{A}$ with subordinations. Furthermore, we provide simple examples for our results. We think it is very important to consider examples of the results.