Some properties for certain classes of univalent functions defined by differential inequalities

Let A be the space of functions analytic in the unit disk D={z: |z| < 1}. Let U denote the set of all functions f ? A satisfying the conditions f(0)=f′(0)?1=0 and $\left|f^{\prime }\left(z\right){\left(\frac{z}{f\left(z\right)}\right)}^2-1\right|<1\left(\left|z\right|<1\right)$. Also, let Ω denote the set of all functions f ? A satisfying the conditions f(0) = f′(0)-1 = 0 and $\left|z{f}^{\prime }\left(z\right)-f\left(z\right)\right|<\frac{1}{2}\left(\left|z\right|<1\right)$. In this article, we discuss the properties of U and Ω.