Some properties of a Subclass of Univalent Functions

Theorem 3.1. Let $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in T_n^{\alpha}(\beta)$ , then for any real parameter $\mu \in \mathbb{R}$ , $$|a_3 - \mu a_2^2| \leq \begin{cases} \frac{2\alpha^{n-1}(1-\beta)}{(\alpha+2)^n} , & \text{if } \mu \leq \frac{\alpha-1}{2}, \\ \frac{2\alpha^{n-1}(1-\beta)}{(\alpha+2)^n} + \frac{2(\alpha-1)\alpha^{2n-2}(1-\beta)^2}{(\alpha+1)^{2n}} [2\mu - (\alpha-1)] , & \text{if } \mu \geq \frac{\alpha-1}{2}. \end{cases}$$

Theorem 4.1. Let $f(z) \in T_n^{\alpha}(\beta)$ and $z = re^{i\theta}$ , where $0 \le \theta \le 2\pi$ , then $$\frac{(r-1)^2 - \alpha^n (1+r)^2}{2r(1+r)} \le \Re\left\{\frac{D^n f^{\alpha}(z)}{z^{\alpha}}\right\} \le \alpha^n \frac{1+r}{1-r}.$$ In order to prove this theorem, we will need the following result which gives a representation of functions in class $T_n^{\alpha}(\beta)$ in terms of an analytic function in the unit disk.

Lemma 4.2. The function f(z), $z \in E$ belongs to the class $T_n^{\alpha}(\beta)$ if and only if there exist a function $\phi(z)$ , analytic in E with the property that $|\phi(z)| \leq 1$ such that $$\frac{D^n f^{\alpha}(z)}{z^{\alpha}} = (2\beta - 1) + \frac{2(1 - \beta)}{1 + z\phi(z)}.$$