Criteria for starlike and convex functions of order α.

Let $\mathcal{A}_{n}$ ( $n\in\mathbb{N}$) be the class of certain analytic functions $f(z)$ in the open unit disk $\mathbb{U}$ and $\mathcal{P}_{n}(\lambda)$ be the subclass of $\mathcal{A}_{n}$ consisting of $f(z)$ which satisfy $|f''(z)| \leqq \lambda$ ( $\lambda> 0$) in $\mathbb{U}$. Some properties for the class $\mathcal{P}_{n}(\lambda)$, which are the improvements of the previous results due to Ponnusamy and Singh (Complex Var. Theory Appl. 34:276-291, 1997), are discussed. [ABSTRACT FROM AUTHOR]