Abstract
Let $\mathcal{A}$ be the class of analytic functions in the unit disk D with the normalization f(0) = f′(0) − 1 = 0. For λ > 0, denote by $\mathcal{M}$(λ) the class of functions f ∈ $\mathcal{A}$ which satisfy the condition
$$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq \lambda,\quad z\in \mathbf{D}.$$
We show that functions in $\mathcal{M}$(1) are univalent in D and we present one parameter family of functions in $\mathcal{M}$(1) that are also starlike in D. In addition to certain inclusion results, we also present characterization formula, necessary and sufficient coefficient conditions for functions in $\mathcal{M}$(λ), and a radius property of $\mathcal{M}$(1).
