Consider the family of locally univalent analytic functions h in the unit disk \(|z|<1\) with the normalization \(h(0)=0\) , \(h'(0)=1\) and satisfying the condition \({{\operatorname {Re}\,}} \left( \frac{z h''(z)}{\alpha h'(z)}\right) <\frac{1}{2} ~ \text{ for } z\in {\mathbb D}\text{, } \) where \(0<\alpha \le 1\) . The aim of this article is to show that this family has several elegant properties such as involving Blaschke products, Schwarzian derivative and univalent harmonic mappings.
