  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 $${\real} \left( \frac{z h''(z)}{\alpha h'(z)}\right) <\frac{1}{2}
~\mbox{ for $z\in \ID$,} $$ where $0<\alpha\leq1$. 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.
