  In this article, we consider the family of functions $f$ analytic in the unit
disk $|z|<1$ with the normalization $f(0)=0=f'(0)-1$ and satisfying the
condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $ for some
$0<\lambda \leq 1$. We denote this class by $\mathcal{U}(\lambda)$ and we are
interested in the relations between $\mathcal{U}(\lambda)$ and other families
of functions holomorphic or harmonic in the unit disk. Our first example in
this direction is the family of functions convex in one direction. Then we are
concerned with the subordinates to the function $1/((1-z)(1-\lambda z))$. We
prove that not all functions $f(z)/z$ $(f \in \mathcal{U}(\lambda))$ belong to
this family. This disproves an assertion from \cite{OPW}. Further, we disprove
a related coefficient conjecture for $\mathcal{U}(\lambda)$. We consider the
intersection of the class of the above subordinates and $\mathcal{U}(\lambda)$
concerning the boundary behaviour of its functions. At last, with the help of
functions from $\mathcal{U}(\lambda)$, we construct functions harmonic and
close-to-convex in the unit disk.
