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In this article, we consider the family of functions $f$ analytic in the unitdisk $|z|<1$ with the normalization $f(0)=0=f'(0)-1$ and satisfying thecondition $\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 areinterested in the relations between $\mathcal{U}(\lambda)$ and other familiesof functions holomorphic or harmonic in the unit disk. Our first example inthis direction is the family of functions convex in one direction. Then we areconcerned with the subordinates to the function $1/((1-z)(1-\lambda z))$. Weprove that not all functions $f(z)/z$ $(f \in \mathcal{U}(\lambda))$ belong tothis family. This disproves an assertion from \cite{OPW}. Further, we disprovea related coefficient conjecture for $\mathcal{U}(\lambda)$. We consider theintersection of the class of the above subordinates and $\mathcal{U}(\lambda)$concerning the boundary behaviour of its functions. At last, with the help offunctions from $\mathcal{U}(\lambda)$, we construct functions harmonic andclose-to-convex in the unit disk.

Relations of the class $\mathcal{U}(\lambda)$ to other families of functions