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

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.