The functional |c₄ + pc₂c₃ + qc³₂| is considered in the class of all univalent holomorphic functions $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{c}_n{z}^n$ in the unit disk. For real values p and q in some regions of the (p,q)-plane the estimates of this functional are obtained by the area method for univalent functions. Some new regions are found where the Koebe function is extremal.

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{z\in ℂ:|z|<1\}$, normalized by $f\left(0\right)=f^{\text{'}}\left(0\right)-1=0$ and such that $\mathrm{Im}z\mathrm{Im}f\left(z\right)\ge 0$ for $z\in \Delta$. In this paper we discuss the class ${\mathrm{T}}_g$ defined as $${\mathrm{T}}_g:=\{\sqrt{f\left(z\right)g\left(z\right)}:f\in \mathrm{T}\},g\in \mathrm{T}.$$ We determine the sets ${\bigcup }_{g\in \mathrm{T}}{\mathrm{T}}_g$ and ${\bigcap }_{g\in \mathrm{T}}{\mathrm{T}}_g$. Moreover, for a fixed $g$, we determine the superdomain of local univalence of ${\mathrm{T}}_g$, the radii of local univalence, of starlikeness and of univalence of ${\mathrm{T}}_g$.