Subclasses of typically real functions determined by some modular inequalities

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\mathrm{\Delta }:=\{z\in \mathbb{C}:|z|<1\}$, normalized by $f(0)=f^{\prime }(0)-1=0$ and such that Im $z$ Im $f(z)$ $\ge 0$ for $z\in \mathrm{\Delta }$. Moreover, let us denote: ${\mathrm{T}}^{(2)}:=\{f\in \mathrm{T}:f(z)=-f(-z)\text{ for }z\in \mathrm{\Delta }\}$ and ${\mathrm{T}}^{M,g}:=\{f\in \mathrm{T}:f\prec Mg\text{ in }\mathrm{\Delta }\}$, where $M>1$, $g\in \mathrm{T}\cap \mathrm{S}$ and $\mathrm{S}$ consists of all analytic functions, normalized and univalent in $\mathrm{\Delta }$.We investigate  classes in which the subordination is replaced with the majorization and the function $g$ is typically real but does not necessarily univalent, i.e. classes $\{f\in \mathrm{T}:f\ll Mg\text{ in }\mathrm{\Delta }\}$, where $M>1$, $g\in \mathrm{T}$, which we denote by ${\mathrm{T}}_{M,g}$. Furthermore, we broaden the class ${\mathrm{T}}_{M,g}$ for the case $M\in (0,1)$ in the following  way:${\mathrm{T}}_{M,g}=\{f\in \mathrm{T}:|f(z)|\ge M|g(z)|\text{ for }z\in \mathrm{\Delta }\}$, $g\in \mathrm{T}$.