Subclasses of typically real functions determined by some modular inequalities

Abstract

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta := \{ z \in \mathbb{C} : |z|<1 \}$, normalized by $f(0)=f'(0)-1=0$ and such that Im $z$ Im $f(z)$ $\geq 0$ for $z \in \Delta$. Moreover, let us denote: $\mathrm{T}^{(2)}:=  \{f \in \mathrm{T}: f(z)=-f(-z) \text{ for } z \in \Delta \}$ and $\mathrm{T}^{M,g} :=  \{ f \in \mathrm{T}: f \prec Mg \text{ in } \Delta \}$, where $M>1$, $g \in \mathrm{T} \cap \mathrm{S}$ and $\mathrm{S}$ consists of all analytic functions, normalized and univalent in $\Delta$.
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 $\{ f \in \mathrm{T}: f \ll Mg \text{ in } \Delta \}$, 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:
$\mathrm{T}_{M,g} = \left\{ f \in \mathrm{T} : |f(z)| \geq M |g(z)| \text{ for } z \in \Delta \right\}$, $g \in \mathrm{T}$.