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Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ:={z∈C:|z|<1}, normalized by f(0)=f′(0)−1=0 and such that Im z Im f(z) ≥0 for z∈Δ. Moreover, let us denote: T(2):={f∈T:f(z)=−f(−z) for z∈Δ} and TM,g:={f∈T:f≺Mg in Δ}, where M>1, g∈T∩S and S consists of all analytic functions, normalized and univalent in Δ.
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∈T:f≪Mg in Δ}, where M>1, g∈T, which we denote by TM,g. Furthermore, we broaden the class TM,g for the case M∈(0,1) in the following way:
TM,g={f∈T:|f(z)|≥M|g(z)| for z∈Δ}, g∈T
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