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Thursday, October 19, 2017

**Abstract:** The family ${\mathcal F}$ of boundary normalized harmonic mappings of the unit disc into itself is considered by the author. An expression of the Schwarz type inequalities concerning the mappings in question will be presented. In particular, introduced by Partyka and the author, the Schwarz range domain $$ \bigcup_{F\in {\mathcal F}} \{ F(z) \colon |z|\leq r\},$$ where $0 \leq r < 1$, is well described within the class ${\mathcal F}$. Previously obtained result says that $|F(0)| \leq \frac{2}{3}$ for $F \in {\mathcal F}$, whereas the present one describes precisely the Schwarz range domain of $F(0)$ for $F \in {\mathcal F} $. This research has been strongly motivated by certain problems of gas flow mechanics. Boundary normalization appears to be naturally applicable to aerodynamics when studying mechanics of the jet outlet stream described by harmonic mappings. Some conjectures coming out from this research, concerning jet engine construction will be also presented.