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We obtain a sharp result on order of certain affine and linear invariant families of harmonic quasiconformal mappings with bounded Schwarzian norm. This problem is motivated by the work of Chuaqui, Hern\'andez and Mart\'in Math. Ann. 367: 1099--1122, 2017. Firstly, for K1, we construct a harmonic K-quasiconformal counterpart of the classical Koebe function and use it to formulate the corresponding conjectures. Then we consider Hardy spaces Hᵖ of harmonic quasiconformal mappings by applying results for quasiconformal mappings obtained by Astala and Koskela Pure Appl. Math. Q. 7: 19--50, 2011. In particular, we determine the optimal order of the family of harmonic quasiconformal mappings with bounded Schwarzian norm to belong to a harmonic Hardy space. This partially solves an open problem posed by Pavlovi\'c in 2014. Finally, we derive pre-Schwarzian and Schwarzian norm estimates of certain harmonic mappings.
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