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In the present article, we discuss about the estimate of the pre-Schwarzianand Schwarzian norms for locally univalent harmonic functions$f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$.In this regard, we first rectify an earlier result of Kanas \emph{et al.} [J.Math. Anal. Appl., {\bf 474}(2) (2019), 931--943] and prove a general resultfor the pre-Schwarzian norm. We also consider a new class $\mathcal{F}_0$consisting of all harmonic functions $f=h+\overline{g}$ in the unit disk$\mathbb{D}$ such that ${\rm Re\,}\left(1+z\frac{h''(z)}{h'(z)}\right)>0$ for$z\in\mathbb{D}$ with dilatation $\omega_f(z)\in Aut(\mathbb{D})$ and obtainbest possible estimates of the pre-Schwarzian and Schwarzian norms forfunctions in the class $\mathcal{F}_0$. Moreover, we obtain the distortion andcoefficient estimates of the co-analytic function $g$ when$f=h+\overline{g}\in\mathcal{F}_0$.

Pre-Schwarzian and Schwarzian norm estimates for harmonic functions with fixed analytic part