  In the present article, we discuss about the estimate of the pre-Schwarzian
and 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 result
for 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 obtain
best possible estimates of the pre-Schwarzian and Schwarzian norms for
functions in the class $\mathcal{F}_0$. Moreover, we obtain the distortion and
coefficient estimates of the co-analytic function $g$ when
$f=h+\overline{g}\in\mathcal{F}_0$.
