$C^m$-extension of subholomorphic functions from plane Jordan domains

We prove that every function $f$ of class $C^m(\,\overline{D}\,)$ subholomorphic in $D$ can be extended to a subholomorphic function of class $C^m$ in the whole $\mathbb C$ with an estimate for the $C^m$-norm, where $m\in(0,2)$ and $D$ is an arbitrary Jordan $B$-domain in $\mathbb C$. We obtain some corollaries and an analogue of the above assertion for the classes $\operatorname{Lip}^m$ with $m\in(0,2]$.