On the inclusion properties for harmonic error functions

For the error functions of the form \begin{equation*} E_{r}\mathfrak{f}(z)=\frac{\sqrt{\pi z}}{2}er\ \mathfrak{f}(\sqrt{z})=z+\Sigma_{n=2}^{\infty} \frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}, \end{equation*}% let $\mathcal{E}S_{\mathcal{H}}(k,\lambda ,\gamma )\,$\ represent the class of harmonic error functions \mathcal{ERF}=\mathcal{ERH}+\overline{\mathcal{% ERG}} in the open unit disk $\mathbb{U}=\left\{ z\in \mathbb{C}:\ \ \left\vert z\right\vert <1\right\}$. The paper attempts to present some basic properties for functions in this class.