Entire functions with an arithmetic sequence of exponents

For a given entire function $f(z)=\sum_{j=0}^{\infty}a_{j}z^{j}$, we study the zero distribution of $f_{r}(z)=\sum_{j\equiv r\pmod m}a_{j}z^{j}$ where $m\in\mathbb{N}$ and $0\le r<m$. We find conditions under which the zeros of $f_{r}(z)$ lie on $m$ radial rays defined by $\Im z^{m}=0$ and $\Re z^{m}\le0$.