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 \mathrm{Im}z^{m}=0 and \Re z^{m}\le0 .