A domain free of the zeros of the partial theta function

We prove that for $q\in (0,1)$, the partial theta function $\theta (q,x):={\sum }_{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{x}^j$ has no zeros in the closed domain $\{\{\mathrm{\mid }x\mathrm{\mid }\le 3\}\cap \{${\rm Re}$x\le 0\}\cap \{\mathrm{\mid }${\rm Im}$x\mathrm{\mid }\le 3\mathrm{/}\sqrt{2}\}\}\subset \mathbb{C}$ and no real zeros $\ge -5$.