On the location of the complex conjugate zeros of the partial theta function

We prove that for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $\theta (q,x):={\sum }_{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{x}^j$ with non-negative real part belong to the half-annulus $\{$Re$(x)\ge 0,1<\mathrm{\mid }x\mathrm{\mid }<5\}$, where the outer radius cannot be replaced by a number smaller than $e^{\pi \mathrm{/}2}=4.810477382\dots$, and that for $q\in (0,{0.2}^{1\mathrm{/}4}=0.6687403050\dots ]$, $\theta (q,\mathrm{.})$ has no zeros with non-negative real part. The complex conjugate pairs of zeros with negative real part belong to the left open half-disk of radius $49.8$ centered at the origin.