A domain free of the zeros of the partial theta function

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