Partial Theta Function and Separation in Modulus Property of its Zeros

We consider the partial theta function theta(q, z) := Sigma(infinity)(j=0) q(j (j+1))/2z(j), where z is an element of C is a variable and q is an element of C, 0 < vertical bar q vertical bar < 1, is a parameter. Set D(a) := {q is an element of C, 0 < vertical bar q vertical bar <= a, arg( q). [pi/2, 3 pi/2]}. We show that for k is an element of N and q is an element of D(0.55), there exists exactly one zero of theta(q, center dot) (which is a simple one) in the open annulus vertical bar q vertical bar(-k+1/2) < z < vertical bar q vertical bar(-k-1/2) (if k >= 2) or in the punctured disk 0 < z < vertical bar q vertical bar|(-3/2) (if k = 1). For k not equal 2, 3, this holds true for q is an element of D(0.6) as well.