It looks to me like $D$ is not bounded.  I have produced [an animation](http://www.math.ubc.ca/~israel/triglike/triglike.gif) using Maple.  For $a$ running from $1$ to $40$ we take $b = 5a$, and plot the curves $\text{Re}(f(x+iy))=0$ (blue) and $\text{Im}(f(x+iy))=0$ (red) for $0 < x,y < 1/2$.  The intersections of red and blue curves are off-axis zeros of $f$.  It certainly appears that there are such zeros, and thus that the ray $b=5a$, $a > 1$, is contained in $D$.  I suspect that it is possible to prove this using the Argument Principle.