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It looks to me like $D$ is not bounded. I have produced an animation 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.