three line theorem when $M_0 = 0$ or $M_1 = 0$

If $M_0 = 0$ then $$ |F(x+iy)|\le \epsilon^{1-x}M_1^x $$ holds for all $\epsilon > 0$, which implies that $F$ is identically zero.

But in this case there is a simpler proof, using the Schwarz reflection principle and the identity theorem. Also the assumption that $F$ is bounded is not needed in this case.

If $f$ is holomophic in the interior of the strip $S$ and continuous on the imaginary axis with $F(iy) = 0$ for all $y \in \Bbb R$ then it can be extended to a holomorphic function on the strip $$ \{ x+iy \mid -1 < x < 1 \} \, , $$ The identity theorem shows that the extended function (and consequently $F$) is identical zero in its domain.