Amplitude bounds of steady rotational water waves

We consider classical steady water waves with a free surface, a flat bottom and constant vorticity $\gamma$. In the adverse case $\gamma >0$ we prove that there is an absolute upper bound on the amplitude, independent of the physical constants, provided that $\gamma$ is sufficiently small. In any favorable case $\gamma \le 0$ we present a new proof of such an absolute bound on the amplitude and prove that the amplitude tends to zero as $\gamma$ tends to negative infinity.