Let $\Omega=\{z, \text{Re}z>0\}$ Suppose that $f$ is continuous in the closure of $\Omega$ and $f$ is holomoprhic on $\Omega$ and there are constants $A<\infty $ and $\alpha<1$ such that $$|f(z)|<Ae^{|z|^\alpha}$$ for all $z\in\Omega$ futhermore, $|f(iy)|\leq1 $ for all real $y$. Prove that $|f(z)|\leq 1$ in $\Omega$.
How would approach this problem? Is the 3 lines theorem helpful?, then we need find a bounded function involving $f$ then apply the 3-line theorem. Any ideas?
Thanks