complex analysis – Prove that $f$ is constant for a holomorphic function with $|f (z)| ≤$ $|Re$ $z|^{-frac{1}{2}}$

Let $f ∈ H (Bbb C)$ with the property
$|f (z)| ≤$ $|Re$ $z|^{-frac{1}{2}}$ off the imaginary axis.

Prove that $f$ is constant.

So I tried by letting $g(z)=(Re$ $z)^{frac{1}{2}}f(z)$ But after this how to proceed to show $f$ is constant I am not understanding. Any theorem I can use to show this or if I am going in the wrong direction?