# 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?