Let $f, g\in O(\mathbb{C})$ (holomorphed in $\mathbb{C}$) and $k\in\mathbb{R}^{\*}$ such that $Re(f(z))\leq kRe(g(z))$, for all $z\in \mathbb{C}$. Show that there are constant $a$ y $b$ such that $f(z)=ag(z)+b$ for all $z\in\mathbb{C}$.

My attempts:

I know that if $f$ is entire and $Re(f(z))\leq M$ for all $z\in\mathbb{C}$ then $f$ is constant so I don't know if I could use this statement with $f(z)-g(z)$ or if there is another way to show it.