Bounded real component, Cauchy-Riemann implications

Question

I was wondering if there is a way to show that an entire function $f(z)=u(x,y)+iv(x,y)$ is bounded, given its real component is bounded, from the Cauchy-Riemann equations. My intuition is that, since the component functions derivatives behave similarly, one's boundedness should follow from the other's, but I am at a loss as how to proceed with the implication using the the Cauchy-Riemann equations, if there is one to be made.