Let f be an entire function. Suppose $\exists$ $M>0$ such that
$\mid$$f(z)$$\mid$ $\leq$ $M$$\mid$$z$$\mid$ $\forall$ $z$$\in$$\Bbb{C}$. Show that $\exists$$\lambda$$\in$$\Bbb{C}$ such that f(z)=$\lambda$$z$ $\forall$ $z$$\in$$\Bbb{C}$.
This problem is related to Liouville's theorem (which is a corollary of the cauchy inequalities) and someone told me use function g(z)=f(z)/z. But since g(z) is not 'entire' function, I don't really know how to use this function to solve this problem. How can I solve this matter?
