The following 2 problems are past exit exam problems for my major. I see that they're worded differently but are asking me to do the same thing. Not sure how they differ much I'd appreciate if anyone filled me in on that.
Can I prove the following 2 problems in the manner done from this post from user17762?: Why: A holomorphic function with constant magnitude must be constant.
1) Suppose $u(x,y)$ is a real valued function which is harmonic on the whole plane such that $|u(x,y)| \le 17$ for every $z=x+iy$ in $\mathbb{C}$. Show that $u$ must be constant
2) Suppose $u: \mathbb{R^2} \to \mathbb{R}$ is harmonic on the whole plane and that $u(x,y)<0$ for all $(x,y)$ in $\mathbb{R^2}$. Show that $u$ must be constant.
For these problems, an approach using the Cauchy-Riemann equations isn't the most convenient. Liouville's theorem is far more straightforward.
An entire (real-valued) harmonic function is the real part of an entire holomorphic function, let's call that $f$.
Both conditions, $\lvert u(x,y)\rvert \leqslant 17$, and $u(x,y) < 0$ imply that
$$\operatorname{Re} f \leqslant M$$
on all of $\mathbb{C}$ for some $M < +\infty$. And that implies that
$$\left\lvert e^{f(z)}\right\rvert \leqslant e^M$$
on all of $\mathbb{C}$, so by Liouville's theorem $e^f$ is constant. Taking the logarithm (which works, since $\mathbb{C}$ is simply connected) we see that $f$ is constant. But then of course $u = \operatorname{Re} f$ is constant too.