Application of Schwarz lemma and Liouville's theorem

Since $h$ is entire with $h(0)=0$, the function $f(z):=h(z)/10z$ is entire. Since $|h(z)|<10$ on the open unit disk, by continuity, $|h(z)|\le 10$ and thus $|f(z)|\le 1$ on the unit circle. By maximum modulus principle, unless $f$ is a constant, the maximum of $|f(z)|$ over closed unit disk can be only achieved on the boundary, that is, the unit circle.

Suppose to a contrary that $f$ is not a constant. Then $|f|\le1$ in the closed unit disk. But we have that $|f(\frac12)|=1$, which asserts that the maximum of $|f|$ over closed unit disk is achieved by an interior point. This gives us a contradiction.

Therefore $f$ must be a constant. Since $f(\frac12)=1$, $f(z)=1$ for all $z$. So $h(z)=10z$.

Let $g(z)= h(z)/10.$ Then $g$ restricted to $\mathbb D,$ let's call it $\tilde g,$ maps $\mathbb D$ into $\mathbb D,$ sends $0$ to $0,$ and takes $1/2$ to $1/2.$ By the Schwarz Lemma, $\tilde g(z)= z.$ That is the same as saying $g(z)=z$ in $\mathbb D.$ By the identity principle, $g(z) = z$ on all of $\mathbb C.$ Therefore $h(z) = 10z$ on $\mathbb C.$ In order then, we see the given statements are false, false, true, and true.