= Solution
Because $f(D)$ avoids the closed negative real axis, its <principal square root> $g=\sqrt f$ is holomorphic, satisfies $g(0)=1$, and has $\operatorname{Re}g>0$. The <Cayley transform between the half-plane and disk>
$$
h=\frac{g-1}{g+1}
$$
maps the right half-plane into the <unit disk> and has $h(0)=0$. By <Schwarz lemma>, $|h(z)|\leq|z|$. When $|z|\leq1/2$,
$$
|g(z)|=\left|\frac{1+h(z)}{1-h(z)}\right|\leq\frac{1+|h(z)|}{1-|h(z)|}\leq3.
$$
Thus
$$
\boxed{|f(z)|\leq9,}
$$
so one may take the universal constant $C=9$.
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