When does equality in the estimation lemma hold? That is, under what circumstances do we have
$\left|\int_\gamma f(z) dz \right|= \text{length}(\gamma)\sup_{z\in \gamma}|f(z)|$
for an entire function $f$?
From the proof of the lemma, $f$ must be such that each of the two inequalities below is an equality:
$\left|\int_\gamma f(z) dz \right|= \left|\int_a^b f(\gamma(t))\gamma'(t)dt \right| \leq \int_a^b|f(\gamma(t))||\gamma'(t)|dt\leq \sup_{t\in[a,b]}|f(t)|\cdot \text{length}(\gamma)$
Is there anything else we can say about such a function $f$?