Is there a lower bound for integration of complex functions?

Question

We are given in our book that the upper bound for complex integrations is $|\int_\gamma f(z)\,dz| \leq mL$ where $L$ is the length of $\gamma$ and $m$ is the $\max(|f(z)|: z\in \gamma)$ and were wondering if there is a lower bound definition with $\gamma=L$ and $M = \min(|f(z)|:z\in \gamma)$ and why it is so.

Answer 1

No. In particular, if $\gamma$ is closed and $f$ is holomorphic (on a neighbourhood of the domain bounded by $\gamma$), $$\int_\gamma f(z)\,dz=0$$ by Cauchy's integral theorem.