I want to find a lower bound for $\left|\int_\gamma f(z)dz\right|$. I know of the estimation lemma and Jordan's lemma for an upper bound, but I don't know of any for a lower bound.
The motivation is that I want to prove that a certain integral diverges on a given contour, and I'm looking for ways to do that. I think that for a given smooth contour $\gamma_R$ that depends on a parameter $R$, and a given function $f(z)$ such that $|f(z)|\rightarrow\infty$ as $R\rightarrow\infty$, one can conclude that $\left|\int_{\gamma_R}f(z)dz\right|\rightarrow\infty$ as $R\rightarrow\infty$. In an attempt to prove that, I use a naive approximation for $\left|\int_{\gamma_R}f\right|$: $\left|\int_{\gamma_R}f\right|=\left|\int_If(\gamma_R(t))\dot{\gamma_R}(t)\right|\geq \left|\sum_{j=1}^nf\left(\gamma_R\left(\frac{j}{n}\right)\right)\dot\gamma_R\left(\frac{j}{n}\right)\right|$ here I use a lower bound of the Riemann integral after the choice of parameterization for $\gamma_R$, without loss of generality and for ease of writing I assumed that the interval of the parameterization is $[0,1]$ and I used the partition of $I$ to $n$ subintervals of length $\frac{1}{n}$. And then using the triangle inequality and the assumption we have that $\left|\int_{\gamma_R}f(z)dz\right|\rightarrow\infty$ (On second thought, we might also need that the derivative of $\gamma_R$ behaves nice enough).
So, to be very specific, the 3 questions I have here: a) is there an "estimation lemma" for a lower bound? b) is my reasoning correct in the above proof? c) are there criteria for contour integral divergence or similar tests?
Thanks