I want to prove the following result:
Let $C$ be a regular closed (planar) curve with rotation index $N\in\mathbb{N}$ and $0<\kappa<K$ for some $K\in\mathbb{R}$. The following inequality holds: $$\text{length}(C)\ge 2\pi\frac{N}{K}.$$
I want to apply the Isoperimetric Inequality in some capacity, as this sort of resembles it: $$ length(C)^2\ge 4\pi\cdot(\pi\cdot (\frac{N}{K})^2). $$
I'm having a difficult time discerning what the quantity $N/K$ represents; all I really know is that $1/K$ is the radius of a circle with curvature $K$ and that the quantity in paranthesis is the area of some circle. I actually don't even know if I'm approaching the question from the right angle. Any help is appreciated.
This isn't the isoperimetric inequality. It's the Hopf Umlaufsatz: $$\int_C \kappa\,ds = 2\pi N.$$