The relationship between the curvature of a curve and a circle

Question

Let $c$ be a circle with constant curvature $k_c$, and the green curve $\alpha$ below with curvature at a point $x$ $k_{\alpha}(x)$

I would like to know if this inequality is true:

$$k_{\alpha}(q)\lt k_c\lt k_{\alpha}(p)$$

is there anything I can say about $k_{\alpha}(r)$?

Answer 1

This inequality is not true in general. You can easily compare the curvature of these two curves at the points where one touches the other.

The curvature of the red curve is greater than of the curvature of the circle at their tangent point whereas the curvature of the curvature of the green one is less than the curvature of the circle at their tangent point.
One more example