Suppose we have two strictly convex closed curves $C_{1}$ and $C_{2}$, $C_{1}$ contains $C_{2}$, then can we conclude $\int_{C_{1}} \kappa_{1}^{p} ds\geq \int_{C_{2}} \kappa_{2}^{p} ds$, $\kappa_{1}$ and $\kappa_{2}$ are corresponding curvatures of $C_{1}$ and $C_{2}$, $p$ is between 0 and 1
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No. Let $C_2$ be the unit circle and $C_1$ be a big square with corners rounded off by circle arcs of radius $r\ll 1$. The integral for $C_2$ is the same as for the $r$circle and this is very small. 

