I'm sure that many readers are already familiar with the well known Bonnesen inequality in the plane for a smooth, connected curve:

$(R_{out} - R_{in})^2 \leq \pi^2 (L^2 - 4\pi A),$

where $R_{out}$ and $R_{in}$ are the outer and inner radii respectively, $L$ is the length of the curve and $A$ is it's area.

This form turns out to often be quite inconvenient for me, and I was wondering if there was a version which read as follows:

$R_{out}^2 - R_{in}^2 \leq C (L^2 - 4\pi A)$,

where now $C$ is a constant which may depend on the support of the curve or other quantities possibly. The main difference is that when $R_{in} - R_{out}$ is small, the second inequality I've written is *strictly stronger* if one assumes the curve is constrainted to a bounded domain. The sacrifice I make though is that I do not care what the constant is, or if it depends on the support of the curve.

This inequality seems quite reasonable but I have thus far been unable to prove it. Has anyone encountered such a version of this inequality?