Let $A$ be a connected compact domain with smooth boundary in the Euclidean 2-plane. Assume its diameter is at most $d$. Assume that the second fundamental form of the boundary is at most $-c$ where $c\geq 0$ (equivalently, the curvature of the boundary is at most $-c$).
Is there an upper estimate on the length of the boundary in terms of $d,c$?
Remark. If $c=0$ then the domain $A$ is convex and hence the length of the boundary is known to be at most $2\pi d$.