Assume that $f''(x),\ g''(x) >0$.

 If $D_f =\{ (x,y)| f(x)\leq y\leq
x \},\ D_g=\{(x,y)|g(x)\leq y\leq x\}$, then $D_g\subset D_f\ \ast$.

And $D_f,\ D_g$ are convex so that from $\ast$, ${\rm length}\
\partial D_f\geq {\rm length}\ \partial D_g$.