Assume that $f''(x),\ g''(x) >0$.
If $D_f =\{ (x,y)| f(x)\leq y\leq k(x) \},\ D_g=\{(x,y)|g(x)\leq y\leq k(x)\}$ where $k(x)=\frac{f(b)-f(a)}{b-a}(x-a)+f(a)$, then $D_g\subset D_f\ \ast$.
And $D_f,\ D_g$ are convex so that from $\ast$, ${\rm length}\ \partial D_f\geq {\rm length}\ \partial D_g$.