Lengths of rectifiable curves in 2-space (Q1098955)

Let (g(t),f(t)) (0\(\leq t\leq 1)\) denote a continuous rectifiable curve in \(R^ 2\). It is known that its length L is \(\geq \int^{1}_{0}((f')^ 2+(g')^ 2)^{1/2}\) and equality holds if and only if f and g are absolute continuous on \([0,1].\) We generalize this result to a more general setting. We find another inequality in which equality must hold when a weaker condition (than absolute continuity) holds. Other variations involving derivatives vanishing at certain points are included.