Let $\alpha_1:[0,l_1]\rightarrow \mathbb{R}^2, \alpha_2:[0,l_2]\rightarrow \mathbb{R}^2$ be two smooth closed curves parametrized by arc-length such that $\alpha_i|_{[0,l_i)}$ is injective and that $\alpha^{\prime}_i(l_i)=\alpha_i^{\prime}(0)$, for each $i=1,2$. Suppose that the (signed) curcature $k_i(s)$ of $\alpha_i$ satisfies $k_i(s) > 0$ for all $s \in [0, l_i]$, and that $k_1(s) \geq k_2(s)$ for all $s \in [0, \min{(l_1, l_2)}]$.
Prove that $l_1 \leq l_2$.
Intuatively this seem clear. However I am struggling to find a proof. Any help whould be appreciated.