Assume that a function $h(\lambda)$ is decreasing and convex given interval $[l,u]$ and has an unique root $\lambda^*\in (l,u)$. Also, assume $|l-\lambda^*| > |\lambda^*-u|$. Consider any $z\in (l,\lambda^*)$. By connecting points $(z,h(z))$ and $(u,h(u))$, we obtain $(w,0)$. Then, I want to show $|z-\lambda^*|>|w-\lambda^*|$.
Any suggestions or comments are welcome.