Strict monotonicity of nonnegative strictly concave function vanishing at the origin

Yuanhong Zhi

In this paper we prove that every nonnegative strictly concave function on the unbounded closed interval $[0,+\infty)$ is strictly increasing, provided it vanishes at the origin. With the help of this result, we then show that the strict monotonicity condition of the theorem concerning the metric transforms is redundant. We also provide a companion version of this result for merely concave nonnegative function which vanishes only at the origin.