Geodesic
Strict monotonicity of nonnegative strictly concave function vanishing at the origin
The Teaching of Mathematics, XIX (2016) no. 2, p. 68

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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.
Classification : 97I20 I25
Keywords: strict concavity, strict monotonicity, metric transform 
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     author = {Yuanhong Zhi},
     title = {Strict monotonicity of nonnegative strictly concave function vanishing at the origin},
     journal = {The Teaching of Mathematics},
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     publisher = {mathdoc},
     volume = {XIX},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM2_2016_XIX_2_a1/}
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Yuanhong Zhi. Strict monotonicity of nonnegative strictly concave function vanishing at the origin. The Teaching of Mathematics, XIX (2016) no. 2, p. 68 . http://geodesic.mathdoc.fr/item/TM2_2016_XIX_2_a1/