Can the elasticity of a concave function be strictly increasing If a real valued function $f$ defined on the positive orthant is strictly concave, can its elasticity, defined as:
$$E(f(x))=\frac{xf'(x)}{f(x)}$$
be strictly increasing in $x$?

Let $f(x) = \sqrt{x+1}$. Then $f$ is strictly concave on $(0,\infty)$. Furthermore,
$$(Ef)(x) = \frac{xf'(x)}{f(x)} = \frac{x\frac{1}{2\sqrt{x+1}}}{\sqrt{x+1}} = \frac{x}{2(x+1)} = \frac{1}{2}\left(1-\frac{1}{x+1}\right)$$
is strictly increasing on $(0,\infty)$.
