I'm trying to prove the following inequality for $0 < x < 1$:
$$\operatorname{erf}\left(\frac{(1+x)\sqrt{\ln{(1+x)}}}{\sqrt{(1+x)^2 - 1}}\right) - \operatorname{erf}\left(\frac{\sqrt{\ln{(1+x)}}}{\sqrt{(1+x)^2 - 1}}\right) \geq \frac{x}{4}$$
Proof by WolframAlpha: http://goo.gl/15mrM
I could also construct a Proof by Mathematica, without too much trouble.
However, I'm looking for a more elegant proof of this inequality. My approach was going to involve showing that this holds for $x = 0$ and $x = 1$, and then show the function is concave. However, taking the second derivative yields the following monstrosity: http://goo.gl/fKxca
Is there a more elegant way to prove this? I wouldn't mind showing a weaker inequality of the form $\geq \frac{x}{c}$ (for some explicit $c$) if the proof was sufficiently simple.