Here's an approach that will establish the inequality, but it doesn't provide any insight into where the inequality came from. Let f(x) be the left side minus the right side, i.e.

f(x) = erfc(x) - \frac{ x \exp(-x^2) }{ \pi(1 + 2x^2) }

Clearly f(x) > 0 and \lim_{x\to\infty} f(x) = 0. A calculation shows that f'(x) < 0 for all x > 0, and so f(x) must be positive for all x > 0. See these notes for details. The notes also state improved bounds but without proof.

Here's an approach that will establish the inequality, but it doesn't provide any insight into where the inequality came from. Let f(x) be the left side minus the right side, i.e.

$f(x) = erfc(x) - \frac{ x \exp(-x^2) }{ \pi(1 + 2x^2) }$

Clearly $f(x) > 0$ and $ \lim_{x\to\infty}$ $f(x) = 0.$ A calculation shows that $f'(x) < 0$ for all $x > 0$, and so $f(x)$ must be positive for all $x > 0$. See these notes for details. The notes also state improved bounds but without proof.

Here's an approach that will establish the inequality, but it doesn't provide any insight into where the inequality came from. Let f(x) be the left side minus the right side, i.e.

f(x) = erfc(x) - \frac{ x \exp(-x^2) }{ \pi(1 + 2x^2) }

Clearly f(x) > 0 and \lim_{x\to\infty} f(x) = 0. A calculation shows that f'(x) < 0 for all x > 0, and so f(x) must be positive for all x > 0.

Here's an approach that will establish the inequality, but it doesn't provide any insight into where the inequality came from. Let f(x) be the left side minus the right side, i.e.

f(x) = erfc(x) - \frac{ x \exp(-x^2) }{ \pi(1 + 2x^2) }

Clearly f(x) > 0 and \lim_{x\to\infty} f(x) = 0. A calculation shows that f'(x) < 0 for all x > 0, and so f(x) must be positive for all x > 0. See these notes for details. The notes also state improved bounds but without proof.