I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following?
$$erfc(x) > \frac{ x \exp(x^2) }{ \pi(1 + 2x^2) }$$
I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following? $$erfc(x) > \frac{ x \exp(x^2) }{ \pi(1 + 2x^2) }$$ 


Durrett, Probability: Theory and Examples, 3rd edition, p. 6 gives $$(x^{1}  x^{3}) e^{x^2/2} \le \int_x^\infty e^{y^2/2} \: dy $$ The proof Durrett gives is from the observation that $$ \int_x^\infty (13y^{4}) e^{y^2/2} \: dy = \left( x^{1} + x^{3} \right) e^{x^2/2} $$ which I suspect can be found by integration by parts, although I haven't written it out; in any case, differentiate it to check. After this, some changes of variables give $$ \left( {1 \over z}  {1 \over 2z^3} \right) e^{z^2}/\sqrt{\pi} \le erfc(z). $$ Finally, $z/(1+2z^2) < 1/z1/(2z^3)$ for $z > 2^{1/4}$, giving your bound for $z > 2^{1/4}$ if $\pi$ is replaced with $\sqrt{\pi}$. Obviously this is a hack trying to get your proposed bound in the form of the bound I already knew, but hopefully it helps. 


Paradoxically, it is quite easier to prove stronger bounds. Let $X$ be a random variable with gaussian distribution and density $$ f(x)=\frac{1}{\sqrt{2\pi}}\exp(x^2/2).$$ Now let, for any $k\in\mathbb{R}^+$, $$A_k = \sqrt{2\pi}\;\exp(k^2/2)\;\mathbb{P}[X>k] = \sqrt{\frac{\pi}{2}}\;\exp(k^2/2)\;\operatorname{Erfc}\left(\frac{k}{\sqrt{2}}\right).$$ Since $\mathbb{E}\left[\left(X\mathbb{E}[X]\right)^2\right]\geq 0$, $\mathbb{E}[X^2]\geq\mathbb{E}[X]^2$, and the same holds for the conditional expected values, under the hypothesis $X>k$. That gives, in terms of $A_k$: $$ (1k\; A_k)^2 \leq A_k\;\left(k+(1+k^2)\; A_k\right), $$ that can be restated as: $$(\heartsuit)\quad A_k^2+k\;A_k1\geq 0.$$ From: $$ A_k \geq \frac{2}{k+\sqrt{k^2+4}}, $$ we immediately have a lower bound for the $\operatorname{erfc}$ function: $$ e^{k^2}\;\operatorname{erfc}(k) \geq \frac{2}{\sqrt{\pi}}\left(\frac{1}{k+\sqrt{k^2+2}}\right).$$ An interesting fact is that the "reverse inequality" $$(\spadesuit)\quad (1k\; A_k)^2 \geq \frac{2}{\pi}\;A_k\;\left(k+(1+k^2)\; A_k\right), $$ equivalent to: $$(\spadesuit_2)\quad\pi\left(\int_{k}^{+\infty}(xk)\;e^{x^2/2}\;dx\right)^2\geq 2\int_{k}^{+\infty}e^{x^2/2}\;dx\int_{k}^{+\infty}(xk)^2\;e^{x^2/2}\;dx,$$ holds, too. Since: $$A_k =\int_{0}^{+\infty}\exp\left(x^2/2kx\right)\,dx, $$ by using the FubiniTonelli theorem and switching to polar coordinates one can see that $(\spadesuit_2)$ is equivalent to: $$\int_{0}^{+\infty}\rho^3 e^{\rho^2/2}\int_{0}^{\pi/4}(\pi\cos(2\theta)2)\; e^{k\rho\sqrt{2}\cos\theta}\;d\theta \; d\rho\geq 0, $$ also equivalent to (by integrating by parts): $$\int_{0}^{+\infty}\rho^3 e^{\rho^2/2}\int_{0}^{\pi/4}(\pi\sin\theta\cos\theta2\theta)\;k\rho\sqrt{2}\sin\theta\; e^{k\rho\sqrt{2}\cos\theta}\;d\theta \; d\rho\geq 0, $$ that is trivial since over $[0,\pi/2]$, by the concavity of the sine function, we have $\sin\phi\geq\frac{2\phi}{\pi}$. Following the line of the previous proof, the $(\spadesuit)$inequality can be used to have tight upper bound for the $e^{k^2}\;\operatorname{erfc}(k)$  function. Moreover, from the $(\spadesuit_2)$inequality the convexity of $\frac{1}{A_k}$ follows. 


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. 

