This is the inequality I have to prove: $$\left| e^{-x^2/2} - e^{-y^2/2} \right| \leqslant \alpha |x-y|,$$ for $\alpha \in [0,1)$.

I'm struggling on where to go with it, I've tried hyperbolic functions and series expansions, but it's not going anywhere.

Any help would be great!

  • 6
    $\begingroup$ Hint: $\frac{e^{-x^2/2}-e^{-y^2/2}}{x-y}$ is the derivative of $e^{-z^2/2}$ at some $z\in(x,y)$. How large can be such derivative? $\endgroup$ – Jack D'Aurizio Mar 1 '17 at 21:41
  • $\begingroup$ ... it follows that your inequality holds for any $\alpha\geq \frac{1}{\sqrt{e}}\approx 0.6$. $\endgroup$ – Jack D'Aurizio Mar 1 '17 at 21:43
  • $\begingroup$ That's great, thank you @JackD'Aurizio! $\endgroup$ – Will Mar 1 '17 at 21:46

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