bounding functions with exponential

Question

I am looking for a technique to bound functions $f(x)$ with $e^{g(x)}$ as I came across such a inequality : $$\frac{1}{\sqrt{1-2\lambda}} \leq e^{2\lambda^{2} + \lambda}, |\lambda| < \frac{1}{4}$$. I don't quite understand how such inequality is derived ? Is there any technique to bound such $\frac{1}{(1-c\lambda)^{\alpha}}$ using exponential functions ?

Answer 1

The inequality is somewhat delicate and a bit tricky. I don't think there is any standard procedure for deriving such inequalities. Here is a proof of the inequality:

The inequality is equivalent to $e^{4\lambda^{2}+2\lambda} (1-2\lambda) \geq 1$. Let us try to find the minimum value of LHS on the interval $(-\frac 1 4, \frac 1 4)$. The derivative is $e^{4\lambda^{2}+2\lambda} [(8\lambda +2)(1-2\lambda)-2]=e^{4\lambda^{2}+2\lambda} (4\lambda-16\lambda^{2})$. This is $0$ only when $\lambda =0$ f or $\lambda =\frac 1 4$. The minimum value can only be attained at these points or the other end point $\lambda =-\frac 1 4$. Now just verify that the inequality $e^{4\lambda^{2}+2\lambda} (1-2\lambda) \geq 1$ holds at these three points.