TITLE: Finding a constant that does not depend on a parameter
QUESTION [0 upvotes]: We consider a parameter $\nu>0$.
We note $b(\xi,x)=\nu(\xi^2+x^2)$ for all $\xi,x\in\mathbb{R}$.
I want to prove that there exists a constant $c>0$ such that
$$\int_0^{+\infty}e^{\Big(tanh(\frac{t}{2})-\frac{t}{2}\Big)b^2}dt\le \frac{c}{\nu^{\frac{2}{3}}}$$ for all $\xi,x\in\mathbb{R}$.
Please help me to do so. Thanks

REPLY [1 votes]: Since 
$$
\int_0^{+\infty}e^{\Big(\tanh(\frac{t}{2})-\frac{t}{2}\Big)b^2}\,dt\ge
\int_0^{+\infty}e^{\Big(-\frac{t}{2}\Big)b^2}\,dt=\frac{2}{b^2}
$$
there is no such constant $c$ independent of $\xi,x\in\mathbb{R}$.
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