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November 3rd, 2017, 04:43 AM   #1
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Inequality with $sinh$ and $cosh$

We consider a parameter $a>>1$ and we note $cosh(\theta)=\sqrt{4a+1}$ and $sinh(\theta)=2\sqrt{a}$

for all $t>0$ we note:

$u(t)=\frac{\sinh(\frac{t}{2}\cosh(\theta))}{\cosh (\theta)}$

$A(t)=\frac{\sqrt{\cosh^2(\theta)u^2(t)+1}-1}{\cosh^2(\theta)}-\Big(\cosh(\frac{t}{2})-1\Big)$

$f(t)=-ln\Big(1-\frac{2A(t)}{u(t)+\sinh(t)+A(t)}\Big)$

I want to find the minimal power $\alpha\ge 0$ such that

$$\frac{\cosh^2(\theta)}{f(t)}e^{-argsh(u(t))}\le \frac{\theta^{\alpha}}{t^3}$$ for all $t>0$
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November 3rd, 2017, 07:24 AM   #2
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Originally Posted by mona123 View Post
We consider a parameter $a>>1$ and we note $cosh(\theta)=\sqrt{4a+1}$ and $sinh(\theta)=2\sqrt{a}$

for all $t>0$ we note:

$u(t)=\frac{\sinh(\frac{t}{2}\cosh(\theta))}{\cosh (\theta)}$

$A(t)=\frac{\sqrt{\cosh^2(\theta)u^2(t)+1}-1}{\cosh^2(\theta)}-\Big(\cosh(\frac{t}{2})-1\Big)$

$f(t)=-ln\Big(1-\frac{2A(t)}{u(t)+\sinh(t)+A(t)}\Big)$

I want to find the minimal power $\alpha\ge 0$ such that

$$\frac{\cosh^2(\theta)}{f(t)}e^{-argsh(u(t))}\le \frac{\theta^{\alpha}}{t^3}$$ for all $t>0$
What is the difference between t and $\displaystyle \theta$? u(t) is defined as a function of t, not $\displaystyle \theta$??

-Dan
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November 3rd, 2017, 09:12 AM   #3
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$\theta$ is fixed from the begining
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November 4th, 2017, 07:32 AM   #4
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@topsquark,

can you help me to prove the iniquality by taking $\alpha=4$?
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