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November 3rd, 2017, 03:43 AM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 101 Thanks: 0  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 $t>0$ 
November 3rd, 2017, 06:24 AM  #2  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,797 Thanks: 715 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
November 3rd, 2017, 08:12 AM  #3 
Senior Member Joined: Jan 2015 From: usa Posts: 101 Thanks: 0 
$\theta$ is fixed from the begining

November 4th, 2017, 06:32 AM  #4 
Senior Member Joined: Jan 2015 From: usa Posts: 101 Thanks: 0 
@topsquark, can you help me to prove the iniquality by taking $\alpha=4$? 

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$cosh$, $sinh$, inequality 
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